Your Information Need: 1334H Math for Marines (MCI)

34h

MARINECORPS INSTITUTE

MATH FOR MARINES

MARINE BARRACKS WASHINGTON, DC

UNITED STATES MARINE CORPS
MARINE CORPS INSTITUTE 912 CHARLES POOR STREET SE WASHINGTON NAVY YARD DC 20391-5680
IN REPLY REFER TO:

Ser 1334H 30 Jun 99 MCI 1334H MATH FOR MARINES 1. Purpose. MCI 1334H, Math for Marines, has been published to provide instruction to all Marines. 2. Scope. This course provides a math review to include algebraic equations, finding area and volume of basic geometric shapes, and use of the Pythagorean Theorem. The course presents the history and principles behind each subject. 3. Applicability. This course is intended for instructional purposes only. This course is designed for all Marines. 4. Recommendations. Comments and recommendations on the contents of the course are invited and will aid in subsequent course revisions. Please complete the course evaluation questionnaire at the end of the final examination. Return the questionnaire and the examination booklet to your proctor.

T.M. FRANUS By direction

Table of Contents
Page Contents ....................................................................................................................... Student Information ...................................................................................................... Study Guide .................................................................................................................. Study Unit 1 Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Study Unit 2 Lesson 1 Lesson 2 Lesson 3 Lesson 4 Study Unit 3 Lesson 1 Lesson 2 Lesson 3 Study Unit 4 Lesson 1 Lesson 2 Lesson 3 Study Unit 5 Lesson 1 Lesson 2 Lesson 3 Numbers and Operations History of Real Numbers............................................................ Numbers and Closure ................................................................. Integers and Rational Numbers................................................... Properties of Numbers................................................................ Other Properties of Numbers ...................................................... Symbols of Grouping ................................................................. Order of Operations.................................................................... Fractions and Percents Basic Principles of Fractions ...................................................... Operations with Fractions........................................................... Decimal Form ............................................................................ Percentage.................................................................................. Algebra Algebraic Expressions ................................................................ Algebraic Sentences ................................................................... Proportion and Percentage.......................................................... Units of Measurement Metric System ............................................................................ U.S. System ............................................................................... Conversion Between U.S. and Metric Units................................ Geometric Forms Angles........................................................................................ Plane Figures.............................................................................. Solid Figures .............................................................................. 5-1 5-6 5-22 4-1 4-10 4-17 3-1 3-7 3-20 2-2 2-4 2-17 2-26 1-1 1-4 1-6 1-7 1-10 1-12 1-13 i iii v

Continued on next page

Table of Contents, Continued

Page Study Unit 6 Lesson 1 Lesson 2 Problem Solving The Logical Plan and Languages ................................................ Solving Word Problems ............................................................. 6-1 6-7 R-1 R-27

Review Lesson.............................................................................................................. Review Lesson Solutions ..............................................................................................

Bibliography ..........................................................................................................................

Student Information

Number and Title

MCI 1334H MATH FOR MARINES

Study Hours

Course Materials

Text Protractor

Review Agency

Marine Corps Institute 912 Charles Poor Street SE Washington Navy Yard, DC 20391-5680

Reserve Retirement Credits (RRC)

ACE

Course submitted for review by the American Council on Education.

Assistance

For administrative assistance, have your training officer or NCO log on to the MCI home page at www.mci.usmc.mil. Marines CONUS may call toll free 1-800-MCI-USMC. Marines worldwide may call commercial (202) 6857596 or DSN 325-7596.

iii

(This page intentionally left blank.)

Study Guide

Congratulations

Congratulations on your enrollment in a distance training course from the Distance Learning Technology Department (DLTD) of the Marine Corps Institute (MCI). Since 1920, the Marine Corps Institute has been helping tens of thousands of hard-charging Marines, like you, improve their technical job performance skills through distance training. By enrolling in this course, you have shown a desire to improve the skills you have and master new skills to enhance your job performance. The distance training course you have chosen, MCI 1334H, Math for Marines, provides a math review to include algabraic equations, finding area and volume of basic geometric shapes, and use of the Pythagorean theorem. The course presents the history and principles behind each subject. YOU ARE PROPERLY MOTIVATED. You have made a positive decision to get training on your own. Self-motivation is perhaps the most important force in learning or achieving anything. Doing whatever is necessary to learn is motivation. You have it! YOU SEEK TO IMPROVE YOURSELF. You are enrolled to improve those skills you already possess, and to learn new skills. When you improve yourself, you improve the Corps! YOU HAVE THE INITIATIVE TO ACT. By acting on your own, you have shown you are a self-starter, willing to reach out for opportunities to learn and grow. YOU ACCEPT CHALLENGES. You have self-confidence and believe in your ability to acquire knowledge and skills. You have the selfconfidence to set goals and the ability to achieve them, enabling you to meet every challenge. YOU ARE ABLE TO SET AND ACCOMPLISH PRACTICAL GOALS. You are willing to commit time, effort, and the resources necessary to set and accomplish your goals. These professional traits will help you successfully complete this distance training course.
Continued on next page

Your Personal Characteristics

Study Guide, Continued

Beginning Your Course

Before you actually begin this course of study, read the student information page. If you find any course materials missing, notify your training officer or training NCO. If you have all the required materials, you are ready to begin. To begin your course of study, familiarize yourself with the structure of the course text. One way to do this is to read the table of contents. Notice the table of contents covers specific areas of study and the order in which they are presented. You will find the text divided into several study units. Each study unit is comprised of two or more lessons, lesson exercises, and finally, a study unit exercise.

Leafing Through the Text

Leaf through the text and look at the course. Read a few lesson exercise questions to get an idea of the type of material in the course. If the course has additional study aids, such as a handbook or plotting board, familiarize yourself with them.

The First Study Unit

Turn to the first page of study unit 1. On this page you will find an introduction to the study unit and generally the first study unit lesson. Study unit lessons contain learning objectives, lesson text, and exercises.

Reading the Learning Objectives

Learning objectives describe in concise terms what the successful learner, you, will be able to do as a result of mastering the content of the lesson text. Read the objectives for each lesson and then read the lesson text. As you read the lesson text, make notes on the points you feel are important.

Completing the Exercises

To determine your mastery of the learning objectives and text, complete the exercises developed for you. Exercises are located at the end of each lesson, and at the end of each study unit. Without referring to the text, complete the exercise questions and then check your responses against those provided.
Continued on next page

Study Guide, Continued

Continuing to March

Continue on to the next lesson, repeating the above process until you have completed all lessons in the study unit. Follow the same procedures for each study unit in the course.

Seeking Assistance

If you have problems with the text or exercise items that you cannot solve, ask your training officer or training NCO for assistance. If they cannot help you, request assistance from your MCI distance training instructor by completing the course content assistance request form located at the back of the course.

Preparing for the Final Exam

To prepare for your final exam, you must review what you learned in the course. The following suggestions will help make the review interesting and challenging. CHALLENGE YOURSELF. Try to recall the entire learning sequence without referring to the text. Can you do it? Now look back at the text to see if you have left anything out. This review should be interesting. Undoubtedly, you find you were not able to recall everything. But with ll a little effort, you be able to recall a great deal of the information. ll USE UNUSED MINUTES. Use your spare moments to review. Read your notes or a part of a study unit, rework exercise items, review again; you can do many of these things during the unused minutes of every day. APPLY WHAT YOU HAVE LEARNED. It is always best to use the skill or knowledge you learned as soon as possible. If it isn possible ve t to actually use the skill or knowledge, at least try to imagine a situation in which you would apply this learning. For example make up and solve your own problems. Or, better still, make up and solve problems that use most of the elements of a study unit.
Continued on next page

vii

Study Guide, Continued
USE THE SHAKEDOWN CRUISE TECHNIQUE. Ask another Marine to lend a hand by asking you questions about the course. Choose a particular study unit and let your buddy fire away. This technique can be interesting and challenging for both of you! MAKE REVIEWS FUN AND BENEFICIAL. Reviews are good habits that enhance learning. They don have to be long and tedious. In act, t some learners find short reviews conducted more often prove more beneficial.

Preparing for the Final Exam, continued

Tackling the Final Exam

When you have completed your study of the course material and are confident with the results attained on your study unit exercises, take the sealed envelope marked FINAL EXAM to your unit training NCO or training officer. Your training NCO or officer will administer the final exam and return the exam the answer sheet to MCI for grading. Before taking your final exam.

Completing Your Course

The sooner you complete your course, the sooner you can better yourself by applying what you learned! HOWEVER--you do have 2 years from the ve date of enrollment to complete this course. If you need an extension, please complete the Student Request/Inquiry Form (MCI-R11) located at the back of the course and deliver it to your training officer or training NCO.

Graduating!

As a graduate of this distance training course and as a dedicated Marine, your job performance skills will improve, benefiting you, your unit, and the Marine Corps.

Semper Fidelis!

viii

STUDY UNIT 1 NUMBER SYSTEMS AND OPERATIONS Introduction. Every Marine, from a scout in a fireteam to the Commandant, uses numbers in some form every day. Numbers are an inescapable item in our way of life. Some common uses of numbers may be any one of the following: All Marines will carry 3 days' rations and ammunition. How much fuel will the tank platoon need to travel 127 miles? Calculate the TNT needed by the engineer platoon to breach the log obstacle. Convert 1230 magnetic azimuth to grid azimuth. If a 20 mile march took 5 hours to complete, how fast did the company travel? These are just a few of the everyday situations involving numbers that Marines deal with. Marines need to know how to work with numbers. This study unit will provide you with the ability to identify components of the real number system and basic properties of numbers. You will also be able to identify symbols of grouping and simplify mathematical expressions. Let's first take a look at the history of real numbers. Lesson 1. HISTORY OF REAL NUMBERS

LEARNING OBJECTIVES 1. Given a list of events, select the single factor that made the Hindu-Arabic number system superior to all others. Given selected purposes, identify the two purposes that numerals or digits have in any number. Systems of Notation

2. 1101.

In order to have an understanding of the systems of notation, you must first look at the principle of one to one correspondence and In model groups. Let's see how Charlie Caveman handled these. ancient times, Charlie had to design a way to keep track of his herd.

The method that he used is basic to our present day mathematics. He simply matched the number of animals with items in a collection of objects. For example, as he let the animals out in the morning he tied a knot in a rope for each animal; when they came back in the evening he matched each knot with an animal. If he had knots left over, he had animals missing. He did the same thing by piling up stones or sticks, each one representing an animal. As Charlie became more rational, he tired of those processes for accounting for his possessions. His next step toward the beginning of numbers was the formation of model Qroups. This was an extension of his one-to-one correspondence or matching process. It involved using well-known items from his environment to symbolize a given collection of objects. Each symbol had a characteristic that was typical of the model group. These were exemplified by: the wings of a bird symbolizing two; the leaves of a clover, three; the legs of a sheep, four; the fingers on a hand, five; the petals on a particular flower, six; and so on. In his mind, Charlie replaced that pile of five stones with the number of fingers on his hand. Naturally, the evolution brought about using the name for the model group instead of the actual symbol. Although it still was not counting, the model group was a major step toward it. It simply provided a recognition of the cardinal (how many) value of a number. Later, when man started to systematize his model groups and organize them in sequence (each succeeding the other by one), he was on his way to counting. Let's look at the different systems of notation and see how this was accomplished. a. Egyptian number system. Anthropologists have shown us that all of the early civilized cultures had number systems and counting. The Egyptians used a system that started with tally marks which were continued until 10 which was replaced by the symbol for a heelbone. Each succeeding power of 10 was replaced by a different symbol. The system was based on addition and repetition. That is, add up the repeated symbols (fig 1-1).

1-2

NUMBER 1 10 100 1,000 10,000 100,000 1,0000000

SYMBOL DESCRIPTION

I
(

TALLY MARK HEELBONE COIL OF ROPE. SCROLL LOTUSFLOWER

r
*

BENT LINE
BURBOT MANIN ASTONISHIMENT

EXAMPLES 1502

OF USE:
cD$
nnnnRRs

286
3451

rnn.rss*:

Fig 1-1.

Symbols for Egyptian Numbers.

Roman number system. Most of us are familiar with the Roman b. system of notation because it is still taught in our schools and is used to some extent for decoration on monuments or buildings, and for dates. Basically, this system is also founded on the principles of addition and repetition. The Roman system greatly reduced the symbolism that was required in the Egyptian system, but it still did not approximate the value of the Hindu-Arabic number system. Hindu-Arabic number system. The Hindu-Arabic system came c. into being possibly as early as 500 or 600 BC. The nomadic Arabian tribes picked up the Hindu notation in their travels and carried it to Europe. It was several hundred years, however, before it replaced the Roman system. This system differs from

the Egyptian and Roman systems in that a different symbol is used for each number. It is similar to the other systems because it
is based on addition, meaning that any number's value is a sum of This simply means that the

its parts.

It is similar to the Egyptian system because there is

a compounding at each power of ten.

Hindu-Arabic system is a decimal system. An important similarity to the Roman system is the position of the numerals. Moving the symbols within the number changes the value of the number. The unique principle of the Hindu-Arabic system was the idea of place value (how many of what) which no system had previously employed. In its early use, the Hindu-Arabic system had no distinct advantage over the number system of any other culture.

1-3

The sinqle factor that made the Hindu-Arabic number system superior to all others was the invention of zero sometime in the 9th century AD. The zero serves as a place holder and a signifier of "no value," or "nothing." When used in this context, the zero becomes significant. The Hindu-Arabic system uses 10 basic symbols: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. In any number, each of the numerals or digits signifying it has two purposes; one is to show the cardinal value (how many) and the other is to show the place value (that is, how many units, tens of units, hundreds of units, thousands of units, and so on). To show the cardinal value, the symbol 0, conveys the concept of absence or emptiness, the symbol 1, the concept of oneness or singleness and so on. To show the place value, with the aid of the principle of addition, we show the next number after nine as 10 which symbolizes one ten and no units. Here the zero is functioning as a placeholder occupying space in the units column. It shows that there are no units involved in this number. The next number is 11 which is one ten and one unit. The next number is 12 which is one ten and two units. This continues until 99 which is nine tens and nine units after which we must use three digits. The numerals 100 signify one hundred, no tens, and no units. The numerals 463 signify four hundreds, six tens, and three units. The numerals 999 signify nine hundreds, nine tens, and nine units. After this, four digits are needed. As you can see from this, the system is unlimited in size. Lesson Summary. This lesson provided you with a brief history of real numbers, the single event that made the Hindu-Arabic number system superior to all others, and the two purposes found in any number. Now, let's move forward to Lesson 2 and take a look at the natural numbers. Lesson 2. NUMBERS AND CLOSURE

LEARNING OBJECTIVES 1. 2. 1201. Given selected numbers, identify the natural numbers in the counting process. Given selected mathematical equations, select the example that illustrates the principle of closure. Natural Numbers

Remember that when Charlie Caveman put the model groups into an organized sequence, this was the forerunner to counting. The successive model groups in his counting system each increased by one the size of the group that preceded it. This was the key to the counting process (increase by one). Now man had the means to count, and the numbers he used to do this counting are called the natural numbers. They are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12. . . (the three dots in mathematics indicate "and so on").

Note that the counting or natural numbers do not include zero (fig 1-2). 1-4

Fig 1-2.

Graphical representation of the natural numbers.

In order to understand the difference between the various systems and why they developed, it is necessary to talk about one important principle of numbers called closure. 1202. Principle of Closure

The principle of closure means that, regardless of any operation (addition, subtraction, multiplication, or division) you perform on a number in the rational number system, the result will always be a rational number. Other number principles will be discussed in a later study unit, but closure must be discussed here in order to progress to the other systems. If you add, multiply, subtract, or divide natural numbers, will the sum, product, difference, or quotient be a natural number? Let's take a look at each of these areas and see how closure affects them. Addition. If you add 6 + 8, 10 + 27, 99 + 236, 9872 + 10463, a. or any two natural numbers, the result will always be a natural number. When this occurs, the system is said to be closed under addition. Multiplication. If you multiply 5 x 6, 33 x 44, 356 x 687, b. 3422 x 22345, or any two natural numbers, the result will always be a natural number. This is because multiplication is a form of repeated additions, which makes the system closed under multiplication. That is, the product of any two natural numbers will always be a natural number. Subtraction. How about subtraction? Is the difference c. between any two natural numbers a natural number? Again look over the system and try a few examples. It should not take you
long to see that problems such as 7 10 and 20 25 cannot be

answered with a natural number. closed under subtraction.

Thus, natural numbers are not

Division. An example of 4 . 3 = 1.33..., is enough to show d. that the natural numbers are not closed under division. What does all of this mean to you? What is the importance of closure? The early cultures had extremely little use for numbers. Counting, mainly to keep track of possessions, was Charlie Caveman had no need for anything their primary use. other than the natural numbers, but with civilized cultures came the development of towns and the use of money which places a premium on knowledge of numbers. This is one of several reasons why closure is important and that there is a need for other number systems. 1-5

Lesson Summary. This lesson provided you with the ability to identify the natural numbers in the counting process and introduced the principle of closure. You learned that natural numbers are not closed under subtraction and division. Closure shows you that the natural numbers are not sufficient to handle all operations. There is a need for other number systems. They will be covered in our next lesson. Lesson 3. INTEGERS AND RATIONAL NUMBERS

LEARNING OBJECTIVES 1. Given various mathematical problems, select the equation that illustrates integers as a system of positive and negative whole numbers. Given a list of selected mathematical terms, select the one that best illustrates the concept of rational numbers. System of Whole Numbers

1301.

a. Integers. The integers are the system of whole numbers. Of course the natural numbers are whole numbers, but not the complete set that man was to need. With the use of subtraction there came a need for an expanded system. If you recall, it was
mentioned that problems such as 7 10 and 20 25 could not be

answered with the natural numbers. The answers to both of these problems require a system that includes positive and negative numbers (fig 1-3). Note that the integers start at and include zero, and extend indefinitely in both directions from that point. -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6

Fig 1-3.

Graphical representation of the integers.

If you perform several checks on this system, you will see that it is closed under addition, multiplication, and subtraction; but again, a simple problem such as 3 2 = 1.5 is enough to show that the integers are not closed under division. b. Rational Numbers. Rational numbers are simply numbers that can be expressed as fractions, that is, as the quotient of two integers. Man must have had a need for the rational numbers before the integers, but the integers were discussed first because of their association with subtraction which man probably used very early in his number career. The rational numbers include the integers (5, 6, 7), terminating and repeating decimal fractions (.25 = 1/4, .333 = 1/3), and all fractional numbers.

1-6

It should be evident that the system is closed under all four of the basic operations: addition, subtraction, multiplication, and division. That is, any of these operations performed on a number in the rational number system will give us a rational number as the answer. Natural numbers, the two number systems that we have just mentioned (integers, rational), and one additional system (irrational numbers) make up the real number system (fig 1-4). REAL NUMBERS

IRRATIONAL

RATIONAL

V2,

NATURAL NUMBERS 1,2,3, .... Fig 1-4.

INTEGERS ...-3-2-1 0 +1+2+3... The real number system.

FRACTIONS 1/2,1/3,1/4

The graphical representation of the real numbers looks the same as the one for rational numbers. For our use, the numbers are the same. The irrational numbers are numbers such as V2, i, and non-terminating, non-repeating decimals. We will discuss them in a later study unit. Lesson Summary. This lesson provided you with the information to identify that integers are a system of whole numbers composed of positive and negative numbers, and rational numbers are numbers that can be expressed as fractions. Lesson 4. PROPERTIES OF NUMBERS

LEARNING OBJECTIVES Given simple equations, and according to established laws of mathematics: 1. 2. 3. 4. Identify the three axioms of equality. Identify the commutative property with respect to addition and multiplication. Identify the associative property with respect to addition and multiplication. Identify the distributive property with respect to multiplication. 1-7

In the pre-1950 era of mathematical instruction, little was ever taught about why you can do some of the things that you do with numbers. Now we are teaching the "why" in addition to the "how." Upon analysis, you will see that much of this "why" is selfevident. This is actually what an "axiom" is -- "a self-evident truth." Since you have been taught how to perform mathematical operations and you now do them without second thought, the introduction of the properties of numbers may seem at first irrelevant and unnecessary. But, this is judgment based on hindsight. Place yourself in the position of the beginner in the study of arithmetic and imagine the value that could be gained by knowing the fundamental properties of a number. Even at this stage of your mathematical education, an understanding of the properties can lead to an increased ability to handle more complicated problems. What you need to understand next are the three axioms of equality. (An equality is simply a statement that two things are equal: a = b.) The axioms of equality are important in the study of equations, and they also have important applications in your work in arithmetic. Let's take a look at them. 1401. Three Axioms of Equality

a. The reflexive Property of equality. This is sometimes referred to as the law of identity. It is simply a long-winded way of saying that any number reflects itself or is equal to itself. Stated symbolically, for any number a, a = a. b. The symmetric property of equality. This states that an equality is reversible. Stated symbolically, for any numbers a
and b, if a = b then b = a.

c. The transitive property of equality. This makes it possible to identify two numbers as being equal to a third quantity. Stated symbolically, for any numbers a, b, and c, if a = b and b
= c, then a = c.

Now that you understand the axioms of equality, let's take a look at the commutative and associative properties. 1402. Commutative and Associative Laws or Properties

There are certain laws that are obeyed by various number systems. In fact, number systems can be categorized by their conformity to these laws. Two of the basic laws are the commutative and the associative. Let's look at how these laws are applied. a. The commutative law or property with respect to addition. This law states that the order of addends (numbers to be added) is immaterial; the sum will be unaffected. It doesn't matter Stated whether we add 7 and 3 by writing 7 + 3 or 3 + 7.
symbolically, for any numbers a and b, a + b = b + a.

1-8

b. The associative law or property with respect to addition. This law states that the order in which addends are grouped or associated is immaterial; again the sum will be unaffected. This pertains to a problem with at least three addends. Addition is a binary operation meaning that only two numbers can be added at one time; however, their sum may be added to another and so on. In the problem 4 + 5 + 8, it doesn't matter whether you add 4 and 5 and then add this sum to 8, whether you add 4 and 8 and this sum to 5, or whether you add 5 and 8 and this sum to 4. The answer will always be 17. Stated symbolically, for any numbers
a, b and c, (a + b) + c = a + (b + c) = (a + c) + b.

c. The commutative law or property with respect to multiplication. This law is similar to the law for addition. It states that the order of factors (numbers to be multiplied) does not affect the product. In the problem 8 x 7, it doesn't matter whether you multiply 8 x 7 or 7 x 8, the result is 56. Stated symbolically, for any numbers a and b, a x b = b x a. d. The associative law or property with respect to multiplication. This law is also similar to the law for addition. It states that in any multiplication problem, the product is unaffected by the combinations in which the factors are grouped. Again, as with addition, multiplication is binary and the groupings will be in pairs. In the problems 4 x 9 x 3, the same product will be found by (4 x 9) x 3, or by 4 x (9 x 3). Stated symbolically, for any numbers a, b and c, (a x b) x c = a
x (b x c) = (a x c) x b.

1403.

Distributive Property for Multiplication

This law or property states that multiplication can be distributed over addition. It can be best explained by use of an example: Multiply 3 times the sum of 4 and 6. One method of doing the problem is to add the 4 and the 6 then multiply this by
3 to get the answer 30: 3 x (4 + 6) = (3 x 4) + 3 x (4 + 6) = = 3 x 10 = 30. Now by the

distributive method, you distribute the 3 to each of the addends:
(3 x 6) 12 + 18 = 30

Although it may not seem like the best method for this particular problem, it is one of the most used properties in algebra. Remember that this property is for multiplication over addition. The converse is not true; addition is not distributive over multiplication. The examples below show the difference between the two:
3 + 3 + (4 x 6) (4 x 6) = 3 + 24 = = (3 + 4) x 27 (3 + 6) = 7 x 9 = 63

Note:
b, and c, a x (b + c)

(27 does not equal 63)
a x b + a x c.

Stated symbolically the distributive law is, for any numbers a,
=

1-9

Lesson Summary. In this lesson you identified the three axioms of equality: reflexive, symmetric, and transitive. You also studied the commutative property for addition and multiplication, the associative property for addition and multiplication, and the distributive property for multiplication. There are many other laws or properties of numbers and all of them are important for various applications; you will cover two more in the next lesson. Lesson 5. OTHER PROPERTIES OF NUMBERS

LEARNING OBJECTIVE Given simple equations, solve the equation using the zero and unity elements. 1501. Zero and Unity Elements

The zero and unity elements are the two additional laws and properties of numbers that you will cover; let's take a look at them. a. Zero element. As pointed out earlier, the invention of zero by the Hindus is what made the Hindu-Arabic number system. Its value as a place holder and a signifier of a nothing value is what set the system apart from others. The zero has several unique properties that come into play in operations with numbers. The first one is in addition. (1) Addition. Zero is referred to as the additive identity or the identity element for addition. This simply means that when zero is added to a number it preserves the
identity of the number: 5 + 0 = 5, 0 + 8 = 8, 10 + 0 =

10, etc. Although fundamental, this property is not always obvious. Let's see how it works with both multiplication and division. (2) Multiplication. Multiplication with the zero is quite another story. Where adding zero to a number preserves its identify, multiplication of a number by zero destroys its identity, for any number times zero is zero. Stated
symbolically, for any number a, a x 0 = 0 x a = 0.

(3) Division. There are two aspects of division with zero. Dividing zero by a number always results in zero. Dividing a number by zero is impossible. These statements can be proved by several examples.
In dividing zero by a number, 0 + 8 = ?, whatever number

replaces the question mark must be a number that, when multiplied by 8, will give zero. Remember that in order to check a division problem, multiply the quotient (answer) times the divisor. This will give the dividend. In the above problem, the only possible replacement for the question mark is zero since it is the only number that multiplied by 8 will give zero.
1-10

You should see that this will be the same for any number divided into zero. Stated symbolically, for any number a, 0 . a = 0. In dividing zero into a number, 6 . 0 ?, you should see that there is no replacement for the question mark that when multiplied by zero will give 6. This would be the same for any number. This operation is said to be impossible or undefined. Stated symbolically, for any number a, a - 0 = impossible. You have just seen the effects of the zero element on numbers; let's now take a look at the effects of the unity element on numbers. b. Unity element. The number one has many significant properties. One aspect that you want to examine is its identity quality. Where zero is the identity element in addition, one is the identity element in multiplication. That is, any number multiplied by one will be equal to that number, or one will preserve the number's identity. This property is sometimes referred to by the important-sounding title, multiplicative identity. Stated symbolically, for any number a, 1 x a a x 1 = a. Division with the number one also has some important aspects. Dividing a number by one preserves its identity in the same manner as multiplying by one. That is, any number divided by one gives us that number. Stated symbolically, for any number a, a - 1 = a. Dividing one by a number is quite a different story. One divided by a number gives the reciprocal of the number. This is a very important term that you will find throughout your mathematical education. You will have occasion to use it several times in this course. For example, the reciprocal of 15 is 1/15, the reciprocal of 3/4, is 4/3. Note: An important fact is that the product of a number and its reciprocal is one; for example: 4 x 1/4 = 1, 15 x 1/15 = 1, 3/4 x 4/3 = 1, etc. Lesson Summary. This lesson taught you two more important laws, the zero and unity elements and how they effect the identity of our number system.

Lesson 6.

SYMBOLS OF GROUPING

LEARNING OBJECTIVE Given simple equations using symbols of grouping, simplify the mathematical statements. 1601. Mathematical Punctuation Marks

The symbols of grouping are the punctuation marks of mathematics. They change the meaning of a group of numbers the same way that commas can change the meaning of a group of words. For example, punctuate this sentence: Pvt. Buttplate said the Gunny is a meathead. If you are a Gunnery Sergeant you probably did it something like this: "Pvt. Buttplate," said the Gunny, "is a meathead." On the other hand, you may have completely changed the meaning by punctuating the sentence like this: Pvt. Buttplate said, "The Gunny is a meathead." You should see from these examples that the placement of punctuation marks is important. Placing symbols of grouping in mathematical equations is equally important. The following examples show the four different symbols of grouping (devices) used: parenthesis ( 6-2 )
brace {6 2} 2]
-

bracket [6 -

bar or vinculum 6

Let's look at a problem and see how the use of these devices can clear it up. What is the answer to this problem: 3 x 7 + 4 = ? Is it 25 or 33? There is a proper order of operations for this type of problem, and you will look into this in the next lesson; however, the problem is clarified when you insert a grouping symbol such as a parenthesis: 3 x (7 + 4) = 33. Moving the parenthesis to another position will produce a different result: (3 x 7) + 4 = 25. In the first example, 3 x (7 + 4), you have a multiplication sign included. When a number directly precedes or follows a grouping symbol, multiplication is indicated. For example, 3(6), (3)(6), and (3)6 all indicate to multiply 3 times 6.

1-12

Grouping symbols are all used in the same manner. problem could be written in four ways: 3 (7 + 4) = 3 x 11 = 33 3 [7 + 4] = 3 x 11 = 33 3 {7 + 4} = 3 x 11 = 33 3 7 + 4 = 3 x 11 = 33

Our original

As you can see, symbols of grouping help simplify mathematical statements, but what happens when you have more than one set of symbols? Our next lesson, Order of Operations, will help answer this question. Lesson Summary. This lesson taught the punctuation marks for mathematics, better known as the symbols of grouping. They are used to simplify mathematical statements. Lesson 7. ORDER OF OPERATIONS

LEARNING OBJECTIVE Given the rules for the order of operations, solve simple equations. 1701. Solving Mathematical Expressions

In solving a mathematical expression it is possible to have more than one answer, but there is only one correct answer. It depends on the order in which you perform the indicated operations of the problem (3 x 7 + 4 = 25 or 33?). In the preceding lesson, you saw how the use of grouping symbols gives a clear indication of the intent of the problem, making it easier to work the problem. Whether or not there are any grouping symbols in a problem, there are rules which dictate the proper sequence of operation. These rules must be followed if you are to arrive at the correct answer: a. When grouping symbols are used, simplify the parts of the expression first, following the order of operation in rules (b) and (c) listed below. b. Perform all multiplication or division operations indicated in the expression in the order you come to them, working from left to right. c. Perform all addition or subtraction operations indicated in the expression in the order you come to them, working from left to right.

1-13

Remember, when you have a problem or equation which contains more than one type of grouping symbol, "work from the inside out." This simply means complete all of the indicated operations within the innermost set of symbols first, the next innermost set of symbols second, and so on. When all of the indicated operations within a grouping symbol have been completed, the symbol may be removed from the problem. This will keep the problem from becoming cluttered with unnecessary symbols which could cause you to make an error. Also, neatness and an orderly step-by-step procedure for simplifying a problem will prevent many errors. Remember, you must do all multiplication and division before any addition and subtraction, and the rules must be applied to each step needed to simplify the problem or equation. Now that you have rules to govern the order of operations, the answer to the expression presented previously is quite easy: 3 x 7 + 4 = ? =
21 + 4 = 25.

Let's look at a few other examples:
6 x 30 8 7 x 2 = (6 x 8) 10)3
-

(7 x 2)
=

= =

48 9

14 = 34

10 x 3 =

(30 -

3 x 3

These examples show you how important the rules for the order of operations are in solving equations. Remember, when grouping symbols are not given, these rules must be followed if you are to arrive at the correct answer. Lesson Summary. This lesson demonstrated how to use the rules for order of operations to solve given equations. Remember the rules: Simplify by removing the grouping symbols, multiply or divide, then add or subtract.

Unit Exercise:

Complete items 1 through 38 by performing the action required. Check your responses against those listed at the end of this study unit.

What single factor made the Hindu-Arabic number system superior? a. b. c. d. Ability to perform addition Compounding of powers of ten Ability to repeat numbers Invention of zero

1-14

The two purposes of the digits in any number is to show the a. b. c. d. cardinal and place values. cardinal and face values. natural and place values. natural and face values.

Which set consists of entirely natural numbers? a. b. 3, 17, 19.5 1/2, 2, 10 c. d. 0, 1, 2 3, 15, 27

Which example illustrates the principle of closure? a. b. 4 - 9= 4 + 9= c. d. 4 9 9= 4=

Which answer to the example below is an integer?
a. b. 3 . 2 12.6 4 c. d. 8 - 12 7 1/4 x 5

Rational numbers can be expressed as a. b. fractions. whole numbers. c. d. decimals. real numbers.

Which example illustrates the reflexive property of equality? a. b. 26 =a a =b c. d. a = a b = 26

When you write a statement such as a = 6, what axiom of equality allows you to change this to 6 = a? a. b. c. d. Commutative property Symmetric property Reflexive property Distributive property If a
=

What axiom of equality applies to this statement?
b and b = c, then a = c.

a. b. c. d.
the

Transitive property Commutative property Reflexive property Distributive property
property.

10. The addition of 27 + 31 can be written 31 + 27 because of a. b. commutative associative c. d. 1-15 reflexive transitive

11. Which example illustrates the associative property for addition? a.
b.

3 + 6 + 9 = 18
18 = 9 + 6 + 3

c. d.
12.

9 + 6 + 3 = 9 + 9 (3 + 6) + 9 = (9 + 3) + 6
9 x 7 illustrates the for

7 x 9 =

multiplication. a. b. c. d. commutative property symmetric property distributive property transitive property

13. Which example illustrates the associative property for multiplication? a. b. c. d. 2 x 9 x 3 =54 54 = 2 x 9 x 3 2 + 9 + 3 =3 + 9 + 2 (3 x 9) x 2 = 3 x (2 x 9)
property.

14. The law that states multiplication can be distributed
over addition is the

a. b. c. d.

distributive commutative transitive symmetric

15. Which example best describes the distributive property?
a. 5 x (2 + 3) = (5 x 2) + (5 x 3)

b. c. d.

5 x (2 + 3) 10 x 15 25

Solve items 16 through 19 by using the zero and unity elements.
16. 10 x 16 -

0 = c. d. 0 1/16

a. b.
17. 7 -

Impossible 16
1
=

a. b.

1 1/7

c. d.

7 O

1-16

18.

58 x 0 a. 0

-580

580

19. 561 x 1 =
a. 56 C. 562

561

5611

Simplify items 20 through 29 by using the symbols of grouping. 20. (8 + 2) + 7

21. 5 + [3 x 2]

22. 8 + ( 3 + 5)

23. 4(3 + 7)

24. 2 {6 + 5}

25.

(8 + 3)

+ 5

26. 2(5 + 3)

27.

[4 x 3] + [4 x 7]

28. 17 15 x 2

29. {17 + 3} 18

1-17

Solve items 30 through 38 by using the rules for order of operations.
30. (7 + 3 + 2) . 3 + 1

31.

(7 + 3 + 2)

. (3 + 1)

32. 5[7 + 9]

4 + 3

33.

7 + 3(5 -

. 6

34. 30 -

3(7 -

35.

27 + 10 -

(11 + 3)

36.

(2 x 3 -

- 3) + 2

37. 6(6 + 8) + 8

38.

(15 -

3 + 8

* 8 x 5

UNIT SUMMARY This study unit has introduced or reintroduced you to the history of the real numbers. You have identified the natural numbers, integers, and the rational numbers within the real number system. You have identified the laws and properties that pertain to the number system. Try to remember these laws, their names, and how they are used with the basic operation of numbers. You can now recognize the symbols of grouping and solve expressions using the rules for order of operations.

1-18

Unit Exercise Solutions Reference 1. 2. 3. 4.
5.

d. a. d. b.
c.

Invention of zero cardinal and place values 3, 15, 27 4 + 9
8 12

llOlc llOlc 1201 1202a
1301a

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
20. 21. 22. 23. 24. 25. 26. 27.

a. c. b. a. a. d. a. d. a. a. a. c. a. b.

fractions a = a Symmetric property Transitive property commutative (3 + 6) + 9 = (9 + 3) + 6 commutative property (3 x 9) x 2 = 3 x (2 x 9) distributive (5 x 2) + (5 x 3) Impossible 7 0 561

1301b 1401a 1401b 1401c 1402a 1402b 1402c 1402d 1403 1403 1501a 1501b 1501a 1501b
1601 1601 1601 1601 1601 1601 1601 1601 1601 1601

(8 . 2) + 7 = 4 + 7 = 11 5 + [3 x 2] = 5 + 6 = 11 8 + (3 + 5) = 8 + 8 = 16 4(3 + 7) = 4(10) = 40 2{6 + 5} = 2{11} = 22 (8 + 3) + 5 = 11 + 5 = 16 2(5 + 3) = 2(8) = 16 [4 x 3] + [4 x 7] = 12 + 28

28. 17 15 x 2 = 17 30 = 510 29. {17 + 3} 18 = {20} 18 = 360 30. (7 + 3 + 2) - 3 + 1 12 . 3 + 1 =

4 + 1= 5
31. (7 + 3 + 2) 12 . 4 . (3 + 1)
=

1701a,b,c

3
32. 5[7 + 9] . 4 + 3 3
= =

1701a,b,c . 4 + 3 = 1701a,b,c
=

5[16] 23

80 - 4 + 20 + 3 =

33. 7 + 3(5 - 1) - 6 7 + 3(4) - 6 = 7 + 12 - 6 =

7 + 2= 9
1-19

1701a,b,c

Reference 34. 30 30 30 15 3(7 - 2) 3(5) = 15 =
=

1701a,b,c (11 + 3) 14 =
=

35.

27 + 10 27 + 10 37 - 14 = 23 (2 x 3 (6 - 4) 2 2= 1
-

1701a, b, c 12 . 3)

36.

2 =
1701a,b,c

2 =

37.

6(6 + 8) 6(14) +8 84 + 8 84 + 8 90 (15 (15 (12 16 2 x 10

+ 8 - 32 . 16 = - 32 . 16 = 32 + 16 = 2 = 1701a, b, c 5

38.

- 3 + 8 - 2) . 8 x - 3 + 4) . 8 x 5 = + 4) . 8 x 5 = . 8 x 5 = 5=

1701a,b,c

1-20

STUDY UNIT 2 FRACTIONS AND PERCENTS Introduction. One day our friend Charlie Caveman and his number one son Calvin were snooping and pooping through the boonies hunting for a tasty brontosaurus for the evening meal. Calvin spotted one and took up a position with good observation and field of fire while Charlie enveloped to the right. Upon arriving at the assault position, Charlie heard a clattering noise and saw Calvin had opened up the base of fire with his slingshot a little too early. Charlie, weary from his envelopment through the heavy brush, charged out and threw his Ml spear at the brontosaurus. His aim was not steady (poor sight picture) and he missed the beast. His spear hit a rock and broke in half. Charlie sat down to catch his breath while the bumbling Calvin went to retrieve the spear. He returned and said, "Dad, your spear broke. Here is the big half. The little half is splintered." Charlie's ears perked up at this unfamiliar terminology. He thought a moment and said, "Calvin, you chased away our evening meal, but you just invented fractions." Of course this is fantasy, but it is used to show that even primitive man, although he primarily used his limited number of model groups, had an awareness that whole things had parts. Undoubtedly, things such as parts of a journey, portions of food, or parts of his herd were recognized for what they are, but it took a more sophisticated culture than Charlie's to actually invent fractions. The Egyptians and Romans had some symbolism to indicate the idea of fractions, but it was the Hindus who actually originated the system that we have today. They took good qualities from both the Egyptians and the Romans and added a vertical system of notation. The system allowed for variation in both the number parts (numerator) and the size of the parts (denominator). Remember that the numerator is the part above the line and the denominator is the part below the line. This study unit will provide you with the skills you need to apply the basic principles of fractions and to perform operations with fractions, the decimal form of fractions, decimals, and percentages. As you are about to see, being able to perform these operations plays an important role in our daily lives as Marines.

2-1

Lesson 1.

BASIC PRINCIPLES OF FRACTIONS

LEARNING OBJECTIVES 1. 2. 2101. Given a series of fractions, use the proper operations to reduce them to the lowest terms. Given a series of fractions, use the proper operations to reduce them to the higher terms. Reducing Fractions

The Golden Rule. The basic principle involving the manipulation of fractions is referred to by many textbooks as the "Golden Rule of Fractions." It is: If the numerator and denominator of a fraction are both multiplied or divided by the same number (other than zero), the value of the fraction is not changed. This principle is fundamental to all of the operations with fractions and there are two important points to be examined concerning it: First, the Golden Rule implies that you must remember to multiply or divide both terms of the fractions. If just the numerator is multiplied, for example, the resulting fraction will not be equal to the initial one. Second, this principle concerns multiplication and division only. If the same number is added to or subtracted from both the numerator and denominator, the value of the resulting fraction will not be the same as the initial one. The importance of this principle lies in the fact that it is often necessary to change the form of a fraction without changing its value. This change in form is known as a reduction. Strangely enough, the term reduction does not necessarily mean to make smaller, because in mathematics we can reduce to lowest terms or reduce to higher terms. The latter appears to be a contradiction, but this is the mathematical terminology. Remember, a reduction is a change in form without a change in value. Let's take a look at both terms. a. Reducing to lowest terms. This is a familiar process that is accomplished by dividing each term of a fraction by a common factor; a process usually referred to as "cancellation." It has come along with the development of mathematics, but it does not accurately describe what is taking place during its use. Cancel implies to eliminate or get rid of, to leave nothing. Our mathematical cancel will always leave at least a value of 1. Let's look at some examples of reducing to lowest terms: 2
4 = 4 6 6 = 2

3 Note: (4 and 6 have the common factor 2) This middle step is usually mental, but it is inserted here to illustrate what is actually being done. 2-2

Reduce 15/30 to lowest terms. Suppose 5 is used as the common factor. In this case dividing each term by 5 gives us: 3
15 = 15 -

9-3 6

3 6

Here, though, 3/6 is not expressed in lowest terms; there is still a common factor 3:

3 = - = 1
6 6 2

2 The highest common factor, 15, would have given the quickest reduction, but sometimes the highest common factor is not readily evident. Always express an answer in lowest terms. When all common factors, other than 1, have been removed from the numerator and denominator, the fraction is in lowest terms. b. Reducing to higher terms. This is the opposite of reducing to lowest terms. Instead of dividing by a common factor, multiply the numerator and denominator by some number greater than one. Reversing the procedure of the examples above:
2= 2 x 2 = 4

3 1
2

2 x 3 15 x 1
15 x 2

6 15
30

Note:

The middle step is usually mental.

It should be evident that to change a fraction to some desired higher term, you must divide the denominator of the smaller fraction into the denominator desired. This quotient is the common factor. For example, change 2/5 to an equivalent fraction with a denominator of 30. 2 = 5 30

2-3

What number must be multiplied times 5 in order to get 30? Since you have memorized the primary multiplication combinations, you know instantly that it is 6. Therefore multiply the numerator by 6 and get:

2 = 12

When the desired denominator reaches a size larger than a product of the primary combinations, then divide as mentioned above. As you know, the resulting fraction is the equivalent of the initial fraction although the form has changed. The ability to reduce to higher terms is essential for adding and subtracting fractions. Lesson Summary. This lesson taught you how to reduce fractions to the lowest terms by dividing and to the highest terms by multiplying. Remember, reducing fractions (reduction) does not change the value of the fraction, only the form. Lesson 2. OPERATIONS WITH FRACTIONS

LEARNING OBJECTIVES 1. 2. 3. 4. 2201. Given a series of fractions, use the operations for addition of fractions to find the sum. Given a series of fractions, use the operations for subtraction of fractions to find the difference. Given a series of fractions, use the operations for multiplication of fractions to find the product. Given a series of fractions, use the operations for division of fractions to find the quotient. Addition of Fractions

a. Adding fractions with like denominators. Do you remember from study unit 1 that the operation of addition of natural numbers is based on several principles, among them the commutative and associative laws and likeness? The addition operation with the rational numbers is also based on these same principles. Of particular concern is the principle of likeness which is controlled by the position of the numerals. In the natural numbers, recall that you add the numbers in the same columns. The column itself provides the likeness: first column is units, second column is tens, third is hundreds, etc. The determining factor of likeness in fractions is the size of the parts to be combined. Of course you know that the size of the part is the denominator. For example, the fractions 4/5 and 4/9, although both 4 of something, are not alike.

2-4

However the fractions 3/5 and 1/5 are alike because although the number of parts is different, the size of the parts is the same. When this is so, there is no difficulty in adding the fractions. The number of parts (numerators) are added to find the total number being considered. Symbolically you could characterize addition of fractions as: For any numbers a, b, and c (except when b = 0): a + c= a + c b b b a practical example would be to add: 3 + 2 + 1 = 3 + 2 + 1 =1 12 12 12 12 2 The middle step is not necessary. It is included to show the mental step that is taken. Note that the answer is expressed in lowest terms. Now, let's examine the operation of adding fractions with unlike denominators. b. Adding fractions with unlike denominators. Take, for example, the addition of 1/3 and 5/6. Clearly, they are not alike. Something must be done to one or both of these fractions to make them alike. Remember that it was mentioned that the process of reduction would be necessary in adding and subtracting fractions. You want to change the form of either 1/3 or 5/6. It is easier to change the 1/3 to sixths without changing the value of the fraction. This is a problem similar to the ones in the last exercise. 1 =v? 3 6 Clearly the answer is 2/6. 5/6 = 7/6 = 1 1/6. Now it is quite simple to add 2/6 +

Again, note that the answer has been reduced to lowest terms. As a review, what kind of fraction is 7/6? How about 1 1/6? If you said improper and mixed number, you are correct. Let's try another example:

Add 2 and 3 2 8

2-5

First 1/2 should be changed to an equivalent fraction with a denominator of 8. This is 4/8 which can be added to 3/8.

4 +

7
8 8

How about when one denominator cannot be reduced to the size of the other one? What is done then? For example:
1 + 3 5 4

The fraction 1/5 cannot be expressed as fourths and 3/4 cannot be expressed as fifths. As you know, a denominator must be found to which each of these denominators can be reduced (a common denominator). The quickest way to find a common denominator is to find the product of the denominators of the fractions to be added. For example, the common denominator for 1/5 and 3/4 is 20. This simply means that each of these fractions can undergo a reduction to 20.

1 =

5 20

while

3 =5

20
19

therefore:

1 + 3 =

+ 15

What is a common denominator of 1/6 and 3/4? If you said 24, you are correct and the fractions would be reduced in this manner:
1 + 3 = 4 + 18 = 22 = 11 6 4 24 24 24 12

Notice that although 24 is a common denominator, there is a better one that could be used. Each of the denominators 6 and 4 can be reduced to 12. Remember this is called the least or lowest common denominator (LCD). See how the problem is worked using this:

1
6

+ 3 =

=11

12 2-6

The last step, reducing to lowest terms, was eliminated by selecting the LCD. Always try to find the LCD, but remember that the product of all of the denominators is a common denominator and can be used. The main disadvantage of this is that you will sometimes have to work with larger numbers and then reduce to lowest terms at the end. For those who like definitions and rules: The sum of two fractions with different denominators is obtained by replacing these fractions with equivalent fractions having a common denominator, and then adding the numerators. Stated symbolically, for any fractions: a and c,a+ b d b
=ad + bc

Note that throughout this study unit, addition has been performed in a horizontal form. It may just as well, and sometimes should, be done in a vertical form. Let's take a look at a couple of examples: 3 2 7
+

3 4 14 14 3 7 = 3 1 14 2
3

9 3 4 2

3 =

+ 1 1 =1 2

4 10 5 = 11 1 4 4

As you can see by the two examples above, adding fractions in a vertical form rather than a horizontal form in most cases will be much easier. Let's now take a look at subtraction. 2202. Subtraction of Fractions

a. Subtraction with like denominators. As stated in lesson 1, subtraction is the opposite of addition. Since the rules of addition of natural numbers apply to rational numbers, it follows that the rules for subtraction of natural numbers apply to the subtraction of rational numbers. In particular, pay attention to the principle of likeness which states that only like fractions may be subtracted. This, as you know, means that only fractions having like denominators may be subtracted. This is done by subtracting the numerators and expressing the result over the common denominator, for example:
11 _ 7 = 11 - 7 = 4

2-7

b. Subtraction with unlike denominators. For fractions with unlike denominators, the process again is similar to addition. Find the LCD and express each fraction as an equivalent fraction with the LCD. Subtract the numerators and reduce the resulting fraction to the lowest terms. For example:
7

12
7

1= 7 6 12
3 = 14

12
9 =

5 12

5
24

c. Subtraction with mixed numbers. There is one other facet of subtraction that should be covered here, that is, subtraction of mixed numbers. At times the process of "borrowing" presents a problem to students. Let's look at several examples. First, fractions with like denominators, and no borrowing:

15 2 3 -61 3
9 1

3 Here the numerators were subtracted and placed over the common denominator and then the integers were subtracted. Now, how about:

15 1 6
12 5

Here the fractional part of the subtrahend (the number being subtracted) is larger than the fractional part of the minuend. This is solved by borrowing 1 from 15, converting it to its fractional form (1 = 6/6) and adding it to 1/6 (often done mentally):

15 1 =

14 + 6 + 1

14 7

- 12

- 12 5 6 2 26 = 2 1 6 3 2-8

Let's take a look at another problem: 4 3 8 -2 3 4 43 8 2 6 8 3 11 8 2 6 8
15

Note that 3/4 was converted to an equivalent fraction with the common denominator 8. This still left the fraction of the subtrahend smaller than that of the minuend. One was borrowed from 4 in the fractional form 8/8 and added to 3/8 totalling 11/8. This gave a workable fraction which could then be subtracted from in the normal manner. A definition of subtraction of fractions has been presented. For any two fractions it can be represented as: a and d' if ad > bc, then:

= ad-bc

Note: the symbol > is read: is greater than. Remember, only like fractions can be subtracted (fractions having like denominators), with unlike fractions, first, find the lowest common denominator, subtract, and reduce the result to the lowest terms. 2203. Multiplication of Fractions

Here again is an operation that is based on the same principles that applied to the natural numbers: the associative, commutative, and distributive laws. Multiplication for fractions is still a rapid or shortened addition process. All you need to do in order to multiply fractions, therefore, is draw on the principles you have already learned. There is one aspect of multiplication of fractions that is a unique process which proves troublesome to the thinking of some students. To multiply usually implies to gain, or to make larger. This is true with integers, but when dealing with multiplication by a fraction, the converse is true; a smaller quantity will result. For example if you take one-half of something, say a dozen eggs, you end up with a smaller quantity, 6 eggs, although you actually performed a multiplication, 1/2 times 12. You should keep this fact in mind when multiplying bv a fraction and have confidence in the answers that are produced. 2-9

a. Multiplying a fraction by a whole number. Multiplication has been defined as a shortened addition of like addends. Therefore, multiplication of fractions is also a shortened addition of like addends. The addends in this case have like numerators and denominators. For example, to add:

In this case, 8/13 which you can see is 4 (the number of addends) times 2 (like numerator) over the like denominator. (Remember, the numerators are added and placed over the like denominator). Stated as a rule: To multiply a whole number times a fraction, multiply the whole number times the numerator and express this product as the numerator of a new fraction whose denominator is the same as the original fraction. Symbolically stated, for any numbers a, b, and c,

a x b = ab c c

Example:

x 3 = 7 x 3 = 21 = 41 5 5 5 5 (* denotes mental step)

One further example by way of a pictorial representation (fig 2-1) should illustrate the rationale of multiplying an integer times a fraction: 5 x 3/4 =?

MLfL UIDW] D[[IlD [[I]D III
I 2 Lk22LL]1 1H EILIL__
4lit4111 1V1[1[ rnrIKTThr11
_

__llLg

111 1
5 units 3 each
4

Fig 2-1.

An integer times a fraction.

2-10

|tt15 units '

each

LIJLIr Z

answer: 3

Fig

2-1.

An integer times a fraction--cont'd.

This example (fig 2-1) is a pictorial representation that illustrates the rationale of multiplying an integer times (x) a fraction. It is sometimes referred to as a "putting-together" operation. b. Multiplying a whole number by a fraction. Because of the commutative law, you know that this process will produce the same result as multiplying a fraction times an integer. The mechanics of the operation are the same: Multiply the numerator of the fraction times the integer and express this product as the numerator of a new fraction having the denominator of the original fraction. Symbolically stated, for any numbers a, b, and c, a x b = ab. For example: Find 2 of 8 acres (fig 2-2). c c 5

2_l ll

acres

acre

5 l acre 5

-__E_EE

]

1 ES~1

Fig 2-2.

A fraction times an integer. 2-11

iZLZZ2 x 8

Fig 2-2.

A fraction times an integer--cont'd.

c. Multiplying a fraction by a fraction (fig 2-3). In this operation you are interested in finding part of a part. The mechanics of the process are the same as the two already discussed except that the denominators are multiplied too. Symbolically stated, for any numbers a, b, c, and d (except when b and d 0), a x c = ac b d bd
Find

of 3 of an acre.

3 x 3= 4 5

1 acre

3 acre 5 Fig 2-3. A fraction times a fraction.

1 acre 5

2-12

4~~~

20 9
_ _

acre

~~~~20

3 5

2=0 of 1 acre

Fig 2-3.

A fraction times a fraction--cont'd.

d. Cancellation. Earlier in this lesson, the term cancellation was used to indicate reducing to lowest terms. Many times the proper use of canceling when multiplying fractions will enable you to work with smaller numbers and find answers more quickly. Look at the example worked first without canceling and then with canceling:

18 18
x

20 20
6

360

(This must now be reduced)

48 - 24 = 2 360 + 24 = 15

2 1 & x 6 = 2 0 -2-G 15 3 5 Note:

(Divide 8 & 20 by 4 and then divide 6 & 18 by 6)

You may cancel the numerator of one fraction with the denominator of another when reducing the individual fraction. From this you should see that there is a definite advantage to canceling before multiplying. Let's see how canceling affects mixed numbers.

2-13

e. Multiplying with mixed numbers. This involves changing the mixed numbers to improper fractions and then proceeding with the rules for multiplying fractions. For example:

4 2 x 5 = 7 9 10
-3-

x -a-0 = 500 921 3

23 1 21

Note: First, change the mixed number to an improper fraction, cancel, then multiply. As you can see canceling with mixed numbers simplifies the equation. 2204. Division of Fractions

There are two computational techniques generally in use for division of fractions: The common denominator method and the simpler and probably more used inverted divisor method. Let's take a look at them. a. Common denominator. In this method, a common denominator is first found just as in addition and subtraction. Each fraction is reduced to an equivalent fraction having the common denominator. The denominators are then disregarded and the numerators are divided. This can be shown with an example:
3. =

4
3 x

3
.1

3 =

Reduce each fraction to an The common denominator is 12. equivalent fraction with a denominator of 12. 9 .4 12 12 Divide 9 by 4 and 12 by 12. 12 12 = 2 1 = 2 4 4
1

2-14

You should see that the divided denominators will always be equal to 1, therefore this step may be eliminated, for example:
5 5_

16
5 16
.

8
-

5 = 5 . 10 8 16 16

5 10

1
2

The common denominator method can also be done another way. This is by multiplying the original fractions by the LCD, for example:
3 .1=v

3 3) (+a x :) 1

4
. (+r

)

9 +4

2 1 4

Notice how LCD 12 cancels out denominators 4 and 3 making them 1. Using either method (division or multiplication) on fractions will provide you the quotient. Now let's see what happens when we invert the divisor. b. Inverted divisor. This is the method that you probably learned in school. Simply stated, you remember it as invert the divisor and multiply. Let us see why this is done. This method is based on the idea of multiplying by the reciprocal of the divisor. Although this sounds complicated, remember from study unit 1 that the reciprocal of a number is 1 divided by the number, or, that the product of the number and its reciprocal is equal to 1. Let's do some examples to illustrate.
3 .1

This example is read 3/4 divided by 1/3. What is the divisor? The divisor is the number by which a dividend is divided, in our case, it's 1/3. If 1/3 is the divisor, what is the reciprocal of the divisor? If you said 3/1, that's correct! Our next step is to multiply by the reciprocal of the divisor (multiply both the divisor and the dividend by this number).

1
(3 x

() 1( -a
1

9) -

1 =

1 4

2-15

Notice that the left fraction becomes 9/4 and the right fraction becomes 1. This will always be the case with the right fraction divisor. Therefore, the right half of the problem can be eliminated. You should see that the left fraction then fits the rule: Invert the divisor and multiply. Let's take a look at another example, first with the extra step and then without:
1 1 x 1 . ( 1 x G)

(ag

-16 2
5

-5 1

6 1

5 1

= 1

1 .5 =

x
is4

1 & =

1
2

-5

This explanation should give you some insight into why the rule is valid and meaningful. There is one more example to discuss: dividing a fraction by an integer. Let's take a look at it. When dividing a fraction by an integer the most important rule to remember is to invert the integer (divisor). For example:
7 .2= 8 2 7 2 =7 x 1 = 7

Remember that 2 is actually the fraction 2/1 so the reciprocal is 1/2. Don't make the mistake of canceling the 2 with the 8. Lesson Summary. This lesson provided you the necessary skill to use the operations for addition of fractions to find the sum, use the operations for subtraction of fractions to find the difference, use the operations for multiplication of fractions to find the product, and how to use the operations for division of fractions to find the quotient.

2-16

Lesson 3. 1.

DECIMAL FORM

Given decimal fractions, apply the operations to convert to common fractions. Given common fractions, apply the operations to convert to decimal fractions. Given a series of decimals, apply the operations for addition of decimals to find the sum. Given decimals, apply the operations for subtraction of decimals to find the difference. Given decimals, apply the operations for multiplication of decimals to find the product. Given decimals, apply the operations for division of decimals to find the quotient.

2. 3. 4. 5. 6.

It may seem odd to some students that decimals are discussed with fractions. On examination, however, you will see that decimals are just another way of writing fractions. Decimal fractions can be converted to common fractions, and vice versa, with very little difficulty. Let's look at the decimal system. 2301. Decimal Numeration

Our decimal system (fig 2-4) as the name implies, is based on the Every number to the left of the decimal point is powers of ten. 10 times the size of its neighbor to the right. Every number to the right of the decimal point is 1/10 the size of its neighbor to the left. For example, the thousands column is 10 times the size of the hundreds column. The tens column is 10 times the size of the units column. The first column to the right of decimal point is 1/10 of the units column or tenths. The next one to the right is 1/10 of the tenths or one-hundredths. The Note that figure 2-4 is only next one is thousandths, and so on. partial since the scale extends indefinitely in both directions. Also note that the system is symmetrical about the units column. That is, one to the left is tens, one to the right is tenths; two to the left is hundreds, two to the right is hundredths; six to the left is millions, six to the right is millionths. It is important to note that the system is not symmetrical about the decimal point. The primary purpose of the decimal point is to separate the whole and fractional parts of a number and to designate the location of the units digit.

2-17

M
I L L 0 I

H u N D B D N

T B N T 0H 0 H 0 d

T 0 U s a A N
A H

H u N D

T N S

s~T

H
A N D S

o u s

Ak
D

2-4 ~i Deia Dia

~~~~D

U N I T S N

H T U E1 N N D T R H B s~~~~~ T

T H 0 u 5 A D

T B N T H 0 U

H N D RK B D T

M I L L

o N
H T

numraton H T
s

A N
H S

H 0
U A N D T H

io6

i1io

o-r

7
Decimal numeration.

77o

Fig 2-4.

A decimal fraction, by definition, is a fraction whose denominator is a power of ten. There will always be a particular number of tenths, hundredths, thousandths, etc, and never any thirds, eighths, twelfths, etc. Other than having these fixed denominators, the only difference between common fractions and decimal fractions is the form in which they are written (fig 25). Also the last digit to the right in a decimal fraction determines the size of the fraction which is the same as the denominator.

= seven tenths = .7
=

10 39 100 421
1000
= fifty-seven ten-thousandths = 57 10000

thirty-nine hundredths

.39

four-hundred twenty-one thousandths
.0057

=.421

Fig 2-5.

Comparison of form for common fractions and decimal fractions.

Notice that zeros are needed for their use as place holders with .0057. Remember, the only difference between common fractions and decimals is the form in which they are written. With this in mind, let's learn how to convert decimals to common fractions.

2-18

2302.

Converting Decimals and Common Fractions

Since decimal fractions and common fractions are simply different forms of the same thing, it should come to mind that one can be changed to the other. Decimal fraction to common fraction. We have already a. discussed the fact that a decimal fraction has a denominator of For example .8 is the same as 8/10 which can some power of ten. Some fractions can be reduced while others, be reduced to 4/5. It is relatively easy to determine by like .33 = 33/100, cannot. Since the inspection whether a fraction can be reduced further. denominators are all powers of ten, you need only try to divide If the fraction is not divisible by 2, 5, or 10 (factors of 10). Let's take a look at by one of these, it is in its lowest terms. some examples and see how this is applied:

.15 = .008=

100
8

20
= 1

.88 = .0017=

100

= 17

1000

125

10000

These examples show how easy it is to convert a decimal fraction to a common fraction. Now let's learn how to convert a common fraction to a decimal fraction. Common fraction to decimal fraction. This operation is a b. little more complicated than the one just mentioned. It involves For example, 9/25 converting the denominator to a power of ten. be converted. What is the most likely power of ten to can easily Now, what Right. which 25 could be converted? Did you say 100? Multiply 25 to convert it to 100? You're right. must be done to If the denominator is Don't forget the Golden Rule. it by 4. multiplied by 4, then the numerator must also be multiplied by 4. Let's look at some examples:

= 4 x

25 20

4 x 25
11 =

100
55 = .55

11 = 5 x

5 x 20

100
136 = 1000 =

17 = 8 x 8 x 125 3 = 2 x

17 = 125 6

136

3 =

2 x 5

2-19

It is hoped that you have noticed that it will not always be possible to multiply the fraction by a whole number to get a power of ten. For example, denominators of 8, 13, 22, and 15 are not factors of integral powers of ten. In cases such as these it is easier to use a different method. This involves dividing the numerator by the denominator. This is called division of decimals which will be covered under operations with decimals. 2303. Operations with Decimals

a. Addition of decimals. As with integers, the addition process with decimals is based on the principles presented in study unit 1. In particular, again only like things may be added: tenths to tenths, hundredths to hundredths, etc. When addends are lined up in this manner, the decimal points fall in a straight line. 2.18 34.35 0.14
+ 4.90

41.57 The columns are added in the usual order and the decimal point of the sum falls directly below the decimal points of the addends. b. Subtraction of decimals. This operation also involves no new principles. Place values are again aligned which causes the decimal points to align too. This alignment is also maintained in the answer, for example: 45.76
31.87 -

8.64000
.00437

13.89

8.63563

The columns are subtracted in the usual order and the decimal point of the difference falls below the decimal points of the subtrahend. c. Multiplication of decimals. Two cases of multiplying decimals will be considered: general multiplication and multiplication by powers of ten. (1) General. Multiplication of decimals can be explained in two ways. The first is more of a common sense type of explanation rather than a statement of principles. For example, if you are multiplying 3.2 x 7.6 you must realize that although there are two digits in each of these numbers, part of each is a fraction. The answer will be close to 21 which is the product of the whole numbers 3 and 7. 2-20

7.6 x 3.2 152 228 24.32 Let's look at a couple more examples: 4.5627 x 10.37 319389 136881 456270 47.315199 126.3 x 12.791 1263 11367 8841 2526 1263 1615.5033

Note:

The non-fractional parts of the numbers in the first example are 10 and 4. Therefore, you know that the product must be close to 40. As you can see, this method is all right as long as the numbers don't get too large. One thing can be observed from the examples. This is that the answer in each case has as many digits following the decimal point as the sum of the digits following the decimal points in the multiplier and multiplicand. In the example 126.3 x 12.791, the multiplicand 126.3 has one digit following the decimal point and the multiplier 12.791 has three for a total of four. Note that the answer also has four.

The other explanation of multiplying decimals is by converting the decimal fractions to common fractions. Let's look at some examples: decimal = .4 x .37 = common fraction fraction
=
=

4 x 37 10 100
148 1000 .148 _ 4316 x 34

decimal

_ 4.316 x 3.4 = common

fraction

fraction
=

1000 10 146744
10000 14.6744

2-21

decimal

.453

x 78.2

= common

453

X 782

fraction

fraction
=

1000 354246
10000 35.4246

Note:

The key is that the number of digits following the decimal point in the answer is the sum of the digits following the decimal points in the multiplier and the multiplicand. This leads us to the general rule for multiplication of decimals: Ignore the decimal points and multiply as though the multiplier and multiplicand were integers. Then locate the decimal point in the product to the left as many places as there are to the right of the decimal points in the multiplier and multiplicand.

(2)

This operation is used enough to warrant Power of ten. special comment. By doing enough examples, you can discover for yourself what happens to the decimal point when you multiply by 10, 100, 1000, etc., or by .1, .01, .001, etc.
10 x 100 x 7 = 70 94 = 9400 1000 x .5891 = 589.1 9.8556 = 98556

10,000 x

Note:

When multiplying by a power of ten, the decimal point is moved to the right as many places as there are zeros in the multiplier. Another aspect of multiplying by powers of ten is multiplying by .1, .01, .001, etc., which are negative powers of ten. This operation is opposite in effect to the one just described.
.01 x 153 = 1.53 .3482 .0001 x .01 x 13.28 = = .001328

.001 x 348.2 =

.0177

.000177

Note:

The effect here is to move the decimal point to the left as many places as there are decimal places (not zeros) in the multiplier.

As mentioned, this operation occurs frequently and you can save time by recognizing the problem and moving the decimal point without multiplying. Practice this and be confident in it and you will save time in computations.

2-22

d. Division of decimals. Four things will be discussed: general division of decimals, division by powers of ten, reducing a common fraction to a decimal fraction, and rounding off. Let's take a look at them. (1) General division of decimals. Here, as in the other discussions, the purpose is not to teach you an operation, but to give you some insight into why the decimal point is moved, or not moved, in dividing decimals. You learned earlier in this lesson that multiplication of a decimal by a power of ten has the effect of moving the decimal to the right. This in effect is what is done in division of decimals. If the divisor can be transformed into a whole number, the division can be accomplished in the same manner as division of integers. Two things are done without actually naming them as such. First, the divisor is actually being multiplied by some power of ten when the decimal is moved to the right to make it a whole number. Second, since the divisor and dividend can be considered as the numerator and denominator of a fraction, if the denominator (divisor) is multiplied by some number, then the numerator (dividend) must be multiplied by the same number in order not to change the value of the fraction. For example:

3.6 F45.75

To make 3.6 a whole number, the decimal point is moved one place to the right which is the effect of multiplying by 10. The same thing must be done to the dividend as was done to the divisor:

36. 457.5

Various methods are used to show that the decimal point has been moved. The most common one is illustrated.

.153 F7.62

.153. 1 t

F7.620. 2-t 3 2-23

Note that a zero was added to the dividend which means the same as multiplying by 1000. Remember also that the decimal point in the quotient (answer) will be directly above the new location of the decimal point in the dividend. The reason for this can be shown several ways. One is to recall the rules for multiplication of decimals. For example in this problem:

.186 2.9. .5.394
1-f X-IT

To check the answer you must multiply the quotient times the divisor. This should equal the dividend. In this case .186 x 29 = 5.394. By working several examples, you can prove to yourself that the decimal point truly ends up in the correct place when it is placed directly over the one in the dividend. (2) Powers of ten. Division by a power of ten is the inverse of multiplying by powers of 10. Where multiplying had an increasing or upgrading effect, division has a decreasing or downgrading effect. Consequently, division will cause the decimal point to "move" in the opposite direction or to the left. Note several examples:
217,000 32,000 + 1000 = 217 10 = 3200 586.37 + 19.79 + 100 = 5.8637 1000 = .01979

The generalization for division by a power of ten is that it has the effect of moving the decimal point in the dividend a number of places to the left equal to the number of zeros in the divisor. (3) Reducing a common fraction to a decimal fraction. Recall that earlier in the study unit it was mentioned that there was another way of reducing a common fraction to a decimal fraction. As mentioned, this method involved division of decimals. Now that this has been discussed, this much-used method may be covered. Since a common fraction is actually a division of the numerator by the denominator, this is what is done:

2-24

.5
1 =

.75 3 = 4 4 3.00 2 8 20 20 .85 17 = 20 F17.00 16 0 20 1 00 1 00

10
1 0

.375 3 = 8 8 3.000 2 4 60 56 40 40

Each of these came out even (no remainder), but this is not always the case. In some instances, you may want to round off your answer to some particular decimal place. If your answer does not come out even (4) Rounding off. (there is a remainder), you will then express your remainder as a fraction. In division of decimals, this expression of the remainder may be eliminated by rounding off the answer to some desired decimal place. Common practice is to carry the division to one degree of accuracy more than is desired and then round off. For example, if the answer to a problem is 17.237 and accuracy to hundredths is all that is desired, the 7 may be eliminated. If the extra digit is 0, 1, 2, 3, or 4, it is usually dropped. If it is 5, 6, 7, 8, or 9, one is added to the digit to the left. Therefore, 17.237 rounded to hundredths would be 17.24. Some other examples are: original answer 36.42793 427.6439 .34625 1.4963 nearest hundredth 36.43 427.64 .35 1.50 nearest tenth 36.4 427.6 .4 1.5

Remember the rule, 0,1,2,3, or 4 round down, and 5,6,7,8, or 9 round up.

2-25

Lesson Summary. This lesson provided you with the necessary skill to apply the operations for converting decimal fractions to common fractions, and common fractions to decimal fractions. Furthermore, you learned how to apply the operations of addition and subtraction of decimals. Lastly, you were able to apply the operations for multiplication and division of decimals. Lesson 4. PERCENmci conferenceE

LEARNING OBJECTIVES 1. 2. 3. 4. 5. 2401. Given a common fraction or a decimal, apply the operations to convert to percent. Given a percent, apply the operations to convert to a common fraction or decimal. Given a number, use the operations to find what percent this number is to another number. Given a percent of a known percent, use the proper operations to find the number this percent represents. Given a percent and the number it represents, use the operations for finding the unknown number. Converting Percents, Fractions, and Decimals

The word "percent" is derived from Latin. It was originally "per centum," which means "by the hundred." Thus, the statement is often made that percent means hundredths. Percentage deals with the groups of decimal fractions whose denominators are 100--that is, fractions of two decimal places. Since hundredths were used so often, the decimal point was dropped and the symbol % was Thus, 0.15 placed after the number and read "percent" (per 100). "15 hundredths," and 15% is read "15 percent." Both is read represent the same value, 15/100 or 15 parts out of 100. a. Changing decimals to percents. To change a decimal to a percent, the first step is to move the decimal point two places to the right. After moving the decimal point two places to the right, add the percent sign (%), for example: .35 = 35% Remember, a decimal point is not shown with whole percents; 63% does not have a decimal point shown. Mentally, however, a decimal point is placed to the right of the number three (63.%).

2-26

Let's look at another example:
.83 = 83%, not 83.%

However, as seen in the next example, fractional percents do have decimal points. 56 2% = 56.5% 20 Now that you know how to change decimals to percents, let's learn how to change fractions to percents. b. Changing fractions to percents. To change a fraction to a percent, first change the fraction to a decimal and then the decimal to a percent, for example:
8 = .375 = 37.5%

Let's try another example:
3 =

.75 = 75%

Remember, change the fraction to a decimal and then change the decimal to a percent. Up to this point we have learned how to change decimals and fractions to percents; now, let's learn how to change percents back to decimals and fractions. c. Changing percents to decimals. Since you do not compute with numbers in the percent form, it is often necessary to change a percent back to its decimal form. The steps are just the opposite of that used in changing decimals to percent. You change a percent to a decimal by dropping the percent sign and moving the decimal point two places to the left. For example, 12.5% becomes 12.5 after dropping the percent sign. Then you move the decimal point two places to the left and 12.5 becomes .125. Let's look at a few more examples:
34% = .34 24.5% = .245 125% = 1.25

2-27

When a fractional percent (3/4%, 2/34%, 9/16%, etc.) is to be changed to a decimal, you must first change the fractional percent to its decimal equivalent percent (.75%, .059%, .56%) before you drop the percent sign and then move the decimal point two places to the left (.0075, .00059, .0056), for example: 5 3% = 5.75% = .0575 4
1% =

.5% =

.005

Now that you have learned how to change percents to decimals, let's learn how to change percents to fractions. d. Changing percents to fractions. To change a percent to a common fraction, first change the percent to a decimal. The second step is to change the decimal to a fraction and reduce to its lowest terms. For example:
80% = .80 = 80 = 4

100
65% = .65 = 65 =

5
13

100
22.5% = .225 = 225

20
= 9

1000

As you can see, changing decimals and fractions to percents is quite easy, but what happens if you need to know the percent of one number to another? Let's take a look and see. e. Percent of one number to another number. You are now going to learn how to solve percentage problems. All percentage problems can be solved by substituting given information into the formula:
small number
=

large number

100%

Percentage numbers are all placed on one side of the equal sign in the formula and all other information on the other side.

2-28

(1) Findinq what percent one number is to another. For example, in the problem: There are 25 Marines in the formation. Of the 25, 5 are NCO's. What percent of the Marines are NCO's? Substitute all known information into the formula.
Step #1 small %
=%

large Step #2 Step #3

100%

5 = x(unknown)% 25 100% 5 = x (Change percents to 25 1.00 decimals and cross multiply)
25x = 5

Step #4

Although this is actually an equation, and you will not be discussing algebra until the next study unit, you will cover enough algebra to complete the formula. Note: The Golden Rule for the multiplicative property states that if two equal numbers are multiplied by the same number, the products are equal. The division property states that if two equal numbers are divided by the same number, the quotients will be equal. Just as addition and subtraction are inverse operations, so are multiplication and division. One operation can undo the other. This rule also applies to solving equations involving the multiplication and division properties of equality. 25x = 5 You are looking for some number that when multiplied by 25 will give a product of 5. The objective in solving an equation is to isolate the variable, X in this case. Here you have multiplication, so the inverse division operation will be applied.
Step #5 25x = 5 (Note that 25x = lx. It is

25 25 x =5 25
Step #6 x = .20 x = 20%

25 customary to drop the 1 since x and lx are the same)

2-29

Let's look at another example: Of the 60 meals picked up by the "platoon guide," 12 of them were turkey loaf. What percent were turkey loaf?
Step #1 small = x%

large Step #2
Step #3

100 (Put known information into the formula)
(Change percents to

12 60 60

x% 100%
x

12 _

1.00

decimals and cross multiply)

Step #4

60x = 12

Step #5

60x = 12 60 60
x = 12

60 Step #6 x = .20
x = 20%

(Change decimal to percent)

If you apply the steps involved, you will have no problem finding what percent one number is to another. You may be asking yourself, why is this important? This is very important to Marines. Percentages are used every day to help account for our men, material, equipment etc. However there is still more to learn. You must also be able to find the number that a percent represents and that is what we are going to learn next. (2) Findinq a percent of a number. This is probably the most used of all percentage problems. A typical problem would Twenty Marines work in the S-1 office. Ten percent be: can go on leave. What is the number of Marines that can go on leave? Here, as in the preceding examples, the known information can be put into the formula:

Step #1

small =

x %

large
Step #2

100%
10%

x(unknown) =

20 Step #3 x = .10 1.00 20

100% (Change percents to decimals and cross multiply)

2-30

Step #4

20 x .10 = 2.0

Step #5

2 Marines can go on leave

Let's try one more example: During the year (365 days), elements of the 1st platoon were on patrol 48 percent of the time. How many days were they on patrol? Step #1
Step #2 Step #3

small number = % large number 100%
x(unknown) = 48%

365
x = .48

100%
(change percents to

365 Step #4 Step #5 lx x
=

1.00

decimals and cross multiply)

365 x .48 175.2 days on patrol

You have been taught how to find what percent one number is to another and how to find the number that a percent represents. You have one more area to learn. You must be able to find the unknown number if you are given a percent. Let's take a look and see how this is done. (3) Finding the unknown number. A typical problem would be: Pvt Pile bought a shirt for $10.50. This was 35 percent of the money he had saved for clothing. How much had he saved? As in the two preceding examples, the problem can be solved by putting the known information into the formula:
Step #1 small number = %

large number
Step #2 Step #3 Step #4 10.50 = .35
1.00

100%
35% (Change percents

x(unknown)
10.50 =
x

100%

to decimals and cross multiply)

.35x = 10.50

Note:

Here you must use the inverse division operation (Step 5). You are looking for some number that when multiplied by .35 will give you a product of 10.50. 2-31

Step #5

.35x = 10.50

(x =

10.50 . .35)

.35
Step #6 x = $30

.35

Let's look at one more example: Cpl Slick spent $52 on liberty in Tokyo last Saturday. This was 65 percent of his bank roll. How much money did he have originally?
Step #1 small number = %

large number Step #2
Step #3

100% (Change percents to decimals and cross multiply)

= 65% 52 100% x(unknown)
.65 52 = x 1.00

Step #4

.65x

Step #5
Step #6

.65x = 52 .65 .65
x = $80

(Use the inverse division
operation: x = 52 . .65)

Remember, the main factor in finding the unknown is the inverse division operation. Lesson Summary. This lesson provided you with the necessary skill to apply the operations of percentage to convert a common fraction or decimal to a percent, to convert a percent to a common fraction or decimal, to find what percent one number is to another, to find the percent of a number, and to find the unknown number.
------------------------------------------------------------__--

Unit Exercise:

Complete items 1 through 74 by performing the action required. Check your responses against those listed at the end of this study unit.

Note:
1.

Complete items 1 through 6 by reducing each fraction to its lowest term.
3_6

48 a. 3 4 b. 4 5 c. 3 8 2-32 d. 6 8

2. a.

56
1 b. 1 c. 1 d. 1

5 3. a.
78 =

102 15 17 b. 13 17 c. 15 19 d. 13 19

99-=,

144 a. 15 16
186 =

13 16

11 16

9 16

5. a.

378 29 63
680

31 63

29 65

31 65

2520
a. 17 b. 19 C. 17 d. 19

Complete items 7 through 12 by reducing each fraction to its higher term by supplying the missing numerator.
7. 4 9
?

a. 8. a.
9.

12
3
?

4 15
7 =

100 b.
?

8 a. 110

240 b. 120 c. 210 d. 220

2-33

10. a. 11. a. 12. a.

5 .? 56 7 30 5= 12 55 b. ? 132 b. 60 C. 66 d. 72 40 c. 42 d. 45

9 .. ' 37 111 18 b. 21 c. 27 d. 99

Complete items 13 through 40 by performing the action required. 13. a. 3 + 1? 3 8 1 24 1
9

8 24

9 24

17 24

14.

1
+ 6

5 12 2
7

5 16

5 18

5 24

15. + a.

1 12 24 64 3 8 7 b. 31 64 C. 24 84 d. 31 84

16.
+

26 40

29 40

26 80

29 80

2-34

17.

3 8 1 12 2 15 3 20 69 120 b. 89 120

89 120

102 120

18.

7 7
8 21 + 3 6

19.

34
72

35
72

36
74

37
74

20.

3 +4 7 1 19

11 + 10 4A 63 9 b. 16 1 21 c. 18 119 d.

21.

7 9 1 -6

17 18

16 18

13 18

d.11 18

22.

3 1
3 5 -6

1 4

1 2

1 4

21 2

2-35

23.

23 8 5 -8

1 4

132 4

1 4

3 4

24.

5 36

1 4 -9 a. 3 25 36 b. 3 1 4 1 25 36 d 11 4

25.

7 2

5 12 7 12 b. 3 5 6

3 3 5

4 3 5

5 6

26.

3 2 3 11 - 12

2 2 3

3 4

212 12

2 1 2

27.

5 2 2 -3

2 1 3

2 2 3

2 1 6

2_ 6

28.

2 49 64
__2

b. 64

117

1 15 64

2-36

29. a. 30. a.

5 x 16 12 x 2 b. 1 2 5 c. 2 3 5 d. 2 2 5 b. 11 c. 10 d. 9

5 1 3 5
2

31.

3 a. 7 9

6 b. 5 9 c. 9 18 d. 7 18

32. a. 33. a. 34. a.

6 2 x 15 3 100 b. 103 C. 300 d. 303

7 1 x 4 2 15 1 b. 2 c. 4 d. 6

5 x 17 2 3 852 3 8. 17 a. 24 85
6

861 3

88 1 3

891 3

35.

3 5 b. 28 85 c. 38 51 d. 40 51

36.

27 a. 35 54

35 b. 37 54 c. 24 35 d. 27 35

2-37

37.

63 360 a. 11 32

28 45 b. 9 32 c. 147 1350 d. 149 1350

38.

6 3 a. 1 14 33 4 b. 1 15 33 c. 2 14 33 d. 2 15 33

39. a.

114 14 1 3

7 12

6 7 8 b. 14 2 3 c. 16 1 3 d. 16 2 3

40.

3 7 a. 2 3

1) 14 b. 1 2

.35 49 c. 1 4 d. 1 8

Complete items 41 through 45 by reducing the decimal fractions to common fractions in their lowest terms. 41. a. .06 3 10 b. 6 10 c. 3 50 d. 3 25

42. a.

.278 278 500 b. 139 500 c. 278 750 d. 139 750

43. a.

.432 216 500 b. 108 275 c. 64 175 d. 54 125

44. a.

.00375 3 80 b. 3 800 c. 5 80 d. 5 800

2-38

45. a.

.8625 69 80 b. 75 80 c. 5 8 d. 7 8

Complete items 46 through 49 by reducing each common fraction to a decimal fraction (power of ten). 46. a. 47. a. 48. a. 49. a. 13 50 .13 8 25 .32 74 125 .925 7 20 .21 b. .27 c. .35 d. .37 b. .592 c. .259 d. .425 b. .24 c. .21 d. .16 b. .16 c. .21 d. .26

Complete items 50 through 61 by performing the action required. 50. 3.68 4.975 1.3 + 16.42 a. 51. 26.375 64.2 .04 18.
+ 17.37

25.375

c. 26.875

25.875

97.61

b. 98.61

99.61

100.61

2-39

52.

8.637 492. .003
+ . 1

400.740

450.740

499.740

500.740

53.
-

46.37
18.48

25.98

25.89

27.89

27.98

54.
-

21.300
2.004

19.692

19.296

18.692

18.296

55.
-

307.00
60.42

264.68

256.58

246.58

244.68

56. x a.

46.37 18.48 856.9176 b. 856.9671 C. 865.9176 d. 865.9671

57.

21.3

x 2.004
a. 42.6852 b. 42.2586 c. 43.6852 d. 43.2586

58.

307.

x 60.42
a. 16548.94 b. 17548.94 C. 18548.94 d. 18458.94

59.

86.3 197.4200

Round to nearest hundredths (.00).

1.12

1.13

1.21

1.31

2-40

60. a.

.031 1.497820 16.05 b.

Round to nearest hundredths (.00). c. 15.05 d. 15.06

16.06

61. a.

2.2 |2.6390 1.20 b. 1.19

Round to nearest hundredths (.00). C. 1.91 d. 1.02

Complete items 62 through 65 by changing each item to a percent. 62. a. 63. a. 64. a. 65. a. .12 .12% 1.25 125% 1 4 14% 9 40 .225% b. 2.25% c. 22.5% d. 225% b. 20% c. .25% d. 25% b. 12.5% c. 1.25% d. .125% b. 1.2% c. 12% d. 120%

Complete items 66 through 68 by changing from percents to fractions (reduced to lowest terms). 66. a. 40% 2
6

2
5

2
4

2
3

67.
a.

12.5%
1 5 b. 1 6 c. 1 7 d. 1 8

2-41

68. a.

32% 16 25 b. 14 25 c. 12 25 d. 8 25

Complete items 69 through 74 by performing the action required. 69. During an equipment inventory, Cpl Pyle found 14 E-Tools missing from the list of 60. What is his percent of missing E-Tools? a. 2.33% b. 23.3% c. 2.43% d. 24.3%

70. Four Marines out of thirty-two from 3rd platoon are on the rifle range. What percent are on the rifle range? a. 12.5% b. 1.25% c. 25% d. 2.5%

71. Of 40 Marines in the platoon, 80% are qualified swimmers. How many Marines are qualified swimmers? a. 23 b. 27 c. 29 d. 32

72. During the month of April (30 days), our company spent 23 percent of its time aboard ship. How many days did the company spend aboard ship? Round to the nearest day. a. 5 b. 6 c. 7 d. 8

73. Ten percent of the platoon are attached to Company A. If there are 4 Marines attached to the Company A, how many Marines are in the platoon? a. 40 b. 38 c. 30 d. 14

74. If 3/4 percent of the battalion are at sickbay (a total of 12 Marines), how many Marines are in the battalion? a. 1200 b. 1400 c. 1600 d. 1800

UNIT SUMMARY This study unit provided you with the basic principles of fractions, the operations with fractions, decimal form and percentage. Study unit 3 will introduce you to algebra. You will then see how these basic principles are applied.

2-42

Unit

Exercise

Solutions Reference

3 4
36 =

2101

3 4

1 2
28

2101

1 2

13 17
78 = 13

2101

102

11 16
99

2101

144

11 16

31 63
186

2101

378

31 63

17 63
680 = 17

2101

2520

12
4 = 12

2101

9 8. 9. 10. 11. d. c. b. a. 75 210 40 55

27 2101 2101 2101 2101 2 -43

Reference 12. 13. c. d. 27 17 24 3= 8 1 = + 3 9 24 2101 2201

8
24 17 24 2201

14.

5 18 1= 9 1 + 6 2 18 3 18 5 18

15.

31 84

2201

2 = 24 84 7 7 1 84 + 12 31 84 2201

16.

29 40

3 = 15 40 8 7 = 14 40 + 20 29 40

2-44

Reference 17. c. 89 120 45 120 = 10 120 = 16 120 = 18 120 89 120 = 2201

3 8 1 12 2 15 3 + 20

18.

1 24 7 8 7 =2
3

2201

24 8 24

21 +
3

5 =

20
24 49 24 14 1 24

19.

35 72 1 6 1 12 1 9 1 + 8 = 12 72 = 6 72 = 8 72 = 9 72 35 72

2201

2-45

Reference 18 1 21 3 27 63 4 11 63 2201

20.

33 7 4i11 63

+049 10

10 268 10~63
17 66 63 183 18 63 1 21

21.

11 18 7 _ 9 1 _ -6 14 18

2202

3
18 11 18 2202

22. d.

2 1 2 31 = 28 6 3 5 5 = 6 6 23 6 2 1 2

23.

2202

4
8 5
-

2 -8

8 5 8

1 6=
8

3 4

2-46

Reference 24. c. 1 25 36 3
-

2202

5 36 4 9

2 4 36 1 16 36 1 25 36

25.

5 6 5 12
12

2202

7
-

6 17 12
12

4 10

4 5

26. b.

3 4
-

2202

2 3 11 -12 3

12 11 12 9 12 2 3 4

27. a.

2 1 3 5 2
-

2202

4 2
3

3 3 2 3
3

2 21

2-47

Reference 28. d. 1 15 64 1 64 64 49 64 15 64 29. c. 10 5 x 8 16 = 8 8 10 2203 2202

2 49 64

30.

1 3 5 5 5

2203

31.

5 9 2 3 5 = 6 10 = 18 5 9

2203

32.

100 6 2x 3 15
=

2203

20 x 15 =300
3 3

100

33.

2 71 x 4 15 15 x 2 4 15 0 =2 30

2203

34.

88 1 3 5 x 17 2 = 5 x 3 53 3 265 3 88 1 3

2203

2-48

Reference 35. d. 40 51 8 17 .3 5 = 8 x 17 5 = 40 3 51 2204

36.

35 54 16 27
.

2204

12 = 35

-6 x 35 = 35
27 +22 54

37.

9 32 9 1 x64- = 9 32 369O-2-& 8 4

2204

63 360

28 =
45

38.

14 33

2204

23
3

= 20 x
3

= So
33

2 14
33

39.

16 2 3 25 1 14 12 172 6 78 = 1375 x +8 3 2 &55 1 = 5 ° = 3 16 2 3

2204

40. b.

1 2 3_ 7 1 +4 2 1 14 7 .4-9 -3-5 7 35 49 7 14 1 2 6 14 1 14 .3 5. *49 5 14
3 35

2204

2-49

Reference 41. c. 3 50 .06 = = 6 100 2302

3
50

42. b.

139 500 .278 = 278 = 139 500 1000

2302

43.

54 125 .432 = 432 = 54 125 1000

2302

44. b.

3 800 .00375 =

2302

3 = 375 800 100000
2302

45.

69 80 .8625 = 8625 = 69 80 10000

46. d.

.26 13 = 26 = 100 50 .26

2302

47. a.

.32 8 = 32 = 100 25 .32

2302

48.

.592 74 = 125 592 = .592 1000

2302

2-50

Reference 49. c. .35 2302

7
20

35
100

= .35

50.

26.375

2303

3.68 4.975 1.3 + 16.42 26.375

51.

99.61 64.2 .04 18 + 17.37 99.61

2303

52.

500.740

2303

8.637 492. .003 .1 + 500.740

53.

27.89 46.37 18.48 27.89

2302

54.

19.296 21.300 2.004 19.296

2303

55.

246.58 307.00 60.42 246.58 2-51

2303

Reference 56. a. 856.9176 46.37 18.48 37096 18548 37096 4637 856.9176 x 2303 2303

57.

a. x

42.6852 21.3 2.004 852 000 000 426 42.6852

58.

18548.94 307. 60.42 614 1228 000 1842 18548.94 x

2303

59.

1.13 1.128 86.3.197.4.200 4--t 4~~t 86 3 11 1 2 86 3 2 4 90 1 7 26 7 640 6 904
=

2303 1.13

2-52

Reference 60. b. 16.06 16.058 .031. 4 f t 1.497.820 L-. t 31 187 186 1 82 1 55 270 248
=

2303 16.06

61.

1.20 1.199 2.2. 12.6.399 2 2 4 3 2 2 2 19 1 98 210 198
=

2303 1.20

62.

12% .12 = 12%

2401

63.

125% 1.25 = 125%

2401

64.

25%

2401

1 = 25%

65.

22.5% 9 40 = 22.5%

2401

2-53

Reference 66. b. 2 5
40% = .40 = 40 = 2

2401

100 67. d. 1 8
12.5% = .125 =

5 2401
125 = 1

1000 68. d. 8 25
32% = .32 = 32 =

8 2401

100 69. b. 23.3%
14 x%

8 25 2401

60
14 _

100%

x
1.00
=

60
60x

60x 60
x =

14 60
.233

x = 23.3%

(missing E-Tools)

2-54

Reference
70. a. 12.5%
4
-t x

2401

32
4 =

100% x 1.00 4 4 32

32
32x = 32x =

32 x = x
=

.125 12.5%

(are on the rifle range)

71. d.

32
x = 80%

2401

40
x =

100%
.80

40 lx
x
=

1.00
=

32 (qualified swimmers)
2401
= 23%

72.

7
x

100%

x = .23 30 1.00 lx
x
=

6.90 6.9
=

(days aboard ship)

2-55

Reference 73. a. 40
4 = 10%

2401 100% .10
1.00
=

x 4
x

.10x

4 4
.10

.lOx = .10

x = 40 (Marines in the platoon)
74. c. 1600 3% 2401

12 =
x 12 =
x

4
100%
.0075 1.00
=

.0075x

12 12

.0075x =

.0075

x = 1600 (Marines in the battalion)

2-56

STUDY UNIT 3 ALGEBRA Introduction. The beginnings of algebra as we know it date from around AD 830 when the Arabs and Hindus became an important part of our number history. The Arabs used algebra in their study of astronomy and it is believed that much of it was borrowed from the Hindus. The word algebra is derived from part of the title of a book written by an Arabian mathematician named Al-Khorwarazmi. The book, "ilm al-jabr wa'l magabalah," became known throughout Europe as "al-jabr" which you can see is similar to algebra. This study unit will enable you to identify and evaluate algebraic expressions, solve simple equations and inequalities, and solve problems involving proportions and percentages. Let's first look at algebraic expressions. Lesson 1. ALGEBRAIC EXPRESSIONS

LEARNING OBJECTIVES 1. 2. 3. 4. 3101. Given an algebraic expression, identify the monomial term. Given an algebraic expression, identify the like term. Given an algebraic expression, identify the polynomial term. Given algebraic expressions, apply the proper operations to evaluate them. General Information

During this study of algebra you will be confronted with two kinds of numbers: The numerical symbols that you have been accustomed to such as 1, 2, 3, 4, 5, etc., and literal numbers, that is, numbers represented by letters of the alphabet. These literal numbers are usually the unknown quantity that you are to find. Since literal numbers can have any value and are not fixed, they are called variables; that is, the value of x, y, z, m, n, etc. can vary from problem to problem. On the other hand, 7 is 7, 6 is 6, 34 is 34, etc., no matter where they are used. The value of these number symbols (numerals) stays constant, and for this reason they are called constants. When a product involves a variable, it is customary to omit the multiplication symbol x so as not to confuse it with the literal number x. Thus, 3 times m is written as 3m and a times b is written ab. Recall that the product of two constants is not written in this manner, but is expressed by using the symbols of grouping that 6 x 4 is not written as 64 but were discussed earlier such as: as 6(4), or (6)(4).

3-1

Another customary practice in algebra is to use the raised dot to signify multiplication (6 * 4, a * b). Like the raised dot, there are additional terms you must become familiar with; let's look at them. 3102. Nomenclature

a. Algebraic expression. An algebraic expression is any combination of numerals, literal numbers, and symbols that represent some number. In its simplest form it is called a term. No part of a term is separated by addition or subtraction signs.

|ExamPles:

x and 173562ab3 c2 m2 p 4 xyY2 are both terms.

Note:

Expressions, as just illustrated, can take a variety of forms.

b. Monomial. An expression of only one term is called a monomial. It can appear in several forms and can have several parts of its own or it may be as simple as a literal number, n. By adding one numeral to this literal number, for example, 5n, you have added new terminology. Five and n are factors of some product which, until you replace n with a constant, can only be known as 5n. Five is known as the coefficient of n. The literal number n is actually a coefficient of 5 also, but it has become conventional to refer to the numerical part of the term as the coefficient.

|Example: In the term 176y3 , 176 is the coefficient of y3 .

This example brings up the need to understand another item, exponent. c. Exponent. This is the small raised number that shows the power of a number or how many times this number, called the base, is to be used as a factor.

3-2

Example:

y7 is y to the seventh power and indicates that y is to be a factor 7 times or yY-y-y-yyy.

Let's look at another example of a different form.

Examnle:

The term 3x4 indicates that the 3 is the coefficient which is a factor of the term. The exponent 4 signifies that x, not 3x, is to be raised to the fourth power or used as a factor 4 times. Broken down, the term would appear as 3>x x x x. Care must be taken so as not to associate the exponent with the wrong base.

Note:

If this example were written: (3x)4 , which at first glance may seem the same, the base would be 3x. The parentheses indicate that the entire quantity 3x is to be used as a factor 4 times as opposed to the single quantity x mentioned previously. This example broken down would appear as 3-xt3-x 3 x 3 x or, rearranging with the cummutative principle for multiplication, 33 3 3 x x x x. The main thing to remember here is to carefully observe the way the
equation is written prior to solving it.

Now that you have a basic knowledge of certain key terms, let's look at how likeness affects these terms. 3103. Property of Likeness

Earlier in the course, the property of likeness was used to perform operations with numbers. This same property applies to terms. As you saw previously, terms can take many different forms and in order for them to be combined they must be the same.

Example:

15ab and 12765ab are similar. The same is true with 17abcdx3 and abcdx3 , 2y4 and 7y 4 , C2 and 25c2 . Two terms such as 8x and 3ab or x and x2 are unlike and cannot be combined or simplified.

3-3

Let's now take a look at how the property of likeness is affected by addition, subtraction, multiplication, and division. a. Addition and subtraction. Remember, every part of the literal portion of the number must be identical for the terms to be similar.

Example:

2 2 2 lOab c - 4ab c 6ab c. If the expression contains like and unlike terms, only the like
ones are combined: 8ab + 4x
-

2ab = 6ab + 4x.

b. Multiplication and division. Whereas adding and subtracting can be done only with like terms, any two terms can be multiplied or divided. We will confine ourselves, however, to like terms. The important difference with multiplication and division is that the letters (variables) are affected as well as the numbers (constants). Note: Two important things to remember: When multiplying like terms, you add the exponents. When dividing like terms, you will have no letters (variables) in the quotient.

Example:

Multiply 7ac times 4ac, Each part of the two terms is multiglied 4 7, aa, and c-c. The
product is 28a c2 .

Let's look at a different example using division.

Exaimple:

Divide lob2

by 5b 2

10b 2 = 2 5b2 Here we divide 10 by 5 and b2 by b2 . With like terms, division of the letters (variables) will always be 1, therefore the answer is simply the quotient of the coefficients (2).

3-4

3104.

Polynomials

There are two types of polynomials that are very common, but before we discuss these, let's review. Do you remember what the algebraic expression of only one term is called? If you said monomial, you are correct. An expression with only one term is called a monomial. Mono means one, therefore it's one term.

Examples:

9m,

-6y3 ,

and

Poly means more than one, so algebraic expressions with more than one term are called polymonials. The two most common polynomial's are binomial and trinomial. Let's take a look at them. a. Binomial. binomial. A polynomial with exactly two terms is called a

Examples:

-9x4

+ 9x3 ,

8m 2 + 6m,

3m5 - 9m 2 .

b. Trinomial. trinomial.

A polynomial with exactly three terms is called a

Examples:

9m3 - 4M2 + 6,

19y2 + By + 5,

-3m5

9m 2 + 2.

Note: The determining factor of how many terms there are in an expression is the number of addition (+) and subtraction (-) symbols. Notice that the binomial example has two terms and only one addition/subtraction sign while the trinomial example has three terms and two addition/subtraction signs. Now that you have a better understanding of the terms in algebraic expressions, let's apply the proper operations to evaluate them.

3-5

3105.

Evaluation

Any algebraic expression can be evaluated to find out what number it represents if the values of the variables are known. It is a matter of arithmetic and following the rules of order of operations that were presented earlier. Your ability to evaluate expressions will be an aid to you in solving formulas and checking equations. The best way to evaluate expressions is to use these three steps: first, write the expression, second, substitute the given values, then do the arithmetic.

Example:

If p - 3, evaluate 4p2 4p2 _ 4(3)2 4(9) = 36 (Note that the 3 is placed in parentheses to indicate multiplication by 4, and that the 3 was squared before being multiplied by 4.)

Let's look at two more examples of a different form.

Example 1:

If m (8m)2

2, evaluate (Sm)2 (8 * 2)2 = (16)2 256

(Note that the entire quantity 8m is to be squared.)
Example 2: If x = 3, evaluate 2x2 + 4x + 5 2x2
+ 4x + 5 =

2(3)2 + 4(3) + 5

2(9) + 12 + 5 Note:

Although some of the steps can be combined and others done mentally, it is recommended that you write each step out and not take any short cuts. The most important thing to remember in evaluation is to follow the three steps: write the expression, substitute the given values, and then do the arithmetic.

Note;:

Lesson Summary. During this lesson on algebraic expressions, you learned about general information, nomenclature, property of likeness, polynomials, and how to evaluate algebraic expressions. In your next lesson, you are going to combine these algebraic expressions to form algebraic sentences which will help you later in solving equations. 3-6

Lesson 2.

ALGEBRAIC SENTENCES

LEARNING OBJECTIVES 1. 2. 3. 3201. Given a number of equations, evaluate if they are true or false. Given a number of equations, use the proper operations to find the correct root. Given a number of inequalities, evaluate if they are true or false. General Information

a. Definition. An algebraic sentence is a mathematical statement composed of algebraic expressions and one of these symbols: = (equals), t (is not equal to), > (is greater than), < (is less than), > (is greater than or equal to), and < (is less than or equal to). Any sentence that uses the equal sign is an equation. Sentences that use the other signs are called inequalities. b. Simple equations. As implied by the statement above, an equation is a statement that two thinqs are equal. It may or may not be true.

Example:

4 + 5 = 9 and 6 + 7 = 10 + 2 are equations. The first is true because 4 + 5 and 9 are symbols for the same number, but the second is false since 6 + 7 and 10 + 2 do not describe the same number.

:Remember, statements such as 4 + 5 - 9 and 6 + 7 10 + 2, whether true or false, are called equations. Everything to the left of the equal sign is called the left member of the equation; everything to the right of the equal sign is the right member of the equation. The solution or answer to an equation is called the "root" of the equation.

Before you examine equations containing variables, let's use some arithmetic ability to determine whether some equations are true or false.

3-7

Examples:

Are the equations below true or false? Eauation #1 9
-

(1 + 8) =

(9 -

1) + 8

9 -9 = 8 + 8

Obviously 0 does not equal 16.

(False)

Equation #2

4(9
4(4)

5) 30
5

18 + 12 = 4 + 3 4 + 6

16- 6 = 10
10
=

(True)

Now that you have determined if an equation is true or false, let's learn how to find the correct root. 3202. Root

Do you remember what root means? That's right, a root is the solution or answer to an equation. You know how to solve an equation such as x + 7 = 13. It can be done by inspection, that is, simply by examining or instinctively knowing that the number to be added to 7 to obtain 13 is 6. This method can be used to solve many simple equations. Of course you realize that all equations cannot be solved by inspection; there must be a mathematical method. Let's look at these methods starting with addition and subtraction. a. Addition and subtraction properties of equality. Can you remember that in study unit 1, some of the properties of numbers were discussed? By taking the properties of equality that were discussed and adding the addition and subtraction properties of equality, you obtain a process for solving equations. Let's look at an illustration.

3-8

Example:

Two Marines receive equal basic pay: Pfo. Hard
=

$845.10 and Pfc. Charger

$845.10.

Each one gets promoted to Lcpl.; a pay raise of $33.00: Lcpl. Hard $845.10 + $33.00 $845.10 + $33.00. $878.10

Lcpl. Charger =

The new basic pay for both Marines Note:

The basic pay has changed, but, Lcpl. Hard's pay is still equal to Lcpl. Charger's pay.

Note:

This example of the addition property of equality shows that if the same number is added to equal numbers, the sums are equal. Symbolically, if a = b, then a + c = b + c. You should see that the subtraction property of equality could be proved in the same way. Symbolically, if a = b, then a - c = b - c. Let's see how these properties can be used to solve equations.

Look at these expressions: 100 - 7, n - 6, 48 + 12, and x + 4. Suppose you wanted to perform some operation that will preserve the quantity on the left. 100 - 7 _ = 100, n - 6 _ = n, 48 + 12 = 48, and x + 4 = x. What can be inserted in the blanks to produce the results on the right? In the first one, if 7 is subtracted from 100, what must be done to return to 100? The answer is to perform the opposite or inverse operation with the same number.

lO0 - 7 + 7 = 100

What must be done to the example n must do the opposite or add 6.

6 to preserve the n?

You

Example:

n - 6,+
3-9

The other equations would look like this:

Examprles:
48 + 12
-

x + 4 -4 =X

You need one more item to enable you to solve equations using the addition and subtraction properties of equality. Do you remember the Golden Rule of Fractions? There is a similar one for equations. It states that whatever operation is performed on one member of an equation must also be performed on the other member. Let's use this principle to solve an equation.

Example:
x 6 - 19

By first using the addition property, you can eliminate the 6 from the left member and leave the x by itself. This is the objective in solving any equation: to isolate the variable in one member of the equation. The left member of the equation looks like this:

Example:
x - 6 + 6-

Since 6 was added to the left member, it must also be added to the right member.

Examole: x - 6 + 6 = 19 + 6 x = 25

3-10

This was a simple equation, one that could actually be solved by inspection; consequently, you probably knew that the answer was 25. Let's now check the answer to be sure that it is correct. The check is part of solving an equation. All you do is substitute your answer (25) for the variable (x) in the original equation.

Example: Check: x - 6 25 6
19

19 19
19

The left member equals the right member making 25 the correct "root" for the equation. Let's look at some other equations. Examples: y + 17
y + 17
-

45
17 = 45 17

y - 28 Check: y + 17
28 +

17 - 45

45 - 45

.8 = 1.1 .8 + 1.9 .8 = 1.1 + .8

x
x

Check:
1.9 1.1 .8 = 1.1

= 1.1

3-11

In the next equation, the variable is on the right, but it is solved in the same manner.

Example:
314 - a + 165
314
-

165 - a + 165 -

165

149 Check: 314

149

165

314 = 314

After solving several equations of this type you may feel that it is not necessary to write in the step involving the addition and subtraction property. Fine, this can be done mentally. However, it is suggested that if you are new to solving equations that you include this step until you have consistently solved each equation correctly and have confidence in your ability to add and subtract "in your head." Let's now look at multiplication and division. b. Multiplication and division properties of equality. The properties are quite simple and to the point. The multiplicative property states that if two equal numbers are multiplied by the same number, the products are equal. Symbolically, if a = b, then ac = bc. The division property states that if two equal numbers are divided by the same number, the quotients will be equal. Symbolically, if a = b, then a/c = b/c. Just as addition and subtraction are inverse operations, so are multiplication and division. One operation can undo the other. The Golden Rule also applies to solving equations involving the multiplication and division properties of equality.

3-12

Example: 6x = 84 6x 6 84 6

x = 14 Check: 6(14) = 84 84 = 84

In the example above, you are looking for some number that, when multiplied by 6, will give a product of 84. Remember that the objective in solving an equation is to isolate the variable, in this case "x". From practice with addition and subtraction, you know that in each case the inverse operation was applied. Here you have multiplication, so the inverse operation of division is applied. Let's look at a few other equations. Examples: 13m = 169 13m - 169 13 13 m = 13 1.5n 3 Check: 1.5(2) = 3 3 n= 2 .7 .7m Check: .7 .7 m= 1 3 Check: 13(13) - 169 169 = 169

1.5n = 3 1.5 1.5

.7 = .7m .7 .7

.7(1)
.7

3-13

In an equation involving division, the multiplication property will be used to find the root. For example, in n/4 = 6, you are looking for some number "n" that, when divided by 4, equals 6.

Example: n 4 4 n 4 n 6 6 24 4 Check: n 4 4 6 6 6

24 = 6

Note:

Since n is divided by 4, multiply it by 4 to get 1 as the coefficient of n. Let's look at a couple more examples. Examples:
c = 6 check:

19 19 19 = 6 * 19
= 114

114
19

6=6
6

a 5 a 5S

Check: 4 = .8 *5 .8 .8 .8

a =4

Note:

As with addition and subtraction, the intermediate step can be mental. You are encouraged to include it until you have confidence in yourself and consistently get correct answers.

3-14

c. Equations involvinq the four operative properties of equality. Often equations may involve a combination of properties such as addition and multiplication, addition and The principles of division, subtraction and multiplication, etc. solving the equations will remain the same; only the number of steps will change. The sequence in which to proceed is based on the principle of isolating the variable.

Example:
8y + 4 = 52

Here you have a multiplication (8y) and an addition (8y + 4). isolate the variable (y), the first step is to eliminate the 4. ;xale: By + 4 8y 8y
iy 8

Check:. 52 52 -4 8y + 4 52

4 -4 48
48

8(6) + 4 - 52

48 + 4
52 - 52

52.

8 y6

Let's look at a few more equations before we move on.

iExample: 9a
-

9 = 27

Check: 9a -9 =27 9(4)
-

9a - 9 + 9 =27 + 9 .9a 9
a
4

9 = 27

ga =3

=27

3-15

gZxamP1e 2p
-

8.5

Check;

; *

Z~~p
2p * .~2 p =

-5 + 5
13.5 2

8.5

-5

-8.5
- 5 5 8.5 8.5

2(6.75) 13.50 -

8.5 = 8.5
6

" xampli: 7n - 3n - 40 + 16 4n 56 Check: 7(14)
-

3(14) 56

40 + 60

AN w 2i 4 4

98 -42 56 56

n = 14

Inequalities. At the beginning of this lesson on algebraic d. sentences, you were given a group of symbols that denote in inequalities. Years ago, inequalities were discussed only school classes. college level math courses and sometimes in high school But now, inequalities are introduced at the elementary between numbers. level. Why? To show the students comparisons 5 An elementary school child can readily interpret the statement 5 this means the number > 3 as 5 is greater than 3 and knows that He also associates these is larger or more than the number 3. that 5 is to the right of 3. numbers on the number line and knows Example:
lI 0 1 2
*

l 4

l 6

l 7

to problems At a more advanced level, inequalities may be applied quality control, and in economics, military tactics, industrial research in general.

3-16

Problems concerning the most profitable division of time on TV programs, the allocation of production facilities in a factory to achieve maximum profit, or the determination of feed mixtures to minimize costs while satisfying standard food requirements are situations that represent a few of the areas to which the mathematics of inequalities can be applied. Most of this is beyond the realm of Math For Marines, but some of the aspects of inequalities are practical and may help you appreciate number relationships. Recall that an inequality is a statement that two quantities or expressions are not equal. Each expression is called a member, and one member may be < (less than), > (greater than), or t (not equal to) the other member. Like an equation, an inequality may or may not be true.

Example: 12 = 8 + (5-3)
12 = 8 + (5-3) 12 = 8 + 2
12 = 10

Is this true or false?

This is false, 12 is not equal to 10.

Let's look at another example.

Example:
11 < 8 + 2

This is read 11 is less than 8 + 2?
11 < 8 + 2 11 < 10

This is false, 11 is not less than 10.

The next step is to extend these concepts to expressions that include variables. What does x > 3 mean? If you are talking about the "positive" whole numbers, this means all of the numbers to the right of 3 on the number line. Note that there is an asterisk above the 3 to show that it is not part of the solution. 3-17

Example:

However, if you are talking about the natural numbers, it would be the "counting" numbers to the right of 3. Example:

Do you remember the difference between whole and natural numbers? If you said zero (0) and negative numbers, your correct. Natural numbers do not include zero and negative numbers. That's the major difference between the two number lines. However, some inequalities are little more complicated and involve the same procedures used to solve equations.

Example: x + 2<5 x + 2 -2 < 5 -2 x < 3

The graph would look like this for the "positive" whole numbers. Example:

O Note:

The double asterisk above the 3 this time is there to show that 3 is part of the solution. This means that any number less than or equal to three is an answer to the inequality. In checking it, 3 or any of the numbers less than 3 can be substituted for the variable.

3-18

Let's look at a few more examples and determine if the inequalities are true or false.

Example: x + 2 < 5 (Substitute 2 9/10 for x)

2 9/10 + 2 < 5 4 9/10 < 5 (This is true; 4 9/10 < 5)

Examples:
2x > 4 2x > 4

x > 2 (This is true, x > 2)
3x + 4 > 13 3x + 4 3x > 9 3x > 9 4 > 13 4

x > 3 (This is true, x > 3)

If you are not sure how to determine whether inequalities are true or false review this section before moving forward. Lesson Summary. During this lesson on algebraic sentences, you have learned how to determine if an equation is true or false, the proper operations to find the correct root, and how to determine if an inequality is true or false. Now that you know the basics, let's learn how to solve problems involving proportions and percentages. 3-19

Lesson 3.

PROPORTION AND PERCENmci conferenceE

LEARNING OBJECTIVES 1. 2. 3301. Given problems involving proportions, solve using the operations for equations. Given problems involving percentages, solve using the operations for equations. General Information

Problems involving both proportion and percentage are readily solved with the use of equations. Although percentage has been discussed previously, it is a topic that has enough use to warrant its being covered again from a different aspect; that of solution by equation. Proportion is another type of equation that can be used to solve many practical problems. Let's take a look at both of these. 3302. Proportion

During the training exercise, a platoon of Marines were clearing obstacles blocking their field of fire, and they had one lone tree left to be removed. The tree had to be dropped toward the east, but there was one small problem; a pipeline was located in the same direction. Cpl Chainsaw (the NCO in charge) didn't know the height of the tree, so he could not determine if the tree would hit the pipeline when it dropped. He knew that the pipeline was 70 feet from the tree. Cpl Chainsaw stared at the shadows cast by members of the platoon. He then whipped out his tape measure and began taking measurements of the Marines and their shadows. Lcpl Handsaw had a shadow length of 48" and his actual height was 72", Pfc Wedge had a shadow length of 42" and his actual height was 63", and Pvt Axe was holding a splitting maul which had a shadow length of 20" and its actual height was 30". Cpl Chainsaw double checked the height of the three objects and the length of the shadows he had measured and then scratched the following information in the dirt: Object Lcpl Handsaw Pfc Wedge Splitting maul Tree Shadow 48 42 20 40 lenqth in. in. in. ft. Height 72 in. 63 in. 30 in. ?

Cpl Chainsaw thought for several minutes and then announced that the tree was 60 ft tall. Was he correct? Let's see how Cpl Chainsaw thought out his answer. First, he noticed that when he compared the shadow lengths to the actual heights, they were all proportional.

3-20

Lcpl Handsaw = 48 = 2

Pfc Wedge = 42 = 2

Splitting = 20 =

maul

This 2/3 is called a ratio. It is the comparison of two things by division, and is read 2 to 3, the fraction line being "to." An older, but still used form is 2:3 with the colon being read "to." Ratios are reduced to lowest terms as indicated by all of the above being reduced to 2/3. Cpl. Chainsaw assumed correctly that the principle would hold true for all objects measured at the same time and place. He then had the problem:
40 = 2.

He needed to find a fraction with a numerator of 40 that is equivalent to 2/3. Do you remember how we solved this back in study unit 2? If you said "inverse division," you're correct. Let's take a look and find out:

Examtnle: 40 ? 2? 2? 2
?=
_

2 3 40 x 3 = 2? - 120 (Cross multiply)
120 (Divide)

2
60

Yes, Cpl Chainsaw was correct. The correct answer is 60 ft. Lcpl Handsaw is now free to cut down the tree without hitting the pipeline because there is a distance of 10 feet to spare. Referring back to our example and the definition of ratio, you see that a ratio compares two numbers. In the example, we had two sets of numbers, one set from the lengths of shadows and one set from the heights. All of the ratios were equal to one another. They are said to be proportional. A proportion is a statement of equality of two ratios.
Thus:

- 32

462

48 = 2,

and 40 = 2

are all proportions.

3-21

The equal sign is read "as." The first one then is read 20 is to Another way of writing a proportion is with a 30 as 2 is to 3. double colon for the "as." 20:30::2:3. Let's now examine an important characteristic of proportions and how we can use the equation-solving techniques learned earlier to solve proportions.
2 3
_

6 9,

3 4

24 32,

5 6

15
18,

3 = 8

375 1000

In each of the above proportions, as in all proportions, there is an important relationship that exists between the two ratios.

-Example:
2 2 9 3 6, 18
=

18,

3
3

9
24, 3 32 4 24, 96 = 96,

4 5

32 15, 5 18
3

15, 90 = 90,
' 375, 3000 3000.

6
3

18
375
'

1000 = 8

1000

If a proportion is true, the products of the numerator of one fraction and the denominator of the other fraction will be equal. This then can be useful in solving more difficult proportions where the outcome is not so obvious. For example, suppose Cpl. Chainsaw measured the shadow of another tree and found it to be 21 ft. How tall is the tree? You have then the length of the shadow (21) is to the height of the tree (x) as 2 is to 3.

21 =2 3 X

From the characteristic just explained, you know the products of the numerator of one fraction and the denominator of the other will be equal.

3-22

Example: x 3

2x = 21(3)

2x = 63

Now you have an equation of the form solved earlier in the study unit.

Example: 2x 2x 2 63 63 2

x = 31 1/2 The height of the tree is 31 1/2 ft.

Let's look at another example of solving proportions.

The truck convoy took 4 hours to travel the first 75 miles of the trip. Assuming the average speed is the same for the remaining 60 miles, how much longer will 'it take to reach the objective? Actually the proportion can be set up in. several ways. If we say that the first ratio is 4 hours to 75 miles, 4/75, then the other would be unknown time, x, to 60 miles, x/60.

3-23

Example--cont'd: 4 75
75x
=

x 60
240

75x = 240

75 x Note:

3.2 hours or 3 hours and 12 minutes.

12. Remember that .2 = 1/5, so 1/5 (60) You could do it another way by setting up ratios of the same units. The ratio of unknown time to known time is x/4. The ratio of distance for unknown time to distance for known time is 60/75.
x 4 75x
=

60 * 75 ** 240

3.2

* Denotes distance at unknown time. ** Denotes distance at known time.

If you had a problem solving these examples, go back and review this section before moving on to percentages. 3303. Percentage

Earlier in the course you were introduced to percentages and you were given the methods to use for solving various percentage problems. Now that the principles of equation solving have been discussed, you will see how they can be applied to percentages. There are three types of percentage problems, sometimes referred to as cases involving: finding a percent of a number, finding what percent one number is of another, and finding a number when the percent of the number is known. It is a relatively simple matter to transform a sentence describing one of the percentage cases into an equation. The two key words in these sentences are "of" and "is." "Of" can be thought of as indicating multiplication and "is" can be replaced by an equal sign.

3-24

a. Findinq a percent of a number. This is probably the most used of the three cases. A typical sentence would be: What is 35 percent of 14? Now, let's take this and make an equation out of it. "What" is the unknown (x), "is" will be replaced by the equal sign, the 35 percent is changed to a decimal, and "of" is replaced by a multiplication symbol.

Examples: What is 35 percent of 14?
x =

.35(14)

x = 4.9 Thus, 35 percent of 14 is 4.9. What is 86 percent of 741?
x = .86(741)

x = 637.26

Thus, 86 percent of 741 is 637.26.

Now that you know how to find a percent of a number, let's find out what percent one number is of another. b. Finding what Percent one number is of another. A typical sentence would be: 20 is what percent of 80? Substitution of the symbols of the equation will be made in the same manner.

Example: Twenty is what percent of 80? 20 - x%(80) Now the equation says that 20 is equal to some unknown percent times 80. One slight adjustment to the equation will put it in a more familiar form. Using the commutative law for multiplication, change the positions of x percent and 80. 3-25

Example--cont'd: 20 - BOx (drop percent sign) 20 a0
.

BOx 80
*

~~1
4

.25= x * 25% x

* These steps are shown but are not

necessary. Check: 20 = 25%(80) 20 = .25(80) 20 = 20

Let's try another example. Forty-five Marines in the company are on the rifle range. There are one-hundred twenty Marines in the company. What percent of the company is on the range?

Example: Forty-five is what percent of 120? 45 = X%(120) 45 = 120X 45 120 .375 37.5% x x x

3-26

Check; 45 = 37.5%(120) 45 = .375(120) 45 45

Now that you know how to find what percent one number is of another, let's learn how to find a number when a percent of the number is known. Finding a number when a percent of the number is known. A c. Forty is 2 percent of what number? typical sentence would be:

Example: 40 = 2%(x) 40 = .02x 40 = .02x .02 .02 2000 = x Check: 40 = 2%(2000) 40 = .02(2000) 40 = 40

Let's take a look at another example. If 120 Marines (80%) in the company are deployed, what is the total number of Marines in the company?

3-27

1xamRle:

120 is 80 percent of what number?
120 = .80x 12_0 x

.80 150 =x Check.
120 = 80%(150}

120 - .80(150)
120 - 120

What is the best way to simplify proportion and percentage word problems? If you said, "Write the statement in words and then substitute the words for mathematical symbols," you're correct. As you have seen from this study unit, writing and solving equations can greatly simplify the solution of percentage problems. Lesson Summary. During this lesson you learned how to solve problems involving proportions and percentages using the operations for equations. It's now time to advance forward and attack the unit exercise.

Unit Exercise:

Complete items 1 through 28 by performing the action required. Check your responses against those listed at the end of this study unit.

Which algebraic expression is a monomial term? a. b.
C.

3x + y + x + y lx (xz)3

OOa
45x3 y

35a

3-28

Which algebraic expression contains like terms? a. b.
c.

135abc , 2

45abc

(xyz)3, xy(z)3 5a , (5a)

d. 3.

16xy, lxy

Which algebraic expression is a polynomial term?
a. 2x lx -

x + 4z 3
y

b. c. d.

37xv 6x + z
xz

46abc

Complete items 4 through 10 by performing the proper operations of evaluation.
4. 8r ,
2

(r = 5)

a. b. c. d. 5.

80 100 200 1600
(y = 5)

(8y) 2, a. b.
c.

1600 200
100

d. 6.

80 (a =5)

a3 - 2a2 + a + 4, a. b. c. d. 584 534 84 29 (x=

x2 +z 2
y

5, y =2, z =3)

a. b. c. d.

4 7 3 1/2 8 1/2

3-29

How many pounds of explosives will you need to cut four 40" diameter trees? Use the algebraic expression: p = D2 750
P = Pounds of explosives

D = Diameter of tree in inches
250 = constant

a. b. c. d. 9.

6.4 pounds 4.6 pounds 26.5 pounds 25.6 pounds

The objective is located 16 1/4 miles (distance) away, and you are moving at a rate of 5 mph. How long will it take you to arrive at the objective? Use the algebraic expression: T = D R
T = Time D = Distance R = Rate

a. b. c. d.

3.25 3.52 2.35 2.53

hours hours hours hours

10. You are in a defensive position with a front of 180 meters. Your platoon is responsible for the tactical wire emplacement of three belts. What is the total length of tactical wire (TAC(W)) required? Use the algebraic expression: TAC(W) = 1.25 x length of front(LOF) x number of belts

(# Belts).
a. b. c. d. 675 657 252 225 meters meters meters meters

Complete items 11 through 13 by determining if the equations are true or false.
11. 3 + 7 a. b. c. d. 10 10 21 21
= = = = =

5 x 2 10 (true) 7 (false) 10 (true) 7 (false)

3-30

12. 4 x 6 1/2 = 6 x 4 1/2
a. b. c. d. 13. 27 = 26 = 26.5 24.5 26 (false) 27 (false) = 26.5 (true) = 24.5 (true) 5 + 11 = 4
-

72 _ 8 a. b. c. d.

82 + 52

41 = 6 (false) 6 = 41 (false) 25 = 20 (true) 20 = 25 (false)

Complete items 14 through 16 by selecting the correct root.
14. 6x 3
=

a. b. c. d.
15.

x= x= x= x=

7 6 5 4

7h + 7.79 = 29

a. b. c. d.

h h h h

= = = =

4.30 4.03 3.30 3.03
=

16. 8b + 2b + 7b + 187

221

a. b. c. d.

b = b = b= b=

3.5 2.5 3 2

Complete items 17 through 19 by determining whether these inequalities are true or false.
17. 3 + 6 < 8 a. b. c. d. 9 9 9 9 < > < > 4 4 4 4 4

(true) (true) (false) (false)

3-31

18.

4(9

8 +

2 t

+ 3

a. b. c. d.
19. 6

10 12 14 10

+ p + p

12 (true) 10 (true) 10 (true) 14 (true)

12 + 8 > 2 * 2

5 a. b.
C.

24 > 4 (true) 24 < 4 (true)
48 > 8 (true)

48 < 8 (true)

Complete items 20 through 28 by performing the action required. 20. The combat engineers reported that the mountain road has a slope of 6 percent (6 feet of rise for every 100 feet of length). How much will the road rise in 1 mile? a. b. c. d. 528 feet per mile 52.8 feet per mile 31.68 feet per mile 316.8 feet per mile

21. You are in a defensive position and have been given 20 40mm rounds (M203 grenade launcher) for every 300 feet of front. How many rounds will you receive for every 450 feet of front? a. b. c. d. 30 29 28 27

22. A total of 135 Marines crossed the stream in 40 minutes. How many Marines will cross the stream in 60 minutes? Note: Round to nearest whole number. a. b.
C.

204 203
202

201

3-32

23. There are 950 Marines in the battalion; 2 1/2 percent are at sickbay. How many Marines are at sickbay? Note: Round to nearest whole number. a. b.
C.

22 23
24

24. The Expeditionary Force has a total of 3248 Marines. It has just been increased by 50 percent. How many Marines make up the new Expeditionary Force? a. b. c. d. 4287 4672 4782 4872

25. Seven Marines out of 12 were wounded while on patrol. What percent was wounded? a. b. c. d. 17.14% 17.33% 58.14% 58.33%

26. If 4.5 tons is 25 percent the breaking strength of the wire rope used for retrieving the HMMWV (High Mobility Multi-Purpose Wheeled Vehicle) from the mud, what is the total (100%) breaking strength of the wire rope? a. b. c. d. 18 19 20 21 tons tons tons tons

27. Sixteen of the 48 rifles inspected had unserviceable stocks. What percent of the rifles had unserviceable stocks? a. b. c. d. 30.33% 33.33% 36.33% 39.33%

28. Eighty Marines (40%) of the range detail qualified expert on the rifle range. What was the total number of Marines on the detail? a. b. c. d. 500 400 300 200

3-33

UNIT SUMMARY In this study unit you were introduced to algebra. You identified a monomial term, like term and polynomial term in an algebraic expression. You applied the proper operations to evaluate an algebraic expression and determined if an equation or an inequality was true or false. When given equations, you used the proper operations to find the correct root and you solved equations involving proportion and percentage. Exercise Solutions Reference 1. 2. 3. 4. d. d. a. c. 45x 3 y 16xy, lxy 2x - x + 4z3 8r 2 gr=5) 8(5) 8(25) 200 (8y)2 ( (8 . 5)
(40)2 =

3102 3103 3104 3105

3105

1600 6.
c. a3 53
-

2a2 + a + 4 (a = 5) 2(5)2 + 5 + 4
2(25) + 5 + 4 50 + 5 + 4

3105

125 125 84 7. d.

x2x z2(x y 52+ 32

3105

2
25 + 9 4 34 4

8 1/2

3-34

dD p d* P
p p

3105

250
402

250
1600

250 P = 6.4 lbs per tree multiplied by 4 trees 25.6 lbs of explosives P 9. a. T D R 3105

T = 16 1/4 5
T

5 T = 3.25 hours or 3 hours and 15 minutes Remember .25 = 1/4 so 1/4 x 60 (60 minutes per hour) is 15 minutes. 10. a. Length of TAC(W) = 1.25 x LOF x # belts
TAC(W) = 1.25 x 180 x TAC(W) = 225 x 3 TAC(W) = 675 Ieters 3

16.25

3105

11. a.
12. b.

3 + 7 = 5 x 2
10 = 10 (true) 4 x 6 1 = 6 x 4 1

3201
3201

2
26 = 13. d. 72
49
-

2
(false) +
+

27 8.5
40

11 = 43 - 82 + 52
64 64 + 25

3201

11 =

9 + 11 = 0 + 25 20 = 25 (false)
14. c. 6x 6x 6x = 6x =

3 + 30 30

27 3 =

3202 27 + 3

6 6 x= 5 15. d. 7h + 7.79 = 29
7h + 7h = 7.79 21.21
-

3202
29 7.79

7.79 =

21.21

7
h
=

7
3.03

3-35

16. d.

8b + 2b + 7b + 187 = 221
17b + 187 17b +187 17b = 34 17b = 34
=

3202

221 187 = 221 -

187

17 17 b = 2
17. c. 3 + 6 < 8 - 4 9 < 4 (false) 4(9 5)
-

3202

18. C.

8 + 2

4 + 3

3202

5
4(4) - 2 t 4 + 6 16 - 2 t 10

14 t
19. a. 6

10 (true)
12 + 8 > 2 2 3202

5
6 4 > 2 *2 24 > 4 (true)

20. d.

6 100

x 5280

3302

lOOx = 31680 1OOx = 31680 100 100 x = 316.8 = The road rises 316.8'

in 1 mile.

21. a.

20 300
300x = 300x =

x 450
9000 9000

3302

300
22. b. 135 = x

300
3302

x = 30 = 30 rounds

40 40
23. c.

60 40
3303
=

40x = 8100 40x = 8100 x = 202.50 = 203 x = .025(950) x = 23.75 2 1/2% of 950 = 23.75

24 Marines

2 1/2% or 2.5% is written as .025. To convert from percent to decimal, move two places to the left.

3-36

24.

x = 1.5(3248)
x = 4872

3303

150% of 3248 is 4872 Marines
25. d. 7 = x%(12) 7 = 12x 7 = 12x 3303

12 26. a.

12 3303

58.33% = x

4.50 = 25%(x)
4.50 = 4.50 = .25x .25x

.25
18 tons

.25
= x

4.50 is 25% of 18
27. b. 16 = x%(48) 16 = 48x 16 = 48x 3303

48
28. d.

48
3303

33.33% = x 80 = 40%(x) 80 = .40x 80 = .40x

.40 x 80 is 40% of 200
200
=

.40

3-37

STUDY UNIT 4 UNITS OF MEASUREMENT Measurement came about primarily because of Introduction. the need for surveying real estate and for measuring quantities in the market place. One of the early problems in measurement is one that still plagues us today. This is the choice of units of measurement. The ancient cultures used body measurements to express various lengths and widths, and a cup, bucket, rock, or piece of iron to express weights or volume. The one major drawback with these choices was standardization. Measures such as the width of the palm (the hand), the distance between the tip of the thumb and the tip of the little finger of a spread hand (the span), the distance from the tip of the elbow to the tip of the longest finger (the cubit), and the distance between the finger tips of each hand when the arms are outstretched (the fathom) were used. As you can see, the quantities would vary with the size of the merchant. We have advanced greatly since ancient cultures; we now have the means to measure much more accurately. In the United States we use both the metric and English (U.S.) systems of measurement. You will be studying both of these in this study unit, and, as a result, you will gain the knowledge and skills needed to convert and compute various units of measurement. Lesson 1. METRIC SYSTEM

LEARNING OBJECTIVES 1. Given a series of situations involving metric linear units, use the decimal system or powers of ten to convert metric units to metric units. Given arithmetical problems involving various linear units, using the decimal system, convert by computing metric units to metric units. Given a series of situations involving metric capacity units, use the decimal system or powers of ten to convert metric units to metric units. Given arithmetical problems involving various metric capacity units, using the decimal system, convert by computing metric units to metric units. Given a series of situations involving metric weight units, use the decimal system or powers of ten to convert metric units to metric units. Given arithmetical problems involving various metric weight units, using the decimal system, convert by computing metric units to metric units. 4-1

4101.

General

Because of the confusion with measuring systems that prevailed throughout the world, a conference of world scientists was called in France during the latter part of the eighteenth century. Based strictly on logic and science, this group developed the metric system. The basic unit of length is the meter which was originally one ten-millionth of the distance from the equator to the north pole. Liquid measure is based on the liter which was defined as the volume of a cube with an edge dimension of onetenth of a meter (a decimeter). The basic unit of weight is the gram which was defined as the weight of a cube of pure water at the temperature of melting ice, the cube being one-hundredth of a meter on an edge (a centimeter). Since 1799, the meter has been standardized as the length of a platinum bar sealed in a case in the Archives of State in France. The standard of weight has been changed to the kilogram (1000 grams) and is represented by a platinum weight which is also in the Archives. With these basics defined, the rest of the metric system is simply based on ratios of powers of ten. Let's first look at metric linear units. 4102. Metric Linear Units

a. Metric length measurement. The basic metric unit for length is the meter. As previously mentioned, the length of the meter was at first defined as one ten-millionth of the distance from the equator to the north pole. It was later found that there had been a slight error in determining this distance. At present, the length of the meter is a bar, called the international meter, which is made of 90 percent platinum and 10 percent iridium. It is preserved at a temperature of 0°C at the International Bureau of Weights and Measures near Paris, France. Two copies of this meter are in the United States, kept at the National institute of Standards and Technology, formerly the Bureau of Standards at One of these is used as the working standard Washington, D.C. and the other one for comparison. To insure still greater accuracy, these are compared at regular intervals with the international meter. b. Tables and terms used. In the English system of measurement there are many different terms and numbers which bear no logical relation to one another. In the metric system only a few terms Let's look at and but one single number. That number is 10. some of the most often used terms (table 4-1).

4-2

Table 4-1.
kilometer (km) = hectometer (hm) = dekameter (dkm) =

Metric Linear Units
1000 100 10 meters = 103 meters = 102 meters = 101

meter

(m)

decimeter (dm) centimeter (cm) millimeter (mm)

= = = =

meter (basic1 unit)
= 10 2 = 102 = 10 3

0.1 meter 0.01 meter 0.001 meter

You should try to memorize the terms in the table. The prefixes kilo, hecto, and deka are attached to the word meter when the measurement is larger than a meter. If the measurement is smaller than a meter, the prefixes deci, centi, and milli are attached to the word meter. The expressions decimeter, dekameter, and hectometer are rarely used in actual work; the value of decimeters is usually expressed in centimeters, the values of dekameters and hectometers in meters or kilometers. c. Operating within the metric system. The powers of ten ratios simplify working within the metric system. For example, a distance of 7 meters 8 decimeters 4 centimeters can be written decimally as 7.84 meters. This can be added to another distance such as 1 meter 2 decimeters 3 centimeters (1.23) without any conversion. Example: 7.84
+ 1.23

9.07 meters or 9 meters 0 decimeters 7 centimeters Any of the four arithmetic operations can be done with the metric system by using the rules for decimals. Converting one unit of measure to another (which is actually either multiplication or division) is done by moving the decimal points in the same manner as was explained in study unit 2 on powers of ten. This is a process that is needed regularly for changing map distances to ground distances and vice versa. Your main concern will be with centimeters, meters, and kilometers and the changing of one to another. Let's see how this is done. (1) To change centimeters to meters, move the decimal point two places to the left. (Note in the linear measure table that meters and centimeters are two places apart.) Example: 125,000 centimeters = 1,250 meters.

4-3

(2) To change meters to kilometers, move the decimal point (Note that meters and three places to the left. kilometers are three places apart in the table.) Example: 1,250 meters = 1.25 kilometers. (3) To change centimeters to kilometers, move the decimal point five places to the left. Example: 125,000 centimeters = 1.25 kilometers. (4) To change kilometers to meters, move the decimal point (Note that when changing from three places to the riQht. large units to smaller units, it may be necessary to add zeros.) Example: 1.25 kilometers = 1,250 meters. (5) To change meters to centimeters, move the decimal point two places to the riQht. Example: 1,250 meters = 125,000 centimeters. (6) To change kilometers to centimeters, move the decimal point five places to the right. Example: 1.25 kilometers = 125,000 centimeters. Let's take another look at an example of a distance and follow it through from kilometers to centimeters. Example: 11 kilometers = 110 hectometers*
= = = = * Not usually used. 1,100 dekameters* 11,000 meters 110,000 decimeters* 1,100,000 centimeters

Inserted to show the progression. 4-4

Let's look at another example, but work in the opposite direction. Example: 50,000 centimeters = 5000 decimeters
= 500 meters = 50 dekameters = 5 hectometers = .5 kilometers

Let's look at a couple of word problems. If the first leg of the forced march was 7 kilometers long, how many centimeters long was the first leg? If your answer was 700,000 centimeters, you're correct. Remember, to change kilometers to centimeters move the decimal point five places to the left. Let's try a harder problem. On a 1:25,000 map, 14.3 centimeters equals how many kilometers? Was your answer 3.575 kilometers? If not, let's see what you did wrong. One centimeter on the map equals 25,000 centimeters on the ground, so 14.3 centimeters on the map equals 25,000 x 14.3, which equals 357,500 centimeters on the ground or 3.575 kilometers. If you had trouble with these problems, review this section before moving to metric capacity units. 4103. Metric Capacity Units

a. Capacity table and terms. The basic metric unit for capacity (fluid) is the liter. The liter is determined by the measurement of a cubic decimeter. The prefixes kilo, hecto, and deka are attached to the word liter when the measurement is larger than a liter. The prefixes deci, centi, and milli are attached to the word liter if the measurement is smaller than the liter (table 4-2). Table 4-2. Metric Capacity Units 1000 liters = 103 100 liters = 102
10 liters = 10

kiloliter (kl) = hectoliter (hl) =
dekaliter (dal) =

liter
deciliter

(1)

1 liter (basic unit)
0.1 liter = 101

(dl) =

centiliter (cl) =

0.01 liter

= 10-2
=

milliliter (ml) = 0.001 liter

10-3

4-5

The expressions or terms kiloliter, dekaliter, deciliter, and centiliter are rarely used in actual day-to-day work. The value of kiloliter is usually expressed in hectoliters, the value of dekaliters in liters, the values of deciliters and centiliters in liters or milliliters. Our main concern will be with hectoliters, liters, and milliliters and the changing of one to another. b. Operatinct within the metric capacity units. The same operations used for metric linear units also apply when working with metric capacity units. Any of the four arithmetic operations can be done using the rules for decimals or the powers-of-ten ratios. Converting one unit to another unit is done by moving the decimal points. Let's see how this is done. (1) To change liters to milliliters, move the decimal point three places to the riqht. (Note in the capacity table that liters and milliliters are three places apart.) Example:
6 liters = 6,000 milliliters.

(2) To change liters to hectoliters, move the decimal point (Note that liters and two places to the left. hectoliters are two places apart in the table.) Example:
450 liters = 4.5 hectoliters.

(3) To change hectoliters to milliliters, move the decimal point five places to the right. Example: 1.25 hectoliters = 125,000 milliliters. (4) To change hectoliters to liters, move the decimal point two places to the riqht. Example:
3.6 hectoliters = 360 liters.

(5) To change milliliters to liters, move the decimal point three places to the left. Example:
750 milliliters = .75 liters.

4-6

(6) To change milliliters to hectoliters, move the decimal point five places to the left. Example: 3,550 milliliters = .0355 hectoliters. Let's take another example of capacity and follow it through from hectoliters to milliliters. Example: 25 hectoliters = 250 dekaliters*
= 2,500 liters = 25,000 deciliters*

= 250,000 centiliters*
= 2,500,000 milliliters

* Not usually used.

Inserted to show the progression.

Let's try a word problem. If the fuel tank holds 755 liters of fuel, how many hectoliters of fuel does it hold? If your answer was 7.55 hectoliters, you're correct. Remember, to change liters to hectoliters move the decimal point two places to the left. Let's try one more. An Ml tank, D7G bulldozer, and a 5 ton truck needed to have their fuel tanks topped off. The fuel bladder has 1892 liters of fuel remaining. The Ml tank needed 7 hectoliters of fuel, the D7G bulldozer needed 500,000 milliliters of fuel, and the 5 ton truck needed 300 liters of fuel. How many liters of fuel remain in the fuel bladder? If your answer was 392 liters, you're correct! If it wasn't, let's see what you did wrong. The Ml tank used 7 hectoliters or 700 liters, the D7G bulldozer used 500,000 milliliters or 500 liters, and the 5 ton truck used 300 liters. Thus, 700 + 500 + 300 equals 1,500 liters. There was 1,892 liters of fuel in the bladder. Subtract 1,500 from 1,892 and your remaining fuel is 392 liters. Note: Keep in mind the operations with powers of ten and try to associate them with the metric prefixes: kilo, hecto, centi, and etc. Remember: to change larger units to smaller units, move the decimal point to the right. To change smaller units to larger units, move the decimal point to the left. 4-7

If you had trouble with these problems, review this section before moving to metric weight units. 4104. Metric Weight Units

a. Definition. The basic metric unit for weight is the gram. The gram is defined as the weight of a cube of pure water at the temperature of melting ice. The cube is one-hundredth of a meter on each side. b. WeiQht table and terms. The prefixes discussed in paragraphs 4102 and 4103 also apply to the metric weight unit, the gram. When the measurement is larger than the gram, the prefixes kilo, hecto, and deka are attached. If the measurement is smaller than the gram, the prefixes deci, centi, and milli are attached to the unit gram. Let's look at some of the most widely used terms. Table 4-3. Tonne (metric ton) kilogram hectogram
dekagram

Metric Weight Units 1,000,000 grams = 106 1,000 grams = 103 100 grams = 10
10 grams = 10

(T) (kg) (hg)
(dag)

gram
decigram

(g)
(dg)

1 gram (basic unit)
0.1 grams = 101

centigram
milligram

(cg)
(mg) =

0.01 grams = 10-2
0.001 grams = 10-3

The term tonne (metric ton) or megagram is the expression used when weighing large or bulky amounts. For measures of weight, the terms gram, kilogram, and tonne are the only three usually employed in practice. The values of hectograms and dekagrams are normally expressed as kilograms. Decigrams, centigrams, and milligrams will usually be expressed in values of the gram. Our main concern will be with grams, kilograms, and tonnes (megagrams). Let's see how they operate. c. OperatinQ within the metric weight units. The powers of ten or rules for decimals also apply when working within metric weight units. Changing from one unit to another unit is as easy as moving the decimal point. If the unit is larger than the gram, move the decimal to the left. When the unit is smaller than the gram, move the decimal point to the right.

4-8

(1) To change grams to kilograms, move the decimal point three places to the left. Example: 9,754 grams = 9.754 kilograms. (2) To change kilograms to tonnes (megagrams), move the decimal point three places to the left. Example: 11,300 kilograms = 11.3 tonnes. (3) To change tonnes to kilograms, move the decimal point three places to the right. Example: 5.75 tonnes = 5,750 kilograms. Let's take a look at another example of a weight and follow it through from tonnes (megagrams) to grams. Example: .5 tonne = 500 kilograms
= 5,000 hectograms = 50,000 dekagrams = 500,000 grams

Here is another example working in the opposite direction. Example: 1,550,000 grams = 155,000 dekagrams
= 15,500 hectograms = 1,550 kilograms = 1.55 tonnes

Now let's try a couple of word problems. The payload of the M923 (5 ton truck) is 9,080 kilograms. How many grams will the M923 carry? If you said 9,080,000, you're correct. Remember, to change kilograms to grams, move the decimal point three places to the right.

4-9

The M923 has a payload of 8,554 kilograms and is towing a trailer with a payload of 136,200 hectograms. What is the total payload in tonnes? If your answer was 22.174 tonnes, you're correct. If not, let's see what you did wrong. The vehicle payload of 8,554 kilograms equals 8.554 tonnes. (Move the decimal point three places to the left.) The trailer payload of 136,200 hectograms equals 13.62 tonnes. (Move the decimal point four places to the left.) Add 8.554 to 13.62 for a total payload of 22.174 tonnes. If you had a difficult time with this area, review the section before moving on. Lesson Summary. In this lesson you gained the skills needed to convert and compute metric units (linear, capacity, and weight). You also saw how easy it is to convert metric units (linear, capacity, or weight) by simply using the powers of ten (movement of the decimal point). This (powers of ten) makes the metric system easy to use. In the next lesson you will study the U.S. system of measurement. It is actually a more difficult system to learn. Lesson 2. U.S. SYSTEM

LEARNING OBJECTIVES 1. Given a series of situations involving U.S. linear measurements, compute using like columns to line up the units of measure. Given situations with various capacity measurements, compute using like columns to line up the units of measure. Given situations with various weight measurements, compute using like columns to line up the units of measure. General

4201.

All of our present day units of measure have some historical background. The backgrounds of measures of capacity and weight vary and at times can be confusing. For example, there is a variance between the U.S. gallon and the British gallon, and there are even differences throughout the U.S. The International Bureau of Weights and Measures says that within the U.S. there are eight different weights for a ton, nine different volumes for a barrel, and a different bushel for charcoal, corn, and potatoes. Out of this chaotic situation, the International Bureau of Weights and Measures has tried to make some order by standardizing measures used throughout the U.S. Let's take a look them.

4-10

4202.

U.S. Linear Measure

a. Units of linear measure. Unlike the metric system where there are only a few terms and but one number to remember, the U.S. system of measuring length contains many terms and numbers. The numbers in the U.S. system bear no relation to one another and can cause a bit of confusion when trying to remember which number goes with which term. Some of the terms (table 4-4) are included just for information (the areas using the rod for measurement) while others are used often enough that they should be common knowledge or committed to memory. Table 4-4. Linear Measure Table
=

12 inches (in.) 36 inches

1 foot (ft)
1 yard (yd)

=
-

3 feet
16 1/2 feet 5 1/2 yards 63,360 inches 5,280 feet 1,760 yards

=
= = =

320 rods

1 1 1 1 1 1 1

yard rod rod mile mile mile mile

(yd) (rd) (rd) (mi) (mi) (mi) (mi)

b. Expressing and computing. You can see from the table that there are, in some cases, several ways of expressing a measurement. The yard is 3 feet or 36 inches; the mile is 5,280 feet, 1,760 yards, or 63,360 inches. Usually though, measurements do not come out evenly and must be expressed in fractional parts or in terms of a smaller unit. For example, 132 inches is 11 feet or 3 2/3 yards. Conversely, 3 2/3 yards can be expressed as 11 feet or 132 inches. The form of expressing any measurement is usually determined by the preference of the user except in cases where something must be ordered in standard packages or sizes or when a standard format is used. For example, when ordering a 2 x 4 (board) that is 8 feet long, you would not ask for one 2 2/3 yards long, nor would you give them the measurement in inches. Computations with these units can be simplified if you relate them to our decimal system. You know that when you add a column of numbers you add like columns: units, tens, hundreds, etc. This in fact is what you do when you add feet and inches. Example: The number (26) is also expressed as 2 tens and 6 units. Thirty-one inches is the same as 2 feet and 7 inches. Additionally 6000 feet can be expressed as 1 mile 240 yards or 1 mile 720 feet.
4-11

To compute with the various measures, then, it is necessary only to line up the proper units. Example: What is the total length of two boards if one is 3 feet 7 inches and the other is 4 feet 8 inches? 3 ft 7 in. 4 ft 8 in. 7 ft 15 in. Since 15 inches is greater than 1 foot (12 inches), the foot is added (or carried) to the foot column and the remainder of the 15 inches is expressed in inches. Example: 7 feet 15 inches
=

8 feet 3 inches

As you can see, by using like columns to line up the units, the computation becomes much easier. Let's try another problem. Example: How many miles were run if the first squad ran 2,520 yards, the second squad ran 2,760 yards, and the third squad ran 2,640 yards? Let's look at two different methods to solve this. Method 1: 2,520 2,760 2,640 7,920 yd yd yd yd

7,920 yards = 4 miles 880 yards or 4 1/2 miles. Method 2:
2,520 yd = 2,760 yd = 2,640 yd = 1 mi 760 yd 1 mi 1,000 yd 1 mi 880 yd

3 mi 2,640 yd 3 miles 2,640 yards = 4 miles 880 yards = 4 1/2 miles Although it is not shown, in both methods division has been used. For example, to change the 7,920 yards to miles, you divided 7,920 yards by 1,760. In method 2, you divided each distance by 1,760 and then added like columns. 4-12

In most cases it is best to change everything to the same unit. When multiplication is necessary to solve a problem, the various units can be changed (carried) as you go along, or they can be changed (carried) at the end. Example: You are preparing supplies for the upcoming patrol. Your squad leader has told you that there are 8 streams you must cross. All of them are 2 yards 2 feet 2 inches wide. You will cross the streams with the aid of a 3/4" fiber rope. The rope can be used only once. In other words, you will need 8 lengths of rope. What is the total length of the 8 pieces of rope? Note: Change (carry) as you multiply.
8 x 2 in. 8 x 2 ft = 8 x 2 yd
= =

16 in. = 1 ft 4 in. (carry 1 ft) 16 ft + 1 ft = 17 ft = 5 yd 2 ft

(carry 5 yd)
16 yd + 5 yd = 21 yd

The total rope length is 21 yards 2 feet 4 inches. If you had trouble with either of these problems, review this section before moving on to capacity measurements. 4203. U.S. Capacity Measure

a. Units of capacity measure. The capacity measure contains many terms and numbers. The term capacity measurement refers to many types of measurement such as liquid measure and dry measure. In this section you will be concerned with liquid measure. The terms in table 4-5 are common knowledge liquid measure. Table 4-5. Capacity Measure Table = 1 cup = 1 pint (pt)
= 1 pint

8 fluid ounces (oz) 16 fluid ounces
2 cups

32 fluid ounces
2 pints 4 quarts

= 1 quart (qt)
= 1 quart = 1 gallon (gal)

128 fluid ounces 31 1/2 gallons

= 1 gallon = 1 barrel

4-13

b. Expressing and computing. Capacity measurements can be expressed in many different ways. For example, the gallon is 4 quarts or 128 fluid ounces, and the quart is 32 fluid ounces or 2 pints. As previously mentioned, measurements do not always come out evenly and must be expressed in fractional parts or in terms of a smaller unit. Computations with liquid capacity measurements are made in the same manner as with linear measurements. Like units are placed in their proper columns. Example: There are two containers filled with diesel fuel. One has a capacity of 2 gallons 3 quarts and the other has 3 gallons 2 quarts. What is the total capacity of both containers? 2 gal 3 qt 3 gal 2 qt
5 gal 5 qt = 6 gal 1 qt

Note:

Since 5 quarts is greater than 1 gallon, the gallon is added (or carried) to the gallon column and the remainder of the 5 quarts is expressed in quarts. Consequently, the total capacity is 6 gallons 1 quart.

Let's take a look at another example problem and two different methods of solving it. Example: How many gallons of water were consumed if the first squad drank 20 pints, the second squad drank 18 pints, and the third squad drank 27 pints? Method 1: 20 18 27 65 pt pt pt pt

65 pt = 32 qt 1 pt 32 qt = 8 gal

65 pt = 8 gal lpt or 8 1/8 gal Method 2:
20 pt = 2 gal 2 qt 18 pt = 2 gal 1 qt

27 pt = 3 gal 1 qt 1 pt
7 gal 4 qt 1 pt = 8 gal 1 pt

4-14

Note:

The total consumed was 8 1/8 gallons (method 1) or 8 gallons 1 pint (method 2). Although it was not indicated, division was used in both methods. There are many problems that you can solve by using just the division step; however, other problems will require both multiplication and division. Let's take a look at one. Each Marine in the patrol drank 3 gallons 2 quarts 1 pint of water during training. There are 8 Marines in the patrol; how much water did the patrol drink? Note: Change (carry) as you multiply. 8 Marines x 3 gal 2 qt 1 pt
8 x 1 pt = 8 pt = 4 qt 8 x 2 qt = 16 qt + 4 qt = 20 qt = 5 gal 8 x 3 gal = 24 gal + 5 gal = 29 gal

Example:

The patrol drank 29 gallons of water. Note: In this example you first multiplied and then divided to obtain the correct portion of pints, quarts, and then gallons.

If you had a difficult time solving either problem, review this section before moving on to weight measurement. 4204. U.S. Weight Measure

a. Units of weiqht measure. The weight of any object is simply the force or pull with which it is attracted toward the center of earth by gravitation. There are three common systems of weights in the U.S.: troy weight, used in weighing jewelry, apothecaries' weight, used in weighing small amounts of drugs, and avoirdupois weight, used for all ordinary purposes. Our primary concern will be with avoirdupois weight. The terms listed in table 4-6 are quite common and should be committed to memory. Table 4-6.
* * 437 1/2 7,000 16 2,000

Weight Measure Table
= = = = 1 1 1 1 ounce (oz) pound (lb) pound ton

grains(gr) grains ounces pounds

2,240
*

pounds

= 1 long ton

A grain was originally a grain of wheat, but it is now described in relation to a cubic inch of distilled water. 4-15

Expressing and computing. Weight measurements can be b. expressed in several different ways. The ton is 2,000 pounds, the pound is 16 ounces or 7,000 grains. Here, as with linear and capacity measurements, the measurements do not always come out evenly and must be expressed in fractional parts or in terms of a smaller unit. For example, 2,500 pounds is 1 1/4 tons or 1 ton 500 pounds. Computations with the weight measurements are made in the same manner as with linear or capacity measurements. To add or subtract, the numbers must be lined up in like columns. Example: Cpl Barbell has two boxes that need to be mailed to Sgt Steel. One of the boxes weighs 10 pounds 14 ounces and the other weighs 19 pounds 6 ounces. The Postmaster said, "You'll save money if the combined weight is less than 35 pounds and you mail just one box." Can Cpl Barbell save any money? 10 lb 14 oz 19 lb 6 oz 29 lb 20 oz = 30 lb 4 oz (Yes, Cpl Barbell can save money because 30 pounds 4 ounces is less than 35 pounds. Note: Since 20 ounces is greater than 1 pound, the pound is added to the pound column and the remainder of the 20 ounces is expressed in ounces. Let's take a look at two more examples.

Example: You have 6 pounds 14 ounces of explosives. You need charges each weighing 1 pound 6 ounces for steel cutting. How many 1 pound 6 ounce charges can you make? 5 1 lb 6 oz 6 lb 14 oz= 22 oz 110 oz
110 oz

You can make five 1 pound 6 ounce charges.

4-16

Example: There are 5 truck loads of supplies. Each truck load weighs 1 ton 500 lb 10 oz. What is the total weight of supplies.
5 x 10 oz
=

50 oz = 3 lb 2 oz

(carry 3 lb)

5 x 500 lb= 2,500 lb + 3 lb = 2,503 lb = 1 ton 503 lb (carry 1 ton)
5 x 1 ton
=

5 tons + 1 ton = 6 tons

The total weight for all supplies is 6 tons 503 pounds 2 ounces. Note: As you can see by each example, division and multiplication are both used in finding the answers. It doesn't matter if you change the various units as you go or at the end.

Lesson Summary. This lesson taught you how to compute U.S. linear, capacity, and weight measurements. You learned that the units of linear measure are inches, feet, yards, rods, and miles, and units of capacity measure are ounces, cups, pints, quarts, gallons, and barrels. In addition, you learned that the units of weight measure are grains, ounces, pounds, and tons. In your next lesson you are going to learn how to convert U.S. and metric units. Lesson 3. CONVERSION BETWEEN U.S. AND METRIC UNITS

LEARNING OBJECTIVES 1. 2. 3. 4. 5. 6. Given linear measurements in metric units, convert to U.S. units. Given linear measurements in U.S. units, convert to metric units. Given capacity measurements in metric units, convert to U.S. units. Given capacity measurements in U.S. units, convert to metric units. Given weight measurements in metric units, convert to U.S. units. Given weight measurements in U.S. units, convert to metric units.

4-17

4301.

Conversion of Linear Measure

General. Hopefully, the time will come when you will no a. longer need to change from the metric system to the U.S. system or vice versa. However, until that time arrives, you need to know how to convert meters to yards, miles to kilometers, inches The most important conversions for Marines to centimeters, etc. involve linear measurements such as centimeters, meters, and kilometers and inches, feet, yards, and miles. Just what is the relationship between a mile and a kilometer, a yard and a meter, or a centimeter and an inch? The relationships are not quite as clear cut as working within the metric system, but with the memorization of a few conversion factors, changing meters to miles or kilometers to yards becomes a simple multiplication or division problem. Convertinq metric linear units to U.S. linear units. As b. previously mentioned, the memorization of a few conversion factors help when changing metric units to U.S. units. Table 4-7 If gives various conversions that will be very useful to you. you memorize them, you will be able to calculate and convert any metric distance to the equivalent U.S. distance. Look at table 4-7 and memorize the factors. Following the table, several examples will be presented illustrating the conversion of the more important units. Table 4-7. Conversions from Metric Units Divide by 2.5 30 0.9 1.6 To Find inches feet yards miles

When you know centimeters centimeters meters kilometers

Note:

All conversion factors are approximations; however, they are accurate enough for use in this course.

Example: The enemy target is 1500 meters from you. Vards away is the target? Note: How many

Since a meter is longer than a yard, you know that the answer will be larger than 1500. It is only If you know that 1 necessary to remember one factor. can be solved by yard = 0.9 meters, the problem division.

4-18

1666.66
0.9

F10

F 15000. 00

The enemy target is 1666.66 yards away. Example: The convoy is moving 300 kilometers north. The first leg of the journey is 100 kilometers. How many miles will the convoy travel on the first leg? From the table you can see that 1 mile = 1.6 kilometers. This problem is easily solved by division.
62.5

1.6

100.00

1000.0

The convoy will travel 62.5 miles on the first leg. Example: The wire obstacle is 185 centimeters high. What is the height of the obstacle in feet and inches? From the table, you know that 1 foot = 30 centimeters and 1 inch = 2.5 centimeters. The problem can be solved by division. 6 2

185.
180 5

2.5

F50

The obstacles's height is 6 feet 2 inches. Note: The remaining 5 centimeters was divided by 2.5 to obtain inches. Now

You just learned how to convert metric units to U.S. units. let's learn how to convert from U.S. units to metric units.

c. Converting U.S. linear units to metric linear units. The conversion of U.S. units to metric units is a simple multiplication problem using the same factors that were used to change metric units to U.S. units. Try to remember the conversion factors that apply to the different units. Several examples will be presented to illustrate the conversion of the more important units.

4-19

Table 4-8.

Conversions from U.S. Units Multiply by 2.5 30 0.9 1.6 To find centimeters centimeters meters kilometers

When you know inches feet yards miles

Example: The bridge to be destroyed is located 5 miles west of the city. How many kilometers west of the city is the bridge? From the chart you should remember that 1 mile = 1.6 kilometers; therefore, 5 miles is 1.6 times 5. x 1.6 5
8.0 km = 5 mi

The bridge is located 8 kilometers west of the city. Example: Hill 7500 on the map is indicated as being 250 feet above sea level. How many centimeters above sea level is the hill? From the chart you know that 1 foot = 30 centimeters; therefore, 250 feet would be 30 times 250. 250 x 30 7500 cm The hill is 7500 centimeters above sea level. Example: Your mission requires you to cross a river that is 45 yards wide. How many meters wide is the river? To obtain meters, multiply 45 yards x .9. 45
x .9

40.5 m The river is 40.5 meters wide.

4-20

Example: The vehicles were protected (hardened) with a layer of sandbags 17 inches high. What is the height of the sandbags in centimeters? 17 x 2.5 42.5 cm The sandbags are 42.5 centimeters high. Up to this point you learned how to convert linear units. Remember, to convert metric linear units to U.S. linear units, divide by the conversion factor, and to change U.S. linear units to metric linear units, multiply by the conversion factor. If you had a difficult time with these conversions, review this section before moving to conversion of capacity measure. 4302. Conversion of Capacity Measure

a. Converting metric capacity units to U.S. capacity units. Conversion between metric capacity and U.S. capacity is done the same way as with linear measure. The only difference is the conversion factors. Changing from metric units to U.S. units is just a simple division problem. Look at table 4-9 and memorize the conversion factors that apply to the different units. Following the table are several examples. Table 4-9. When You Know milliliters liters liters liters Conversions from Metric Units Divide by 30 0.47 0.95 3.8 To Find fluid ounces pints quarts gallons

Example: The motor transport operator added 150 milliliters of brake fluid to the master cylinder. How many fluid ounces of brake fluid did he add? To convert milliliters into fluid ounces, divide by 30.

4-21

5 30 1 150 The operator added 5 fluid ounces of brake fluid. Example: The guard detail drank 35 liters of milk during chow. How many pints did the guard detail drink? From the chart you see that 1 pint = 0.47 liters. The problem can be solved by dividing 35 by 0.47 75.46

0.47

47 1 3500.

The detail drank 75.46 pints of milk. Example: After rebuilding the transmission on the M929 dump truck, the mechanic added 17 liters of transmission fluid. How many quarts of transmission fluid did he add? 17.89

0.95

95 11700.

The mechanic added 17.89 quarts of transmission fluid. Example: During the first day of training, the platoon drank 83.6 liters of water. How many gallons of water did the platoon drink? 22.

3.8

F83.

38 1836.

The platoon drank 22 gallons of water.

4-22

You can see by these examples that conversion from metric capacity units to U.S. capacity units is accomplished through division. If you had difficulty with these conversions, review this section before moving forward. b. ConvertinQ U.S. capacity units to metric capacity units. To change U.S. capacity units to metric units, simple multiplication is used. The approximate conversion factors are the same as the ones used to convert metric capacity to U.S. capacity. You should memorize the conversion factors to apply them to their respective units. Several examples follow the conversion table. Table 4-10. When you know fluid ounces pints quarts gallons Conversions from U.S. Units Multiply by 30 0.47 0.95 3.8 To find milliliters liters liters liters

Example: The squad used 50 fluid lubricant preservative) many milliliters of CLP convert 50 fluid ounces by 30. 50 x 30 1500 The squad used 1500 milliliters of CLP. Example: The engineers' chainsaw requires 3 pints of twocycle oil for every three gallons of gasoline. How many liters of two-cycle oil are required? From the table you see that 1 pint = 0.47 liters. 0.47 3 1.41 ounces of CLP (cleaner to clean their weapons. How did the squad use? To to milliliters, multiply 50

The chainsaw requires 1.41 liters of two-cycle oil for every three gallons of gasoline.

4-23

Example: Each Marine carried 6 quarts of water while in the desert. How many liters of water did each Marine carry? Multiply 6 by 0.95 to obtain liters. 1 quart = 0.95 liters. 0.95 6 5.70

Each Marine carried 5.7 liters of water while in the desert. Example: The fireteam had one 5 gallon water jug. How many liters of water did they have? From the table, 1
gallon = 3.8 liters.

3.8 5
19. 0

The fireteam had 19 liters of water. Remember, converting U.S. capacity units to metric capacity units is just the opposite of converting metric capacity units to U.S. capacity units. If you had a difficult time with any of these conversions, review this section before moving to conversion of weight measure. 4303. Conversion of Weight Measure

a. Converting metric weight to U.S. weight. Converting metric weight to U.S. weight is done the same way that was covered in the two preceding lessons. The conversion is done using division and some very important conversion factors. (The conversion factors in the table are approximations and should not be used for accurate weight conversions.) Examples follow the factor conversion table. Table 4-11. When vou know grams kilograms tonne Conversions from Metric Units Divide by 28 0.45 0.9 To find ounces pounds short ton (2000 lb.)

4-24

Example: The material for the bunker weighed 16 tonnes. How many tons does the material weigh? 1 short ton = .09 tonne. 17.77

0.9

9 1 160.00

The material for the bunker weighs 17.77 short tons. Example: It requires 99 kilograms of explosives to breach the obstacle. How many pounds of explosives are required? 1 pound = 0.45 kilograms. 220 0.45 F99
=

9900

Two hundred twenty pounds of explosives are required to breach the obstacle. Example: The compass weighs 167 grams. How many ounces does the compass weigh? 1 ounce = 28 grams. 5.96

F167

28 1 167.00

The compass weighs 5.96 ounces. As shown by the examples, conversion from metric weight to U.S. weight is done by division. If you had a difficult time with these conversions, review this section before moving forward. Converting U.S. weight to metric weiqht. To convert U.S. b. weight to metric weight, you multiply the U.S. weight by the conversion factor (Table 4-12). Keep in mind that the conversions are only approximations.

4-25

Table 4-12.

Conversion from U.S. Units Multiply by 28 0.45 0.9 To find grams kilograms tonnes

When you know

ounces pounds short tons (2000 lb.)

Example: The mail clerk said the MCI course weighs 12 ounces. How many grams does the MCI course weigh?
1 ounce = 28 grams.

12 x 28 336 The MCI course weighs 336 grams. Example: The Marine weighed in at 160 pounds. How many kilograms does the Marine weigh? 1 pound = 0.45 kilograms. 160 x 0.45 72 The Marine weighs 72 kilograms. Example: The tank weighs 60 short tons. How many tonnes does the tank weigh? 1 short ton = 0.9 tonnes. 60
x 0.9

54 The tank weighs 54 tonnes. Note: It would be a good idea to commit to memory the conversion factors and the units to which they apply. Lesson Summary. In this lesson you converted metric linear, capacity, and weight units to U.S. units and vice versa. Remember, to change from metric to U.S. units, divide by the conversion factor and to change U.S. units to metric units, multiply by the conversion factor. Now it's time to test your knowledge. Attack the unit exercise! 4-26

Unit Exercise:

Complete items 1 through 51 by performing the action required. Check your responses against those listed at the end of this study unit.

The first leg on the land navigation course was 43,000 centimeters long. How many meters long was the first leg? a. b. 4,300 430 c. d. 43.0 4.30

The enemy is located 15 kilometers west of our position. How many meters away is the enemy? a. b. 1.5 150 c. d. 1,500 15,000

The explosion was seen 600 meters north of the OP (observation post). How many kilometers away was the explosion seen? a. b. 60 6.0 c. d. .6 .06

The first leg of the forced march was 7 kilometers long. How many centimeters long was the first leg? a. b. 700,000 70,000 c. d. 7,000 700

On a map with a scale of 1:50,000, the distance between point A and B is 18 centimeters. What is the actual ground distance in meters? a. b. 900 9,000 c. d. 90,000 900,000

On a 1:25,000 map, 14.3 centimeters equals how many kilometers on the ground? a. b. 3.757 3.577 c. d. 37.57 35.77

The platoon drank 155 liters of water. How many milliliters of water did the platoon drink?
a. 1,550 C. 155,000

15,500

1,550,000

4-27

The fuel bladder has 755 liters of diesel fuel remaining. How many hectoliters of diesel fuel remain? a. b. 75.5 7.55 c. d. .755 .0755

It takes 4,834 milliliters of fuel to prime the diesel engine. How many liters of fuel are needed to prime the engine?
a. 48,340 C. 48.34

483.4

4.834

10. Of 6.5 hectoliters of fuel, the private spilled 350 milliliters. How many liters of fuel did the Private spill? a. b. .35 3.5 c. d. .65 6.5

11. If 750 milliliters of oil were poured from the 4.75 liter container, how many liters of oil remain?
a. 746.25 C. 4

.74625

12. You have 36 liters of fuel and just added 4,000 milliliters more. How many hectoliters of fuel do you have? a. b. 400 40 c. d. 4 .4

13. A 3,450 kilogram pallet of ammo equals how many tonnes? a. b. 34.5 3.45 c. d. .345 .0345 How many kilograms does

14. The grenade weighs 500 grams. the grenade weigh? a. b. .5 5 c. d. 50 55

15. If the ship weighs 430 tonnes, how many kilograms does the ship weigh?
a. 4,300 C. 430,000

43,000

4,300,000 How many

16. The pallet of tents weighs 750,000 grams. tonnes do the tents weigh?
a. 750 C. 7.5

d. 4-28

.75

17. You just removed 60,000 grams of 5.56 ammo from the stockpile of 167 kilograms. How many kilograms of ammo remain?
a. 161 C. 60

107

10.7

18. You have 37 tonnes plus 650,000 grams of C-4 explosives. How many kilograms of explosives do you have?
a. 37.650 C. 3,765.0

376.50

37,650

19. There are 4 boards measuring 3 feet 4 inches, 27 inches, 1 1/2 yards, and 2 3/4 feet. What is the total length of all 4 boards? a. b. 10 ft 10 in. 10 ft 22 in. c. d. 11 ft 10 in. 12 ft 10 in.

20. Each Marine in the patrol has a piece of fiber rope 5 yards 2 feet 27 inches long. There are 19 Marines in the patrol. What is the total length of all 19 segments of rope? a. b.
C.

119 yd 3 ft 6 in. 120 yd 2 ft 9 in.
121 yd 2 ft 9 in.

122 yd 3 ft 6 in.

21. The minefield is 23 yards 1 foot wide. The engineer platoon had 11 centerline markers. The markers were placed at an equal distance apart throughout the minefield. What is the distance between each marker? a. b. 6 ft 2 in. 7 ft 2 in. c. d. 1 yd 2 ft 2 yd 1 ft

22. The first container had 3 gallons 3 quarts 1 pint of diesel fuel, the second container had 5 gallons 2 quarts of diesel fuel, and the third container had 6 gallons 1 quart 1 pint of diesel fuel. What is the total quantity of diesel fuel? a. b. 14 gal 2 qt 14 gal 3 qt C. d. 15 gal 2 qt 15 gal 3 qt

23. Each heater stove at the Mountain Warfare Training Camp There burns 2 gallons 3 quarts of diesel fuel per day. are 417 stoves in the camp. How much diesel fuel is burned daily? a. b. 1,146 gal 3 qt 1,416 gal 3 qt c. d. 4-29 1,246 gal 3 qt 1,426 gal 3 qt

24. You have been given 5 gallons 1 quart 2 pints of water for 11 Marines. How much water will each Marine receive? a. b. 4 qt 3 qt 1 pt c. d. 2 qt 1 pt 2 qt

25. You have 3 pallets loaded with supplies. One pallet weighs 1 ton, 600 pounds 15 ounces, another pallet weighs 6 ton 35 pounds 4 ounces, and the last pallet weighs 1,900 pounds 12 ounces. What is the total weight of the 3 pallets? a. b. c. d. 8 8 7 7 ton ton ton ton 536 356 535 356 lb lb lb lb 15 15 31 31 oz oz oz oz

26. Each Marine has a light marching pack weighing 25 pounds 6 ounces. There were 437 Marines on the march. What is the total weight of all the packs? a. b. c. d. 6 6 5 5 ton ton ton ton 925 lb 14 oz 1,088 lb 14 oz 925 lb 14 oz 1,088 lb 14 oz

27. The utilities platoon crated five reciprocating water pumps with a total weight of 490 pounds 10 ounces. What is the weight of one pump? a. b. 97 lb 2 oz 97 lb 4 oz c. d. 98 lb 2 oz 98 lb 4 oz

28. The distance that male Marines run for the PFT (Physical Fitness Test) is 3 miles. How many kilometers is this? a. b. 1.6 3.2 c. d. 4.8 6.4

29. One of the combat conditioning courses is 3000 meters long. How long is the combat conditioning course in yards? a. 31,333.33 c. 3,133.33

33,333.33

3,333.33
How many inche:

30. The tank ditch is 430 centimeters deep. deep is the ditch? a. b. 162 172 c. d. 182 192

4-30

31. The wire obstacle is 430 centimeters wide. wide is the obstacle? a. b. 12.33 13.33 c. d. 14.33 15.33

How many feet

32. The enemy is located 65 miles east of us. kilometers away is the enemy? a. b. 101 102 c. d. 103 104

How many

33. Pvt. Demolition found a landmine that was 7 1/2 inches in diameter. How many centimeters in diameter is the landmine? a. b. 18.75 18.85 c. d. 19.75 19.85 How many centimeters wide is 320 360

34. The bunker is 8 feet wide. the bunker? a. b. 200 240 c. d.

35. All Marines had to carry (firemans carry) their partners 500 yards during the evacuation drill. How many meters did they carry their partners? a. b. 430 440 c. d. 450 460

36. It took 475 milliliters of boiling water to purify the canteen cup. How many fluid ounces was needed to purify the cup? a. b. 18.53 17.53 c. d. 16.83 15.83 How many

37. The Marine drank 15 liters of water a day. pints did the Marine drink? a. b. 91.31 31.91 c. d. 13.91 11.31

38. The fuel can holds 6 liters of diesel. does the fuel can hold? a. b. 4.31 5.31 c. d. 6.31 7.31

How many quarts

4-31

39. Each Marine living in the barracks uses approximately 48 liters of water a day. How many gallons of water does each Marine use? a. b. 11.36 11.63 c. d. 12.36 12.63 How many

40. The engine burns 64 ounces of oil every day. milliliters of oil does the engine burn? a. b. 1,290 1,920 c. d. 2,019 2,120

41. It took 4.5 pints of chlorine to purify the water in the holding tank. How many liters of chlorine was needed? a. b. 1.215 1.115 c. d. 2.215 2.115

42. It took 8.25 quarts of paint to touch up the 5 ton truck. How many liters of paint did it take to touch up the truck? a. b. 6.3785 6.8375 c. d. 7.3785 7.8375

43. The maintenance shop uses one 55 gallon drum of 30 weight oil daily. How many liters does the shop use. a. b. 207 208 c. d. 209 210 How many

44. The field jacket liner weighs 1,064 grams. ounces does the liner weigh? a. b. 32 34 c. d. 36 38

45. The M203 grenade launcher has a total weight (with rifle) of 4.275 kilograms. How many pounds does the launcher weigh? a. b. 10 lb. 9.5 lb. c. d. 9 lb. 8.5 lb.

46. The LVS (Logistic Vehicle System) was hauling a load of supplies weighing 18 tonnes. What was the weight of the load in short tons? a. b. 23 22 c. d. 21 20

4-32

47. The messhall cook used 54 ounces of butter to make pastries. How many grams of butter did the cook use? a. b. 1,078 1,088 c. d. 1,512 1,688

48. The combat engineers used 1,750 pounds of C-4 to breach the minefield. How many kilograms of explosives did they use? a. b. 767.5 777.5 c. d. 787.5 797.5

49. The Marines flew in 21 short tons of supplies for the refugees. How many tonnes of supplies did they fly in? a. b. 19.8 19.9 c. d. 18.8 18.9 How many

50. The MK19 40mm machinegun weighs 75.6 pounds. kilograms does the machinegun weigh? a. b. 32.04 33.02 c. d. 34.02 35.04

51. The weight of the M9 9mm service pistol with a 15 round magazine is 1,145 grams. How many ounces does the pistol weigh? a. b. 49.80 oz 48.89 oz c. d. 44.89 oz 40.89 oz

Unit Summary This study unit provided a brief historical account of units of measurement, and the knowledge needed to perform computations using the metric and U.S. systems. Additionally, you were taught how to perform conversions between the U.S. and metric units. Exercise Solutions Reference 1. 2.
3.

b. d.
c.

43,000 cm = 430 m 15 km = 15,000 m
600 m = .6 km

4102 4102
4102

4. 5.

a. b.

7 km = 700,000 cm 1 cm (map) = 50,000 cm (ground) 18 cm (map) = 50,000 x 18
= 900,000 cm = 9,000 m

4102 4102

4-33

Reference 6. a. 1 cm (map) = 25,000 cm (ground) 14.3 cm (map) = 25,000 x 14.3 (ground)
= 3,575,000 cm = 3.575 km 7. c. 155 1 = 155,000 ml 4103

4102

8. b. 9. d. 10. a.
11. c.

755 1 = 7.55 hl 4,834 ml = 4.834 1 350 ml = .35 1
4.75 1
=
-

4103 4103 4103
4103

750 ml
.75 1

4.75 1 -

=4 1 12. d. 36 1 = .36 hl
4,000 ml = .04 hl

4103

36 1 + .04 1 =.4 hl 13. 14. 15. 16. b. a. c. d. 3,450 kg = 3.45 T 500 g = .5 kg 430 T = 430,000 kg 750,000 g = .75 T
167 kg 60 kg = 107 kg

4104 4104 4104 4104
4104

17. b.

18. d.

37 T = 37,000 kg 650,000 g = 650 kg 37,000 kg + 650 kg = 37,650 kg 3 ft 4 in.
27 in. 1 1/2 yd + 2 3/4 ft

4104

19. d.

=
= = =

3 ft 4 in.
2 ft 3 in. 4 ft 6 in. 2 ft 9 in.

4202

11 ft 22 in. = 12 ft 10 in.
20. c. 19 x 5 yd 2 ft 27 in. = 4202

19 x 27 in. = 42 ft 9 in.

(carry 42 ft)

19 x 2 ft = 38 ft + 42 ft = 80 ft

= 26 yd 2 ft (carry 26 yd)
19 x 5 yd = 95 yd + 26 yd = 121 yd

total = 121 yd 2 ft 9 in. 21. d. Divide 23 yd 1 ft by 10 Note: There are 10 spaces between 11 markers.
23 yd = 69 ft + 1 ft = 70 ft 70 ft . 10 = 7 ft or 2 yd 1 ft

4202

22. d.

3 5 + 6 14

gal gal cral gal

3 2 1 6

qt 1 pt qt ct 1 pt qt 2 pt = 15 gal 3 qt

4203

4-34

Reference 23. a. 2 gal 3 qt x 417 =
417 x 3 qt
= =

4203

1,251 qt = 312 gal 3 qt 1146 gal 3 qt

417 x 2 gal = 834 gal + 312 gal 24. d. Divide 5 gal 1 qt 2 pt by 11 5 gal 1 qt 2 pt = 44 pt
44 pt
-

4203

11 = 4 pt or 2 qt

25. a.

1 ton 600 lb 15 oz 6 ton 35 lb 4 oz + 900 lb 12 oz 7 ton 2,535 lb 31 oz = 8 ton 536 lb 15 oz
437 x 6 oz = 2,622 oz = 163 lb 14 oz 437 x 25 = 10,925 lb + 163 lb =

4204

26.

4204

11,088 lb = 5 tons 1,088 lb 14 oz 27. c.
28. 29. 30. 31. c. d. b. c.

490 lb 10 oz = 7,850 oz
7,850 oz - 5 = 1,570 oz = 98 lb 2 oz 3 mi x 1.6 = 4.8 km 3,000 m - 0.9 = 3,333.33 yd 430 cm - 2.5 = 172 in. 430 cm - 30 = 14.33 ft

4204
4301 4301 4301 4301

32. 33. 34. 35.
36. 37. 38. 39.

d. a. b. c.
d. b. c. d.

65 mi x 1.6 = 104 km 7.5 in. x 2.5 = 18.75 cm 8 ft x 30 = 240 cm 500 yd x 0.9 = 450 m
475 ml - 30 = 15.83 fl oz 15 1 . 0.47 = 31.91 pt 6 1 - 0.95 = 6.31 qt 48 1 + 3.8 = 12.63 gal

4301 4301 4301 4301
4302 4302 4302 4302

40. b. 41. d. 42. d.
43. c. 44. d.

64 fl oz x 30 = 1,920 ml 4.5 pt x 0.47 = 2,115 1 8.25 qt. x 0.95 = 7.8375 1
55 gal x 3.8 = 209 1 1,064 g . 28 = 38 oz

4302 4302 4302
4302 4303

45. a.
46. 48. d. c.

4.275 kg . 0.45 =10 lb
18 t
+

4303
4303

0.9 = 20 short ton

47. c. 49. d. 50. c.
51. d.

54 oz x 28 = 1,512 g
1,750 lb x 0.45 = 787.5 kg

4303
4303

21 ton x 0.9 = 18.9 t 75.6 lb x 0.45 = 34.02
1,145 g + 28 = 40.89 oz

4303 4303
4303

4-35

STUDY UNIT 5 GEOMETRIC FORMS Introduction. Geometry is the study of the measurement and relationships of lines, angles, plane (flat) figures, and solid figures. In this study unit you will perform calculations to determine degrees, area, perimeter, volume, and circumference. Throughout your career, you will use the measurements and relationships studied here to perform a variety of tasks such as map reading and land navigation, determining fields of fire, calculating the amount of water and fuel for training operations, and determining the storage area for supplies. In fact the list really could go on. The first topic in our study is angles. Let's take a look at them. Lesson 1. ANGLES

LEARNING OBJECTIVES 1. 2. 3. Given a series of angles, apply the operations for measuring angles with a protractor to obtain degrees. Given a series of angles and a protractor, measure the angles to classify them. Given two or more angle readings, compute into degrees, minutes, and seconds, using the operations for computation of angles. Measuring Angles

5101.

a. Definition. An angle is a set of points consisting of two rays (sides) and a common end-point (fig 5-1). Angle BAC is abbreviated BAC. It can also be written as angle CAB; they are the same angle. Point A which is the middle letter in the angle is called the vertex of the angle.

-0 *

Fig 5-1.

Ray AB + ray AC = angle BAC.

5-1

An angle is referred to in one of several ways (fig 5-2). It may be called angle B, angle ABC or CBA, or it may be labeled with a small letter b or numeral 2.

Fig 5-2.

Labeling an angle.

Now that you know how to label angles, let's find out how they are measured. b. Measurement. Although the angle is the set of points on the two rays, the measure of the angle is the amount of "opening" between the rays in the interior of the angle. To measure an angle, we must have a unit of measure and a tool inscribed with
those units. The unit is the degree (0) and the tool is the

protractor. The protractor which comes with this course has two scales. Let's take a look at it. The inner scale measurement is in degrees and the outer scale measurement is in mils. We will only be using the inner scale. To avoid confusing the two scales, it is recommended that you cut off the mils (outer) scale. Looking at your protractor, you will see that the degrees range from 0-360 (360 is also positioned at 0), making a complete circle. The degrees on your protractor are determined by a set of 360 rays drawn from the same center point. These rays which are numbered from 0 to 360 make 360 angles all the same size. One of these angles is the standard unit or degree for measuring angles. Figure 5-3 illustrates a scale from 0 to 180° and the
actual size of 10.

70 60 50

10U ~~~~~~~~120 130

4014

-u8

Fig 5-3.

Measuring scale for angles. 5-2

Note:

This protractor is normally used with the "0" pointed in the direction of north when used for land navigation. You won't be using it that way in this course. When determining the degrees of an angle, the 0/180 line becomes the base line. The base line is positioned on the angle so that you can read the scale in a clockwise direction.

To measure an angle with the protractor, place it on the angle so that the center point (crosshairs of the protractor) is on the vertex (A) of the angle and position the 0/180 line so that you read the angle measurement in a clockwise direction from zero up the scale towards 180 (fig 5-4). Observe the degree reading where the protractor scale lines up with the other ray. The number that corresponds to this ray is the measurement, in degrees, of the angle. Angle "A" in this illustration measures 35

A Fig 5-4. Measuring an angle.

Now use the protractor provided to measure the angles below. If the rays are not long enough, lay the edge of your protractor along the side of the angle or carefully extend the rays with a pencil. Example:

This is a 900 angle. When you measured this angle with your protractor, you should have placed it on the angle so that the center point (crosshairs of the protractor) was on the vertex of the angle. You should have positioned the 0/180 line so that you could read the angle measurement from zero in a clockwise direction up the scale towards 180. The degree reading on the protractor which lies on the other ray
is 90°.

5-3

Let's try another angle. Example:

Measure angle C in the example below.

Angle "C" is a 1140 angle. Remember, you must place the center point of the protractor on the vertex of the angle and position the 0/180 line to read up the scale. Example: I)

Was your answer for angle "D" 790?

If not you need to go

back and review the section. 5102. Angle Classification

Classifications. Angles are classified according to the number of degrees they contain (fig 5-5). An angle of 900 is called a right angle and the rays are said to be perpendicular. An angle less than 900 is called an acute angle. An angle whose measure is more than 900 and less than 1800 is called an obtuse angle. An angle of exactly 1800 is called a straight angle.

LESSTHAN900 LESS TUA
9A

MORE THAN 902

Fig 5-5.

Angle classifications. 5-4

Can you look at an angle and classify it? If you can, you're ready to start computing angle readings. If not, take a few minutes to study these angles (fig 5-5). Note how they look and approximately how many degrees they measure. 5103. Computation of Angle Readings

Computations. So far, all of the angles discussed have been measured in even degrees. As a Marine, you land navigate using readings on a compass in degrees and occasionally use parts of a degree to compute yearly declination on older maps. All Marines must know how to add and subtract various angles to change compass headings, figure back azimuths, compute yearly declination, and to accurately plot on a map. Here you will learn a little more than just degrees. You will learn about minutes (') and seconds (") too! Each degree contains 60 minutes and each minute contains 60 seconds. Let's work a few problems involving minutes and seconds. Addition and subtraction of angles is done the same way as other units of measure using the decimal system. Keep the proper units in the same column and carry or borrow as needed. Example: 12014'361"
+ 22056'4211 3407017811 = 35°11'18"

Note:

You should see that 78" is 1' with 18" remaining.
70' + 1' is 71' plus 10 is 350. which is 10 with 11' remaining. Your answer is 3501111811 340

Example: + 120°45'22" 16017'381"
13606216011
=

137031

Example:
42012'30" = 12030'501 =
-

41071'90"1 12030150" 29041'40",

Note:

Be careful with this problem. You cannot subtract 50" from 30" nor 30' from 12' so you must borrow 1' or 60" from 12' and add it to 30" and borrow 10 or 60' from 420 and add it to 12' before you subtract. Let's look at another example.

5-5

Example:
-

112°4'25" 106°5'24"

= = -

111064'25" 1060 5'24"
5059

Note:

You should see from the preceding examples that, as long as you keep the proper units in the same column and borrow or carry as needed, the problem is easily solved.

Lesson Summary. During this lesson you were provided with the procedures used to measure angles, classify angles, and compute angle readings in degrees, minutes and seconds. In the next lesson you will determine dimensions of plane geometric figures. Lesson 2. PLANE FIGURES

LEARNING OBJECTIVES 1. 2. 3. 4. 5201. Given the dimensions of plane geometric figures, use the proper formula to compute the perimeter. Given the dimensions of plane geometric figures, use the proper formula to compute the area. Given the radius of various circles, use the proper formula to compute the circumference. Given the radius of various circles, use the proper formula to compute the area. Perimeter and Area of Squares

a. General. There is an unlimited number of figures or shapes in a plane. A plane has no boundaries although we usually visualize it as looking like a piece of paper. For our purposes, we will think of a plane simply as being a flat surface with two dimensions. This piece of paper, if we forget that it actually has a thickness, represents a plane. If you took a pair of scissors and cut out various shapes from this page, each would be a plane figure. Some of these would have practical uses, others would not due to having too many sides that are too irregular in shape. The majority of the objects around you have some regular geometric shape. Let's look at some of them. b. Square. This familiar figure has four sides of equal length and a right angle at each corner. Opposite sides are parallel. Two lines are parallel if they do not intersect, no matter how far extended. There are two important measures of a square: the perimeter and the area.

5-6

(1) Perimeter. To find the perimeter or distance around a square, add the four sides or even more simply: Perimeter (P) = 4s (4 x length of 1 side). Example: What is the perimeter of a square with a side that measures 16 in.?
P = 4s P = 4(16)

P = 64 The perimeter is 64 inches. Using the equation-solving techniques, if the perimeter is known, you can determine the length of a side. For example, if the perimeter of a square is 62 inches, what is the length of one side? Example:
P = 4s 62 = 4s 62 = 4s

4 62 4
15.5
=

4 s s

Therefore, each side is 15 1/2 inches long. (2) Area. Area, not only for a square, but for plane figures, is described or measured in terms of square units. A square with sides of 3 inches contains 9 squares, each 1 inch on a side (fig 5-6).

31"

T
3,'

4 7 Fig 5-6.

5 8

6 9

Area of a square with s = 3.

5-7

You could prove to yourself, by drawing squares of various sizes and marking them off into grids, that the area of a square is equal to the product of two of the sides or the square of one side. A = s x s or A = s Example: Grenade range 303 has a square parking lot with one side that is 32 feet long. What is the area of the parking lot?
A A A
=S2 = =

322 1024

The area is 1024 square feet. Do you remember how to find the perimeter of a square? If you said P = 4s, you're right! How about the area of a square? If your answer was A = s2, you're right again! If you had a difficult time understanding perimeter or area, review this section before moving on. 5202. Perimeter and Area of Rectangles

a. Rectangle. This is another familiar figure. Each pair of opposite sides are equal in length and parallel, and there is a right angle at each corner. The two important measures are perimeter and area. Let's look at them. (1) As with the square, you can find the perimeter by adding up the measures of the four sides. The longer dimension is called the length (1) and the shorter is called the width (w). There are two lengths and two widths to a rectangle. The formula is: P = 21 (2 x length) + 2w (2 x width). Example: The engineers cut a sheet of plywood 7 feet long and 4 feet wide. What is the perimeter of the sheet of plywood? P
P
= = = =

P
P

21 + 2w 2(7) + 2(4) 14 + 8
22

The perimeter is 22 feet. Note: You can also find a missing dimension if you know the perimeter and one of the two dimensions. Let's see how to do this.

5-8

Example: The perimeter of the momat (fiberglass matting) staging lot is 82 feet and the length is 31 feet. What is the width?
P = 21 + 2w 82 = 2(31) + 2w 82 = 62 + 2w 82 62 = 62 62 + 2w

20 = 2w 20 = 2w 2 2
10 = w

The width is 10 feet. Example: In front of the barracks there is a rectangular shaped concrete slab with a perimeter of 476 feet and a width of 70 feet. What is the length of the concrete slab?
P 476 476 140 336 336 = = = = = = 21 21 21 21 21 21 + + + + 2w 2(70) 140 140 -

476 -

140

2
168 =

2 1

The length is 168 feet. (2) Area. Area is described and measured in terms of square units. As with the square, you can segment a rectangle using a grid and count the number of units (fig 5-7).

6 1 7 13 19 2 8 14 20 3 9 15 21 4 10 16 22 5 11 17 23 6 12 4 18 24

Fig 5-7.

Area of a rectangle.

5-9

You can see in figure 5-7 that the number of square units is equal to the product of the two dimensions of the rectangle. This gives the formula A = lw. Example: What is the area of a rectangle that is 13.5 inches long and 6.2 inches wide? A = lw
A = (13.5)(6.2) A = 83.7

The area is 83.7 square inches. Again, as you did before, you can find an unknown dimension if you know the area and one of the dimensions. Example: The top of the grenade box has an area of 176 square inches with a length of 16 inches. What is the width? A = lw
176 = 16w 176 = 16w

11 = w

The width is 11 inches. Example: The area of the foundation for the bunker is 417 square feet with a width of 12 feet. How long is the foundation? A = lw
417 = 1(12) 417 = 1(12)

12
34.75
=

12 1

The length is 34.75 feet. Before you move on to parallelograms, do you recall the formulas used to compute the perimeter and area of a
rectangle? If you said P = 21 + 2w and A = lw you're

right! If you didn't, you should review this section before moving on.

5-10

5203.

Perimeter and Area of Parallelograms

a. Parallelogram. A parallelogram is a quadrilateral (four sided figure) with each pair of sides equal and parallel. It looks like a rectangle that has been shoved sideways (fig 5-8). Opposite angles are also equal; angle A equals angle C and angle B equals angle D. The long dimension of a parallelogram is usually referred to as the base (b). The perpendicular distance between the bases is called the height (h).

A4; I

~~~b(base)l
AB = CD and AB//CD BC = AD and BC//AD

Note:

// means parallel Parallelogram. the perimeter the sides are slanted are = 2b + 2w.

Fig 5-8.

b. Perimeter. As with the other quadrilaterals, is the sum of the lengths of the 4 sides. Two of called bases (b) and the other two sides that are called widths (w), so the formula would then be P Example:

What is the perimeter of a parallelogram that has a base of 22 inches and a width of 12 inches?
P P P P = = = = 2b + 2w 2(22) + 2(12) 44 + 24 68

The perimeter is 68 inches. c. Area. If you placed a grid over a parallelogram as was done with the square and rectangle, you would get fractional squares By cutting off the three due to the sloping sides (fig 5-9). fractional squares on one side and joining them with the parts of squares on the other side, you get three complete squares and a complete rectangle.

5-11

<111111
3 Figure 5-9.

Making a parallelogram into a rectangle.

Now you can borrow the formula for the area of a rectangle and substitute the letters used for parallelograms (fig 5-10). b
,

,1
//.

Area = base x height A = bh

Fig 5-10. Example:

Formula for the area of a parallelogram.

The bottom of the enemy trenchline is shaped like a parallelogram with a base of 50.2 feet and a height of 15.3 feet. How much area does the bottom of the trenchline cover? A = bh
A = 50.2(15.3) A = 768.06

The area is 768.06 square feet. Remember, to find the perimeter of a parallelogram, use the formula: P = 2b + 2w, and to find the area, use the formula: A = bh. 5204. Perimeter and Area of Trapezoids

Trapezoid. This is a quadrilateral with only one pair of a. opposite sides parallel (they are not equal). As with the parallelogram, the "top" and "bottom" of a trapezoid are referred to as bases. Since they are of different lengths, the subscripts 1 (b ) and 2 (b ) are used to identify them (fig 5-11). 1 2 5-12

The perpendicular distance between the parallel bases is called the height (h).

A / i

~~b2ED
Trapezoid.

Fig 5-11.

b. Perimeter. To find the perimeter, all you have to do is add the lengths of the four sides. P = a + b + c + d. Example: A trapezoid with side "a" 24 inches, side "b" 12 inches, side "c" 18 inches, and side "d" 12 inches has a perimeter of how many inches? P =a + b + c + d
P P
=

24 + 12 + 18 + 12 66

The perimeter is 66 inches. c. Area. The formula for the area of a trapezoid is derived from the formula for the area of a triangle. A trapezoid can be divided into two triangles of unequal area (fig 5-12).

B bIC

AAb

Fig 5-12.

Dividing a trapezoid into two unequal triangles.

Follow along closely. To find the area of trapezoid ABCD (fig 5-12), you could add the area of triangle ABC to the area of triangle ACD. To find the area of EFGH, add triangle EFH to triangle FGH. But instead of finding two separate areas and then

adding them together, a little algebraic manipulation with the formula for the area of a triangle gives us a formula for the area of a trapezoid. The triangle formula A = 1/2bh will be explained later in the study unit but will be used here without explanation. 5-13

1 In figure 5-12, the area (A) of ABCD is equal to 1/2b1 h + l/2h(b 1 be written as A = l/2h(b + b2). This is beyond + b2 ) or could the algebra that was covered in study unit 3, but if you remember how to factor, you should see how it was derived. Let's look at an example using this formula.

Example: The enemy bunker was constructed in the shape of the trapezoid illustrated. What is the area of the trapezoid?
11

~~~~~~I
16

A = l/2h(b1 + b2 ) A = 1/2(6)(11 + 16) A = 3(27)

A = 81 The area is 81 square feet. So, to find the perimeter of a trapezoid use the formula P = a + Now, how do you compute the area? If your said, use b + c + d. formula A = l/2h(b1 + b2 ) you're absolutely correct! the 5205. Triangles. Triangle ABC is the set of points A, B, and C and all points of AB on a line between A and B, all points of AC on a line between A and C, and all points of BC on a line between The points A, B, and C are called the vertices (plural B and C. of vertex) of the triangle. A triangle may also be thought of as points A, B, and C, and the line segments that connect them. Note that in both of these descriptions, no mention is made of the inside or outside areas. The triangle is only that part represented by the line. Everything inside is the interior of the triangle; everything outside is the exterior (fig 5-13).

AAC
RIO '' EXTERIOR':'''',,','EXTERI,
A triangle, its interior, and its exterior. 5-14 Fig 5-13.

a. Types: A triangle can be classified according to the lengths of its sides or the measures of its angles. The sum of the measures of the angles of any triangle is always 1800. Let's look at the most common types of triangles. (1) Equilateral. As the name implies, all three sides are the same length. The equilateral triangle is also called equiangular (equal angles). If all of the sides (A,B,C) are the same length, then all of the angles are the same measure (600). Example:
B

A/\c
(2) Isosceles. A triangle with two equal sides (y), and consequently, two equal angles (x). The equal angles are opposite the equal sides. Example:

(3) Scalene. A triangle in which all three sides (x,y & z) are of different length. All three angles (1,2 & 3) are also of different measure. Example:

5-15

(4) Right triangle. measures 900. Example:

A triangle in which one of the angles

90X

(5) Acute. A triangle in which all three angles have measures less than a right angle. Example:

(6) Obtuse. A triangle in which one of the three angles has a measure greater than a right angle. In this case, angle A is greater than 900. Example:

b. Perimeter. The perimeter of a triangle is found by adding the lengths of the three sides: P = a + b + c. Example: What is the perimeter of a triangular shaped field that measures 15 feet by 15 feet by 20 feet? P =a + b + c P = 15 + 15 + 20 P 50 The perimeter is 50 feet. 5-16

c. Area. The formula for the area of a triangle can be derived from the area of a parallelogram. Any two equal triangles can be placed together to form a parallelogram. As illustrated below, the parallelogram has been divided into two equal triangles ABC and ADC. Example:
B C

AD The formula used to determine the area of a parallelogram is A = bh. Since a parallelogram contains two equal triangles, the area of one of them would be half that of the parallelogram or 1/2bh. Example: In the illustration below, if b is used as the base, 2 is the height; if c is the base, 3 is the height; and if a is the base, then 1 is the height. It doesn't matter which dimension you call the base or which the height. With real dimensions, any of the combinations would give the area of the triangle. B
C

a 2 2
b

A Example:

Find the area of the triangle illustrated. as the base and 6 as the height. B 8 A A A
A
= = =

Use 11

I C

1/2bh 1/2(11 x 6)
33

The area of the triangle is 33 square units. 5-17

Do you remember how to find the perimeter and area of a triangle? To find the perimeter of a triangle use the formula P = a + b + C. To find the area of a triangle use the formula A = 1/2bh. Make sure you know how to use these formulas before continuing. 5206. Circles. A circle is a closed curve all of whose points are the same distance from a fixed point called the center. The center is usually labeled with a capital letter and the circle can be referred to by this letter. As with the other plane figures, a circle is only the points on the curve that represents the circle. Everything outside is the exterior; everything inside the closed curve is the interior. Let's take a look at the nomenclature of a circle (fig 5-14).
ovWCUMFE,k4.

DVUETER

CHORD/

Fig 5-14.

Nomenclature of a circle.

a. Nomenclature. The set of points that make up the circle is called the circumference. This is to the circle as the perimeter is to quadrilaterals. The fixed distance from the center to any point on the circle is called the radius. A line segment having its endpoints on the circle is called a chord. As figure 5-14 shows, a chord does not necessarily pass through the center. If it does, it is called the diameter of the circle. The diameter An important is equal to the length of two radii (d = 2r). concept about circles is that the ratio of the circumference of a circle to its diameter is a constant value 3.141592. It is symbolized by the Greek letter ff (pi); this number has been known and used since ancient times. You could derive this value for yourself by measuring the circumference and the diameter of an object and then dividing the circumference by the diameter: c/d = R. Pi is usually rounded off to the decimal 3.14 or the fractions 3 1/7 or 22/7. b. Circumference. Since circumference has been defined and its relationship to 71 and the diameter has been shown, it should be easy to determine the formula for computing the circumference (C) of a circle. If c/d = n, then you can manipulate this equation = C = rd d x c id, to find the value of c. Since d = 2r, the formula could also be C = 27r. these two formulas are used. 5-18 Let's see how

Example: Find the circumference of a water bladder (storage tank for water) that has a radius of 5 feet.
C = 2vr C = 2(3.14)5 C = 31.4

The circumference is 31.4 feet. Example: Find the circumference of the 155 howitzer barrel. It has a diameter of 155mm.
C = lrd C = 3.14(155) C = 486.7

The circumference is 486.7mm. The following example shows how to find the radius when the circumference is known. Example: One of the trees used for the log obstacle has a circumference of 56.52 inches. What is the radius of the tree. C = 2vr 56.52 = 2(3.14)r 56.52 = 6.28r 56.52 6.28
9 =r

6.28r 6.28

The radius of the tree is 9 inches. Let's test your memory. If you know the radius of a circle and want to find out its circumference, what is the formula used? If your answer was C = 2irr, you're correct! How would you determine the circumference when you know its diameter? If your answer is C = 7d, you're right again! If your answers were different from those given, review this section before moving on to area of circles. 5207. Area of circles. The area of a circle is always measured in square units. It would be extremely difficult to determine the number of whole units by placing a grid over the circle as with rectangles; therefore, a formula is used. The formula was derived from the rectangle area formula, and can be described using an illustration (fig 5-15).

5-19

Divide a circle into any number of equal-size pie shaped sections.

Open the circle so that the sections are in a line.

Interlock the sections so that a figure approaching a rectangle is formed.

If the circle could be separated into enough sections, the smaller they got, the flatter the arcs would be. Eventually the figure would be a rectangle. Fig 5-15. Deriving the area of a circle. 5-20

If you substitute the appropriate symbols for parts of the circle into the formula for the area of a rectangle, you will get the formula for the area of a circle. The length of the rectangle that has been formed is related to the circumference of the circle in that the top and bottom of the rectangle are the arcs that were originally the circumference (fig 5-15). Since the circumference of a circle can be expressed by the formula 27r, the length of a rectangle would be half of this or 1/2 x 2vr. The width of the rectangle is the radius of the circle. You can now substitute the variables into the formula for the area of a rectangle. A= 1 x w A = 1 x 27r x r = 1 x a 7or x r 2 -a A = 7r x r A =or

Example:
What is the size of the casualty area for the M16A1 antipersonnel (AP) mine if the radius of the area is 30m? A = 7Tr2 A = 3.14(302) A = 3.14(900)
A = 2,826

The area is 2,826 square meters. Example: The diameter of the bull's-eye on the Able target at the rifle range is 12 inches. What is the area of the bull's-eye?
A = 7rr2 A = 3.14(62) A = 3.14(36) A = 113.04

The area is 113.04 square inches. Note: The diameter was used in this example. the radius is 1/2 the diameter. Remember that

Example: During physical training (PT), the squad formed a circle with a radius of 10 feet.

5-21

What is the area of the PT circle?
A A A A = = = = 7rr2 3.14 (102) 3.14(100) 314

The area is 314 square feet. Lesson Summary. In this lesson you studied and applied the formulas used to determine the dimensions of plane geometric figures. In the next lesson you will apply the formulas used to determine dimensions of solid figures. Lesson 3. SOLID FIGURES

LEARNING OBJECTIVES 1. 2. 3. 4. 5. Given the dimensions of solid geometric figures, use the proper formula to find the surface area. Given the dimensions of solid geometric figures, use the proper formula to compute the volume. Given a number, use the proper operations to find the square root. Given the dimensions of two sides of a triangle, use the proper formula to find the dimension of the unknown side. Given the dimensions of two sides of one triangle and the dimensions of one side of another triangle, use the proper operations to find the dimensions of the missing corresponding side. Surface Area and Volume of Solid Geometric Figures

5301.

a. General. A more appropriate title for this next group of geometric figures might be three dimensional figures but they are called solid geometric figures. Even though they may be hollow, there are points in this "hollow" area that help to form the solid figure. The unit of measure for a solid figure is a cube, 1-unit of length on each edge (fig 5-16). When you find the volume of a solid figure, you are finding out how many 1-unit cubes or cubic units are contained in the figure. Notice that a third dimension has been added.

5-22

In plane figures, on the other hand, you dealt with base and height or length and width. You now have length(l), width(w), and height(h) contained in one figure.

I
1

I~~~~~~

II W

Fig 5-16.

One unit cube.

Cube. The familiar cube is a geometric figure with six b. square faces. The edges all have the same measure. The sum of the areas of the six faces is called the surface area. Example: The cube illustrated represents a company supply There are 6 faces, each measuring 4 feet by 4 box. What is the surface area (SA) for the feet (4 x 4). box shown below?

4 4

SA = 6(face2
SA = 6(4 2)

SA = 6(16) SA = 96

The surface area equals 96 square feet. To find the volume (V) of a cube you must find the number of unit cubes it contains.

5-23

Example: 44

If you constructed a cube as shown above and then counted the unit cubes, it would not take long to generalize the formula for the volume of a cube: V = lwh. Remember, though, that a cube has the unique characteristic of having square faces that are all the same size. The l,w, and h (the edges) are the same. Consequently, the formula can be changed to V = f x f x f or V = f . Example: Suppose your mission requires you to transport several supply boxes (4' x 4' x 4') using a 5 ton truck. To do this you must know how many cubic feet the 5 ton truck can hold and how many cubic feet are in each supply box. Note: In this example, we'll figure the cubic feet for only one box. V
=f3

V= V =

43 64

The volume is 64 cubic feet. Let's try another example. Example: The figure illustrated represents a cube shaped storage box that you will use to store equipment during the training exercise. Find the volume of the storage box.

12 12 12~~1 V =f V = 123 V = 1728 The volume is 1728 cubic feet. 5-24

Let's test your memory. Do you recall the formula for computing the surface area? If your answer was SA = 6(face ), you're correct! How about finding the volume? If you said, V = f If you had a problem using these formulas, you're correct again! review this section again before moving on. b. Rectangular solids. Rectangular solids are similar to cubes except that their faces are not necessarily squares. The sum of the area of the six sides equals the surface area, so the surface area is found by adding the area of the two ends, the two sides, the top and the bottom. The formula for finding the surface area Volume is found by is: SA = 2(h x w) + 2(h x 1) + 2(w x 1). using the original formula that was presented in discussing the
cube: V = lwh.

Example: Find the surface area and the volume of a milvan (a milvan is large steel box) 20 feet long 8 feet wide 8 feet high.
2(h x w) + 2(h x 1) + 2(w x 1) SA = 2(8 x 8) + 2(8 x 20) + 2(8 x 20) SA = SA = 2(64) + 2(160) + 2(160) SA = 128 + 320 + 320 SA = 768

The surface area is 768 square feet. V = lwh
V = 20 x 8 x 8 V = 1,280

The volume of the milvan is 1,280 cubic feet. Now, what is the volume in cubic inches? There are 1,728 cubic inches in 1 cubic foot. To obtain cubic inches, simply multiply the total cubic feet by 1,728. Example:
cu in = 1,728 x cu ft cu in = 1,728 x 1,280 cu in = 2,211,840

The volume is 2,211,840 cubic inches. Let's look at another problem.

5-25

Example: Pvt. Layaway has a box of uniforms stored in supply. The box is 30 inches high 27 inches wide and 48 inches long. What is the volume of the box? V = lwh
V = 48 x 27 x 30 V = 38,880

The volume is 38,880 cubic inches. Let's change the cubic inches to cubic feet. Remember, there are 1,728 cubic inches in 1 cubic foot. Example:
cu ft = cu in + 1,728 cu ft = 38,880 - 1,728 cu ft = 22.5

The volume is 22.5 cubic feet. Remember, to compute the surface area for a rectangular solid, Now, what the formula is SA = 2(h x w) + 2(h x 1) + 2(w x 1). is used to compute the volume of a rectangular solid? formula That's correct, it is V = lwh. Make sure you can use these formulas before moving on. Cylinder. This geometric figure is best typified by a tin c. can. There are two kinds of cylinders, right cylinders and oblique cylinders (fig 5-17). Since we rarely see oblique cylinders, this discussion will be limited to right cylinders.

RIGHT CYLINDERS Fig 5-17.

OBLIQUE CYLINDERS Cylinder.

A right cylinder has two equal circular bases and a side or lateral surface perpendicular to the bases. We are interested in three facts about cylinders: lateral surface area, total surface area, and volume. All three are based on principles that have been previously discussed. Lateral surface area does not include the top and bottom of the cylinder. 5-26

If the cylinder is cut vertically and opened up, it will form a rectangle with h for one dimension and 2vr (the circumference of the top or bottom) for the other. Lateral surface, then, would be equal to 2frh. Example: Find the lateral surface area (LSA) of a cylinder with a radius of 7 inches and a height of 14 inches.
LSA = 27rrh LSA = 2 x 3.14 x 7 x LSA = 615.44 14

The lateral surface area is 615.44 square inches. Total surface area is equal to the lateral surface plus the area of both bases (top and bottom). The formula is TSA = 27rrh + 2(7r2). Through algebraic manipulation, this can be simplified
to TSA = 2vr(r + h).

Example: Find the total surface area of a cylinder; the radius is 7 inches and the height 14 inches.

2 Tr (7)
14

TSA TSA TSA TSA

= = = =

27r(r + h) 2 x 3.14 x 7(7 + 43.96(21) 923.16

14)

The total surface area is 923.16 square inches. Now that you have found the total surface area, let's find the volume of a cylinder. The procedure is comparable to the one used to compute the volume of a rectangular solid. Remember that, when using lwh, you are finding the area (1w) of the base of a rectangle. Now, multiply the area by the number of units of the height(h). To apply this formula to cylinders, the area of the base is orr. Therefore, to find the volume of a cylinder use V = 7rr2h.

5-27

Example: Find the volume of a cylinder. The radius is 7 inches and the height is 14 inches.
V =
7ir2h

V = 3.14 x 72 x 14
V = 3.14 x 49 x 14 V = 2154.04

The volume is 2154.04 cubic inches. What is the formula used to find the lateral surface area for a cylinder? You're right, it's LSA = 27rh. How about total surface area? Yes, it's TSA = 2fr(r + h). Now what about the volume of a cylinder? of course, it's V = ir2h. 5302. Square Root and Triangles

a. General. Back in study unit 3, in the discussion of proportion, Cpl Chainsaw discovered a method for finding the height of a tree by measuring its shadow and relating it to the height and the length of the shadow of some known object. Cpl Chainsaw used a proportion to solve his problems, and unknowingly he was applying a principle that involves the relationship between sides and angles of similar triangles. There are several relationships between triangles, particularly right triangles, that do not involve difficult mathematics but which can be of use to you in many situations. One is the earlier mentioned relationship of similar triangles; the other is the Pythagorean Property. The pythagorean law or property is a basic tool in geometric and trigonometric (relating to trigonometry) calculations. It has been known since approximately 500 BC when the Greek philosopher-mathematicianPythagoras proved the relationship of the sides of a right triangle. The scuttlebutt is that one day Pythagoras was staring at a mosaic floor. The patterns so intrigued him that he started breaking them up in his mind's eye into right triangles. He noticed that when a square was constructed on each side of a triangle that the areas of the squares had a particular relationship (fig 5-18).

5-28

Fig 5-18.

Visions of Pythagoras.

What Pythagoras noticed was that if he added the squares of the two shorter sides of a right triangle, the sum was equal to the square of the long side (hypotenuse) of the triangle. This is symbolized in an equation as a2 + b2 = c2 where a and b are the lengths of the sides of a right triangle and c is the hypotenuse. This relationship was mentioned at the beginning of this study unit. In ancient times, there was a group of Egyptian surveyors called "rope stretchers." These Egyptian surveyors used the Pythagorean theorem applied to a piece of rope to establish right angles or corners or property. Their rope had knots tied at even intervals. Using the relationship 3:4:5, they could form a right triangle and a perfect 9 0 0 angle (fig 5-19).

5 3

Fig 5-19.

Knots in rope used to form a 909 angle.

Before you can solve problems using the Pythagorean property, you must know how to find the square root of a number. Let's see how this is done.

5-29

b. Sauare root. The square root of any number (x), is that number which, when multiplied by itself, gives (x). It can be found in several ways. Some square roots are used so often that they become a matter of common knowledge just as the addition and multiplication tables. For example 144 is the square of 12, which makes 12 the square root of 144 (12 x 12 = 144). The squares of integers from 1 to 20 are used enough to be memorized. If you know the squares, then consequently you know the square root of these squares (fig 5-20). n 1 - ------------------3 -------- ----- ---4 ---------- -----5 6 7 8 9 ____________---10

n2 1 9 16 25 36 49 64 81
100

11 12 13 14 15 16 17 18 19 20 Fig 5-20.

121 144 169 196 225 256 289 324 361 400
-

Squares of the numbers (n) from 1

20.

There are tables in many textbooks that give the square root of every number up to 1,000, some go even higher. When tables are not available, the computational method is used. There is no "new" method for computing square root, just a fairly rigorous method that has been used for years but needs practice to be maintained. Example: What is the square root of 1156? (1) The square root symbol (V) over the number is called a radical sign. (2) Starting at the decimal point, group the digits in pairs. Vii 56.

5-30

(3) There will be one digit in the answer for every group under the radical. Over the first group, put the largest number whose square does not exceed 11. For example, three (3) squared is 9, 42 is 16. The largest that can be used is 3. Square 3 and write the product under the first group. 3 11l 56
9

(4) Subtract the 9 from 11 and bring down the next group. 3 -Vll 56
9

2 56 (5) Double the part of the root already found (3 doubled is 6) and add a zero to form a trial divisor. Place this next to 256. 3 Vll 56
9

256

(6) Divide 256 by the trial divisor (60) and write the quotient (4) above the second group as the probable second figure of the root. 3 4 V11 56
9

256

(7) Correct the trial divisor (60) by adding to it the second number of the root (4). Then multiply the corrected divisor (64) by the second number of the root (4). Next, subtract the product from the remainder and write any remainder that's left over below it. In this case, it comes out even. The square root of 1156 is 34. This can be checked by squaring 34. 3 4 Vll 56
9

-6G

256

64 Check:

256

34 x 34 = 1,156

The square root of 1,156 is 34. 5-31

What happens when the square root does not come out even? find out. Note:

Let's

Some people like to keep the "square" whole and bring down each group of numbers as needed. Your next example demonstrates this technique. What is the square root of 2. We know that 2 is not a perfect square and (1) decide to carry the calculation to three decimal places. This means that you must add three groups of zeros. The decimal point is placed directly over its original place. V2.000000 (2) The highest number whose square does not exceed Square 1 and write The square of 1 is 1. 2 is 1. Subtract 1 from the product under the first group. 2 and bring down the next group.
1.

Example:

V2.000000
1 1 00

Double the part of the root found (1 doubled is (3) 2) and add zero to form a trial divisor of 20. Place this next to 100.
1.

V2.000000
1

100

It goes Divide 100 by the trial divisor 20. (4) 5 times, but 5 will not work. Why? evenly Remember, you must take the trial divisor (20) and You add it to the second number of the root (5). then multiply the corrected divisor (25) by the So, 25 x 5 = 125. second number of the root (5). This exceeds 100; therefore, 5 cannot be used as Place this the quotient. The correct number is 4. as the second figure in the root and add it to the trial divisor. 1. 4 V2.000000
1 2-G 100

24 5-32

(5) Multiply the corrected divisor (24) by the second number of the root (4) and subtract the product from 100. Write any remainder left below and bring down the next group. 1. 4 V2.000000
1 2-G 100

96 400

(6) Again, double the part of the root already found (14 doubled is 28) and add a zero to form a trial divisor (280). Place this next to 400. Divide 400 by the trial divisor (280). Write the quotient (1) above the next group as the probable third figure of the root (1 x 280 = 280); this does not exceed 400. Next, correct the trial divisor (280) by adding to it the third figure of the root (1). Multiply the corrected divisor (281) by the third number of the root (1). Subtract the product (281 x 1 = 281) from the remainder and write any remainder left below. Bring down the next group. 1. 4 1 12.000000
1 2-G 100

24 96 28e 400 281 281
11900

(7) Double the part of the root already found (141 doubled = 282) and add a zero to form a trial divisor (2820). Place this next to 11900. Divide 11900 by the trial divisor (2820). Write the quotient (4) above the next group as the probable fourth figure of the root (4 x 2820 = 11280) which does not exceed 11900. Place the 4 as the fourth figure of the root. Correct the trial divisor (2820) by adding to it the fourth figure in the root (4).

5-33

Multiply the divisor by 4. 1. 4 1 4 V2.000000
1 -G 100

24 96 28-G 400 281 281
2-82 11900

2824 11296 604 Note: You would then subtract the product (2824 x 4 = 11296) from the remainder (11900) and be left with a remainder of 604. Since you only need to find the root to three places, no further computations are necessary. So, the square root of 2 is 1.414. It could be carried further. Keep in mind that if you square 1.414, you will not get 2. It will be close though. The more decimal places that the root is carried, the closer to 2 you will get when you square the root. For our purposes, three places is close enough. Let's try one more example. Example: Find the square root of 912.04. (1) Starting at the decimal point, group the digits in pairs. Notice that the left-most group (9) has only one digit. Note: To group each problem with or without a space between each group is a matter of personal preference. As you become more experienced with square roots, you may decide not to place a space between each group as noted with the example below. -9 12. 04 Start with the first group under the radical, (2) in this case (9). Over the first group, put the largest number whose square does not exceed 9. The highest number whose square does not exceed 9 is 3. Square the 3 and subtract it from 9.

5-34

Bring down the next group. Double the part of the root already found (3 doubled is 6) and add a zero to form a real divisor (60). Notice however, that this trial divisor (6) is too large for 12. 3 V912.04
9

(3) In this case, there is no trial divisor so you must place a zero as the second figure of the root and bring down the next group. Double the part of the root found (30 doubled is 60) and add a zero to form a trial divisor of 600. This is an important step that is often overlooked. Divide 1204 by the trial divisor (600). Write the quotient (2) above the next group as the probable third figure of the root. Correct the trial divisor (600) by adding to it the third figure in the root (2). Multiply the corrected divisor (602) by the third number of the root (2). 3 0. 2 V912.04
9

6-G 12 04 6-@G 12 04 602 Subtract the product from the remainder and write any remainder left over. In this case the answer comes out evenly. The square root of 912.04 is 30.2. This can be checked by squaring 30.2. Check: 30.2 x 30.2 912.04 The square root of 912.04 is 30.2. You have just reviewed three detailed examples on how to compute the square root of any number. If you had a difficult time with any of these examples, review this section before moving on to right triangles. c. Unknown side of triangles. Now that we have discussed the principles of the pythagorean property and the computation of square roots, let's put them together to solve problems involving the sides of right triangles. Just for review, the pythagorean property stated in words is: The sum of the squares of two sides of a right triangle equals the square of the hypotenuse.

5-35

Example: Two sides of a right triangle are 9 inches (a) and Find the length of the hypotenuse 12 inches (b). (third side).
a2 +
b2 = c2

+ 12 2 = c 2
225 = c2

81 + 144 = c2

Note:

To find c you must take the square root of
c. 2

225 = c2

15 = c

The hypotenuse is 15 inches.

Let's try another example. Example: The illustration below represents a softball field. What is the distance from home plate (HP) to second base (2)? The bases are 60 feet apart. 2

,>~~~6
c TIP
C2 2 c

a2 2 60 +

b2 = 2 =

3600 + 3600 = c2
7200 = c2

V7200 = c
84.85 = c

The distance from home plate to second base is 84.85 feet. The last example shows you that if the hypotenuse of a right triangle is known, the unknown side can be found. Let's see how this is done.

5-36

Example: The hypotenuse of a right triangle is 13 feet and one side is 5 feet. Find the third side. a
2

2 a + 5 = 13 a2 + 25= 169 2 a = V144 a = 12

+ b

The third side is 12 feet long. You have just used the proper formulas to compute unknown distances of right triangles. Next, you will solve unknown dimensions of similar triangles. d. Similar triangles. If the corresponding angles of two triangles are equal, the triangles are said to be similar. The triangles do not have to be the same size or have sides the same length; only the angles need to be the same (fig 5-21). E B

450

Fig 5-21.

Similar triangles.

In similar triangles, corresponding sides are proportional. If two sides of one triangle and one side of another triangle are known, the length of the missing corresponding side can be found. This can be illustrated by referring to the problem concerning the height of the tree in study unit 3. Example: In the illustration provided (fig 5-22), the tree and the individual measured are assumed to be at right angles to the ground. The sun hits both the tree and individual at the same angle. Since the angles are the same measure, the sides are proportional. The lengths of the shadows correspond to the sides of a triangle. The length of the other side is the height of the object (tree or individual). Therefore, a proportion is set up and you can find the missing dimension. To restate, if corresponding angles of two triangles are equal, the triangles are similar and the corresponding sides are proportional. 5-37

4C~~~~~~~~~~L

72in )

. /

~~~48in

6ft

4ft

40ft

Fig 5-22.
4 =

Illustration of similar triangles.

6
x
240 240 4 60

40
4x = 4x = 4 x =

The tree is 60 feet tall. Let's work another problem that's a little more difficult.

5-38

Example: Find side x if triangle ABC is similar to triangle DEF. B

Triangle ABC
7 = 15

Triangle DEF
7 =

x
28 15

x
15x 15x
=
_

28
196 196 or

15 15
x
=

15x = 196 15x = 196 13 1/15

15 x
=

15
13 1/15

Side x equals 13 1/15. Let's try another problem that's a little more difficult. Example: Find the missing sides in triangle ACF and triangle ADG. D

l14
8
F G

13
A 12 E

Since angle A is part of each triangle and each has a right angle, the third angle also must be equal. Therefore the triangles are similar.

5-39

The three sides of triangle ABE are given as are one side of triangle ACF and triangle ADG, so proportions can be set up to find the missing sides.
13 = AC C

5
5AC = 5AC =

8
104

5
AC

104 5
F

=20.8
A C

12 = AF 5 8 5AF 96
5AF
_96

20 F
A 19.5

5 5 AF = 19.2
13 = AD

14
14

5AD = 182 5AD = 182

5 5 AD= 36.4

A D

12
5 5AG
5AG

AG
14 168
= 168

36.4 A 33.6

5
AG
=

5
33.6

As you can see, finding the missing sides is easy using proportions. Lesson Summary. In this lesson you calculated the surface area and volume of a cube, rectangular solid, and cylinder. You also used the proper operations to find the square root of a number, the unknown side of right triangles as well as the length of triangles. You should now be ready to complete the unit exercise.
-------------------------------------------------------------__-

5-40

Unit Exercise:

Complete items 1 through 64 by performing the action required. Check your responses against those listed at the end of this study unit.