Math For The Trades_Penn Foster's Examination Number. 086097

Front 1

Introduction

What most skilled trades work requires?

Back 1

Most skilled-trades work requires a strong working knowledge of practical mathematics.

Front 2

Introduction

How this study unit will supports you in mathematics?

Back 2

This study unit will reinforcewhat you already know about math and will show you howto apply your math skills to solve real-world trades-relatedproblems.

Front 3

Introduction

This study unit deals with what?

Back 3

In this study unit, You’ll encounter a method of expressing part of a whole number—fractions.

Front 4

Introduction

Why it is required to study fractions?

Back 4

Not all things are measured in wholes. Therefore, we have a way of mathematically dealing with partial quantities. That’s where fractions come in. For example, in electrical work you’ll find heaters with ratings expressed in watts, and on blueprints you’ll often find measurements that include fractions such as 3⁄4 inch. In fact, a good working knowledge of fractions is a must for any trade.

Front 5

Introduction

What will you learn in this study unit?

Back 5

In trades work you’ll often have to measure things to find their lengths, areas, volumes, angles, and other dimensions.The most common measurement involves distance or length.In this study unit, you’ll learn about units of length and angles, as well as methods of measuring lengths and angles.

In this study unit you’ll also be introduced to the concept of solving problems with formulas. You’ll learn how to set up formulas to find a needed quantity and how to solve the formula, once it’s set up.

Front 6

Introduction

When you complete this study unit, you'll be able to?

Back 6

1-Simplify, add, subtract, multiply, and divide fractions

2-Change fractions to decimals and decimals to fractions

3- Solve problems involving percent, ratios, and proportions

4-Measure using both English and metric units of length

5-Calculate the areas and perimeters of commonly encountered shapes

6-Use a protractor to measure angles

7-Explain the use of variables in formulas

8- Prepare and use formulas to solve problems

9-Work with square roots

Front 7

Contents

What are the main headings in this study unit?

Back 7

1-FRACTIONS

2-OPERATIONS WITH FRACTIONS

3-DECIMALS AND PERCENTS

4-RATIOS AND PROPORTIONS

5-MEASUREMENTS AND UNITS OF LENGTH

6-DEVICES FOR MEASURING DISTANCE

7-ANGLES

8-SIMPLE FORMULAS

Front 8

Fractions

What are the subheadings of fractions?

Back 8

What Is a Fraction?

Proper and Improper Fractions

Reading Fractions

Uses of Fractions

Equivalent Fractions

Reducing Fractions

Solving a Simple Industrial Problem

Changing Improper Fractions to Mixed Numbers

Changing Mixed Numbers to Improper Fractions

Front 9

Fractions_What is a Fraction?

Define what is a fraction?

Back 9

The word fraction means “broken.” A fraction represents partof a whole that’s been broken into equal-size pieces.

Front 10

Fractions_What is a Fraction?

Give an example of a fraction from every day life?

Back 10

Fractions crop up so often in our everyday dealings thatsometimes we use them without really thinking about them.Even small children use fractions. When one child has a cookie, for example, and a friend wants part of the cookie,the child must break the cookie in half in order to share it.Each child will get one half (1⁄2) of the cookie, and 1⁄2 is afraction.

Front 11

Fractions_What is a Fraction?

How fractions are written?

Back 11

All of these ways are correct. It doesn’t matter which way you write your fractions. As long as you write the numerator,then a line, then the denominator, you’ll be correct.

Front 12

Fractions_What is a Fraction?

Describe the fraction 1/5.

Back 12

Let’s take a look at the fraction 1⁄5 (one fifth). The top number(1) is the numerator, and the bottom number (5) is thedenominator. The line that separates the two numbers isthe fraction bar. The.

Whole 1 is divided in to five equal parts where 1/5 is the 5th equal part of this division.

Front 13

Fractions_What is a Fraction?

Define the role of numerator and denominator.

Back 13

The denominator tells you how many equalparts the whole unit is divided into. In the fraction 1⁄5, the denominator of 5 tells you that the whole has been dividedinto 5 equal parts. The numerator tells you how many ofthese equal parts are represented by the fraction. The fractionrepresents one of the five parts.

Front 14

Fractions_What is a Fraction?

Graphically represent the fraction 1/5.

Back 14

Front 15

Fractions_Uses of Fractions

Graphically represent the fraction 2/5.

Back 15

Front 16

Fractions_Proper and Improper Fractions

Graphically represent the fraction 3/5.

Back 16

Front 17

Fractions_Proper and Improper Fractions

Graphically represent the fraction 4/5.

Back 17

Front 18

Fractions_Proper and Improper Fractions

Graphically represent the fraction 5/5.

Back 18

Front 19

Fractions_Proper and Improper Fractions

Define the kinds of fractions.

Back 19

There are two kinds of fractions: proper and improper

Another kind of fraction is a mixed fraction.

Front 20

Fractions_Proper and Improper Fractions

Define what is proper fraction?

Back 20

If the numerator of a fraction is less than its denominator, then the fraction is less than 1 and is called a proper fraction.

Front 21

Fractions_Proper and Improper Fractions

Define what is improper fraction?

Back 21

If thenumerator is equal to or greater than its denominator, thefraction is an improper fraction. If the numerator of a fractionequals its denominator, the fraction equals 1. If the numeratoris greater than the denominator, the fraction represents anamount greater than 1.

Front 22

Fractions_Proper and Improper Fractions

Graphically represent the fractions 2/3 also tell if it is a proper or improper fraction?

Back 22

Front 23

Fractions_Proper and Improper Fractions

Graphically represent the fractions 5/3 also tell if it is a proper or improper fraction?

Back 23

Front 24

Fractions_Proper and Improper Fractions

Graphically represent the fractions 3/3 also tell if it is a proper or improper fraction?

Back 24

Front 25

Fractions_Proper and Improper Fractions

Graphically represent the fractions 1 1/3 also tell if it is a proper or improper fraction?

Back 25

Front 26

Fractions_Proper and Improper Fractions

Define mixed fractions.

Back 26

A third type of fraction that you’ll encounter is a mixed fraction,which consists of both a whole number and a fraction, such as 1 1⁄3.

Front 27

Fractions_Reading Fractions

How to read fractions?

Back 27

To read a fraction properly, simply say the number in the numerator, and then say the number of parts into which it’s been divided (the denominator).

Example: 1/5 one fifth.

Front 28

Fractions_Reading Fractions

How to read fractions with large amounts?

Back 28

Sometimes, you may be trying to read a fraction with a very large denominator, such as 95 ⁄ 167. Instead of reading “ninety five one hundred sixty-sevenths,” it’s generally easier and clearer to read “ninety-five over one hundred sixty-seven.”

Front 29

Fractions_Reading Fractions

Give some examples, how to read fractions

Back 29

1/2 one half. 1/3 one third, 2/3 two third, 1/4 one fourth, 3/4 three fourths, 1/5 one fifth, 1/6 one sixth, 1/7 one seventh, 1/8 one eighth, 1/9 one ninth, 1/10 one tenth, 1/12 one twelfth, 1/13 one thirteenth, 1/14 one fourteenth, 1/15 one fifteenth, 1/16 one sixteenth, 1/17 one seventeenth, 1/18 one eighteenth, 1/19 one nineteenth, 1/20 one twentieth, 1/30 one thirtieth, 1/100 one hundredth, 1/1000 one thousandth

Front 30

Fractions_Uses of Fractions

What are the uses of fractions?

Back 30

Fractions have three common uses in everyday mathematics:

1. To stand for part of one whole thing

2. To stand for part of a group

3. To show division

Front 31

Fractions_Uses of Fractions

Describe the first use of fractions "To stand for part of one whole thing"

Back 31

This illustrationshows a screw being measuredwith a ruler. The topedge of the ruler showsinches, and each inch isdivided into eight equalparts, or eighths. You cansee that this screwmeasures 5⁄8 inch long.

Front 32

Fractions_Uses of Fractions

Describe the second use of fractions "To stand for part of a group"

Back 32

Imagine that your desk drawer contains five tools,as shown below. Three of the tools are calculators, and twoare calipers. What fraction of the tools are calculators?

The answer is three-fifths, or 3 ⁄ 5. The fraction 3 ⁄ 5 stands for thepart of the tools that are calculators.

Front 33

Fractions_Uses of Fractions

Describe the third use of fractions "To show division".

Back 33

When you write a fraction by placing one number overanother number with a line between them, you’re actuallyindicating a division problem.

Front 34

Fractions_Uses of Fractions

How to read following fractions if they are the division problems?

12/3, 1/4, 5/5

Back 34

In these fractions, the denominator of the fraction becomesthe divisor in the division problem, and the numeratorbecomes the dividend. So, the fraction 12⁄3 can be read as“12 divided by 3,” the fraction 1⁄4 can be read as “1 dividedby 4,” and the fraction 5⁄5 can be read as “5 divided by 5.”

Front 35

Fractions_Equivalent Fractions

Define equivalent fractions with example?

Back 35

Sometimes two fractions have very different numerators anddenominators, but still have the same value. For example,consider the fractions 32⁄64, 16⁄32, 4⁄8, and 8⁄16. Although thesefour fractions look very different, they all represent exactlythe same value. In other words, these fractions are said to beequivalent, or equal. Equivalent fractions may have differentnumerators and denominators, but represent the same value.

Front 36

Fractions_Equivalent Fractions

Show graphical representation of the following values. 32/64, 16/32, 4/8, 8/16,

Back 36

The two rulers have marking for eighths, sixteenths, thirty-seconds, and sixty-fourths. Thevertical line shows that the fractions 4 ⁄ 8, 8 ⁄ 16, 16 ⁄ 32, and 32 ⁄ 64 all represent the same length on the rulers.

Front 37

Fractions_Equivalent Fractions

If you multiply the numerator and denominator of a fraction by the same number then what change will occur.

Back 37

If you multiply the numerator and denominator of a fraction bythe same number, the value of the fraction stays the same.

Example:

4/8 = 4 x 2 / 8 x 2 = 8 / 16

4/8 = 4 x 4 / 8 x 4 = 16 / 32

8/16 = 8 x 4 / 16 x 4 = 32 / 64

From these examples, you can see that the fractions 32⁄64, 16⁄32,4⁄8, and 8⁄16 are all equivalent fractions that represent the exactsame value.

Front 38

Fractions_Equivalent Fractions

How can you multiply the numerator and denominator by the same number and still have the same value?

Back 38

It's easy to understand if you remember two rules:

• Any number divided by itself equals 1.

• Any number multiplied by 1 equals that same number.

When you multiply the numerator and denominator by the same number, say 5, you’re reallymultiplying by 5 ⁄ 5. The fraction 5 ⁄ 5 means 5 divided by 5, or 1. Therefore, when you're multiplying by 5 ⁄ 5,you're really multiplying by 1—and when you multiply by 1, you don't change the value of thenumber being multiplied.

Front 39

Fractions_Equivalent Fractions

Define what is "Reducing Fractions"?

Back 39

If you divide the numerator and denominator of a fraction bythe same number, the value of the fraction stays the same. when you use division to determine equivalentfractions, the numbers become smaller. Using division to findan equivalent fraction is called reducing a fraction. Reducingfractions makes them easier to understand and to work with.

Front 40

Fractions_Equivalent Fractions

Reduce the fraction 18 / 54.

Back 40

18/54 = 18 ÷ 2 / 54 ÷ 2 = 9 / 27

9/27 = 9 ÷ 9 / 27 ÷ 9 = 1 / 3

Front 41

Fractions_Equivalent Fractions

Define what is lowest term of a fraction?

Back 41

When a fraction can’t be reduced anyfurther, it’s said to be in its lowest terms.

Front 42

Fractions_Equivalent Fractions

Reduce following fractions in to their lowest terms.

9/24, 21/77, 20/24

Back 42

9/24= 9 ÷ 3 / 24 ÷ 3 = 3 / 8

21/77= 21÷ 7 / 77 ÷ 7 = 3/11

20/24 = 20 ÷ 4 / 24 / 4 = 5/6

Front 43

Fractions_Equivalent Fractions

To convert a fraction in to an equivalent fraction that has a specific denominator what should do?

Back 43

1. Look at the two fractions. Divide the larger denominatorby the smaller denominator.

2. Multiply the resulting quotient by the first numerator.

The answer is the numerator of the second fraction.

Front 44

Fractions_Equivalent Fractions

Example: The fraction 2⁄3 is equal to what fraction with a denominator of 21?

Back 44

Setup:

2 / 3 = ? / 21

Divide the larger denominator (21) by thesmaller one (3). The result is 7.

21 ÷ 3 = 7

Multiply the resulting quotient (7) by thenumerator of the first fraction (2).

7 × 2 =14

The answer (14) is the missing numeratorof the second fraction. Therefore, thefraction 2⁄3 is equal to 14⁄21.

2 / 3 = 14 / 21

Front 45

Fractions_Equivalent Fractions

Example: What fraction with a denominator of 24 is equal to 5 ⁄ 12?

Back 45

Setup:

5 / 12 = ? / 24

Divide the larger denominator (24) by thesmaller one (12). The result is 2.

24 ÷ 12 = 2

Multiply the resulting quotient (2) by thenumerator of the first fraction (5).

2 × 5 =10

The answer (10) is the missing numeratorof the second fraction. Therefore, thefraction 5⁄12 is equal to 10⁄24.

5 / 12 = 10 / 24

Front 46

Fractions_Solving a simple Industrial problem

Is 1 ⁄ 16 inch smaller or larger than 3 ⁄ 64 inch?

Back 46

However, if you change the fractions so they both have the same denominator, you’ll be able tosee which fractional measurement is larger. Since thefraction 3⁄64 is already in its lowest terms, we can’t reduce it tosixteenths. Therefore, we’ll have to change the fraction 1⁄16 tosixty-fourths. Thus, you’ll need to find the fraction that’sequal to 1⁄16, but that has the denominator of 64.

1 / 16 = ? / 64

64 ÷ 16 = 4

4 ×1 = 4

1 / 16 = 4 / 64

Youcan see that 4⁄64 inch is larger than 3⁄64 inch, so the original1⁄16-inch fillet is greater than the 3⁄64-inch fillet specified.

Front 47

Fractions_Changing Improper Fractions to Mixed Numbers

Define what is improper fraction?

Back 47

You’ll recall that an improper fraction is one in which thenumerator is larger than the denominator—for example, 64 ⁄ 7.

Front 48

Fractions_Changing Improper Fractions to Mixed Numbers

Define what is mixed number?

Back 48

A mixed number is a value that has a whole number and afraction together. For example, the mixed number 11⁄2 is read“one and one-half.” The 1 is the whole number part, and the1⁄2 is the fraction part.

Front 49

Fractions_Changing Improper Fractions to Mixed Numbers

How to change an improper fraction to a mixed number?

Back 49

1. Divide the numerator of the fraction by the denominator.The resulting quotient is the whole-number part of youranswer.

2. Write the remainder as the numerator of the fractionpart of your answer.

3. Write the divisor as the denominator of the fraction partof your answer.

Front 50

Fractions_Changing Improper Fractions to Mixed Numbers

Example: Change the improper fraction 64 ⁄ 7 to a mixed number by using division.

Back 50

1- Set up

2- Carry out the division. You get a quotientof 9, and a remainder of 1.

3- Write the quotient (9) as the whole-numberpart of your answer. Write the remainder (1)as the numerator of the fraction part ofyour answer. Write the divisor (7) as thedenominator of the fraction part. Youranswer is 91⁄7.

Front 51

Fractions_Changing Improper Fractions to Mixed Numbers

Example: Change the improper fraction 15 ⁄ 2 to a mixed number.

Back 51

Front 52

Fractions_Changing Mixed Numbers to Improper Fractions

How to change mixed numbers to improper fractions?

Back 52

To change a mixed number to an improper fraction, you’llneed to reverse the procedure you just used. This meansthat you’ll use multiplication, which makes sense becausemultiplication is the inverse, or opposite, of division.

Front 53

Fractions_Changing Mixed Numbers to Improper Fractions

Example: Change 91⁄7 to an improper fraction.

Back 53

1-Setup

2- Multiply the denominator of the fraction(7) by the whole number (9). You get aproduct of 63.

3- Add the numerator of the fraction (1)to the product obtained in the precedingstep (63).

4- Place the sum (64) over the denominatorof the fraction (7). The answer is 64⁄7.The mixed number 91⁄7 is equal to theimproper fraction 64⁄7.

Front 54

Fractions_Changing Mixed Numbers to Improper Fractions

Example: Change the mixed number 71⁄2 to an improper fraction.

Back 54

Front 55

Fractions_Self Check-1

Match the fractions in the left-hand column with their equivalent fractions in the right-hand column.

1. 1/2 a. 9/12

2. 3/4 b. 4/12

3. 5/6 c. 15/24

4. 12/20 d. 8/16

5. 5/8 e. 3/5

6. 1/3 f. 10/12

Back 55

1. 1/2 = 8/16

2. 3/4 = 9/12

3. 5/6 = 10/12

4. 12/20 = 3/5

5. 5/8 = 15/24

6. 1/3 = 4/12

Front 56

Fractions_Self Check-1

7. Describe the three common uses of fractions.

Back 56

1. To stand for part of one whole thing

2. To stand for part of a group

3. To show division

Front 57

Fractions_Self Check-1

8. Multiplying the numerator and the denominator of a fraction by the same number is the same as multiplying the fraction by the whole number _______.

Back 57

Ans: 1

Front 58

Fractions_Self Check-1

9. Reduce the following fractions to their lowest terms.

a. 9/12

b. 8/12

c. 15/20

d. 9/18

e. 24/30

f. 25/40

Back 58

a. 9/12 = 9 ÷ 3 / 12 ÷ 3 = 3 / 4

b. 8/12 = 8 ÷ 4 / 12 ÷ 4 = 2 / 3

c. 15/20 = 15 ÷ 5 / 20 ÷ 5 = 3 / 4

d. 9/18 = 9÷ 9 / 18÷ 9 = 1 / 2

e. 24/30 = 24÷ 6 / 30÷ 6 = 4 / 5

f. 25/40 = 25÷ 5 / 40÷ 5 = 5 / 8

Front 59

Fractions_Self Check-1

10. Indicate whether the following fractions are proper or improper.

a. 1 ⁄ 10

b. 16 ⁄ 14

c. 25 ⁄ 20

d. 9 ⁄ 13

e. 6 ⁄ 6

f. 14 ⁄ 11

Back 59

a. 1 ⁄ 10 = Proper

b. 16 ⁄ 14 = Improper

c. 25 ⁄ 20 = Improper

d. 9 ⁄ 13 = Proper

e. 6 ⁄ 6 = proper

f. 14 ⁄ 11 = Improper

Front 60

Fractions_Self Check-1

11. Change the following improper fractions to mixed numbers and express your answer in thelowest terms.

a. 4/3

b. 10/4

c. 12/5

d. 24/18

e. 29/6

f. 55/10

Back 60

Front 61

Fractions_Self Check-1

12. Change the following mixed numbers to improper fractions.

Back 61

a. 8 x 6 = 48, 48 + 1= 49, 49 / 8

b. 5 x 2 = 10, 10 + 3 = 13, 13 / 5

c. 9 x 10 = 90, 90 + 1 = 91, 91 / 9

d. 4 x 8 = 32, 32 + 3 = 35, 35 / 4

e.16 x 6 = 96, 96 + 5 = 101, 101 / 16

f. 32 x 3 = 96, 96 + 3 = 99, 99 / 32

Front 62

Operations with Fractions

What are the meanings of operations with fractions?

Back 62

Operations are the procedures ofadding, subtracting, multiplying, and dividing.

Front 63

Operations with Fractions

Describe the meanings of like fractions.

Back 63

Fractions that have the same denominators are called likefractions. The fractions 2 ⁄ 9, 4 ⁄ 9, 5 ⁄ 9 and 7 ⁄ 9 are examples of likefractions, because they all have a denominator of 9.

Front 64

Operations with Fractions

Describe the meanings of unlike fractions.

Back 64

Fractions that have different denominators are called unlike fractions. The fractions 1⁄2, 3⁄4, 5⁄8, and 7⁄10 are examples of unlike fractions because their denominators (2, 4, 8, and 10)are all different.

Front 65

Operations with Fractions_Adding and Subtracting Like Fractions

How to add and subtract like fractions.

Back 65

1. Add or subtract the numerators. The result is the numerator of your answer. The denominator remains the same.

2. If the answer is an improper fraction, change it to a mixed number.

3. Reduce the fraction part of your answer to its lowest terms.

Front 66

Operations with Fractions_Adding and Subtracting Like Fractions

Example: A board is 3 ⁄ 8 inch thick, and has

a 1 ⁄ 8-inch layer of paint on it, as shown in Figure 7. What’s the total thickness of the board?

Back 66

This illustrations hows the the board and its coating of paint, with the dimensions labeled.As long as fractions have the same denominator,it’s easy to find their sum.Remember, you should always reduce the answer to its lowest terms.

Front 67

Operations with Fractions_Adding and Subtracting Like Fractions

A drawing of a part is shown below. The dimensionsof the part are given in fractions of an inch. What’s thedimension of the part that’s marked with a question mark?

Back 67

Front 68

Operations with Fractions_Adding and Subtracting Like Fractions

Example: The drawing below shows two dimensions of apart, in inches. Find the total length of the side shown bythe question mark.

Back 68

To find this dimension, you must add the fractions 7⁄8 and 5⁄8.

Front 69

Operations with Fractions_Adding and Subtracting Unlike Fractions

Define what are the unlike fraction?

Back 69

The fractions that have different denominators are called unlike fractions. For example, the fractions 1⁄4, 3⁄5,and 7⁄10 are unlike fractions because their denominators(4, 5, and 10) are all different.

Front 70

Operations with Fractions_Adding and Subtracting Unlike Fractions

How to add and subtract unlike fractions?

Back 70

To add and subtract unlikefractions, you must first change the fractions so that theyall have the same denominator. The denominator should bethe lowest common denominator, or LCD, which is the smallestnumber that can be divided (without a remainder) by allof the denominators.

Front 71

Operations with Fractions_Adding and Subtracting Unlike Fractions

Find the lowest common denominator for threefractions 1⁄2, 1⁄4, and 3⁄8, and then convert them to equivalentfractions with that denominator.

Back 71

1. Unlike fractions

2. Finding Least common denominator

3. Changing in to equivalent fraction.

Front 72

Operations with Fractions_Adding and Subtracting Unlike Fractions

Find the lowest common denominator for the fractions 3⁄4, 2⁄5, and 1⁄10, and then convert them to equivalent fractions with that denominator.

Back 72

Front 73

Operations with Fractions_Adding and Subtracting Unlike Fractions

Example: If you worked 1⁄2 hour overtime on Monday and3⁄4 hour overtime on Tuesday, how many hours overtime didyou work in the two days together?

Back 73

Front 74

Operations with Fractions_Adding and Subtracting Unlike Fractions

Example: Suppose you have a 7⁄8-inch piece of rod thatyou’re going to use as a spacer. It must fit flush with the faceof the coupling. When you put it in place, you find that itsticks out 3⁄32 inch, so you’ll need to cut it to fit. Your problemis to find what the final length of the spacer should be.

Back 74

Front 75

Operations with Fractions_Adding and Subtracting Mixed Numbers

Define what is called mixed number?

Back 75

Remember that a mixed number is a value that has a wholenumber and a fraction together. For example, the mixednumber 11⁄2 is read “one and one-half.” The 1 is the wholenumber part, and the 1⁄2 is the fraction part.

Front 76

Operations with Fractions_Adding and Subtracting Mixed Numbers

Define the method to add and subtract mixed numbers.

Back 76

To add or subtract two or more mixed numbers, you’ll needto add the fraction parts of the numbers first, and then thewhole-number parts.

Front 77

Operations with Fractions_Adding and Subtracting Mixed Numbers

Describe adding and subtracting mixed number's method in steps.

Back 77

1. Look at the fraction parts of the mixed numbers. If the fractions are like, you can add or subtract them directly.However, if the fractions are unlike, find the lowest common denominator and change the unlike fractions to equivalent fractions with that denominator.

2. Add or subtract the fractions.

3. Add or subtract the whole numbers.

4. Reduce your answer to lowest terms if necessary.

Front 78

Operations with Fractions_Adding and Subtracting Mixed Numbers

Solve the following question.

Two pieces of pipe, one that’s 31⁄4 inches long andanother that’s 21⁄4 inches long, have been welded together.What’s the total length of the pipe?

Back 78

Front 79

Operations with Fractions_Adding and Subtracting Mixed Numbers

Subtract the mixed number 21⁄7 from 53⁄7.

Back 79

Front 80

Operations with Fractions_Adding and Subtracting Mixed Numbers

The drawing below is a simplified drawing of a pulley with the dimensions given in inches. Find thedimension marked by the question mark.

Back 80

The answer is 11⁄16. The dimension marked by the questionmark in the figure above is 11⁄16 inch.

Front 81

Operations with Fractions_Adding and Subtracting Mixed Numbers

Example: Add the mixed numbers 1 3⁄4 and 3 3⁄4.

Back 81

The answer 4 6⁄4 contains an improperfraction that can be reduced. Changethe fraction part of the answer (6⁄4) tothe mixed number 1 2⁄4. Then, reducethe fraction 2⁄4 to its lowest terms (1⁄2).

Front 82

Operations with Fractions_Adding and Subtracting Mixed Numbers

Imagine that you have a piece of pipe that’s41⁄4 feet long, and you cut a piece 23⁄4 feet long from it. Whatis the length of the pipe that remains?

Back 82

Front 83

Operations with Fractions_Adding and Subtracting Mixed Numbers

How to borrow in subtraction of mixed numbers?

Back 83

If you need to borrow when you’re subtracting fractions, be sure to change the borrowed wholenumber 1 into a fraction that has the proper denominator. For example, if the fractions you’resubtracting are fifths, the borrowed 1 will become 5⁄5. If the fractions are sixths, the borrowed 1will become 6⁄6, and so on.

Front 84

Operations with Fractions_Multiplying Fractions

Is it necessary to change fractions in to like fractions for multiplication?

Back 84

We can multiply both like andunlike fractions without changing their denominators.

Front 85

Operations with Fractions_Multiplying Fractions

Define the method of fractions multiplication in steps.

Back 85

1. Multiply the numerator of the first fraction by thenumerator of the second fraction. The resulting productis the numerator of the answer.

2. Multiply the denominator of the first fraction by thedenominator of the second fraction. The resultingproduct is the denominator of the answer.

3. Simplify the answer if necessary.

Front 86

Operations with Fractions_Multiplying Fractions

Multiply 3⁄4 by 2⁄3.

Back 86

Front 87

Operations with Fractions_Multiplying Fractions by WholeNumbers

How to convert a whole number in to equivalent fraction?

Back 87

Every whole number can be converted to an equivalent fraction,simply by placing the whole number over the number 1.For example, the whole number 5 is equal to the fraction 5⁄1,the number 12 is equal to the fraction 12⁄1, and the number432 is equal to the fraction 432⁄1.

Front 88

Operations with Fractions_Multiplying Fractions by Whole Numbers

How to multiply a whole number by a fraction?

Back 88

Now, suppose that you need to multiply a fraction by a wholenumber. To do this, you’ll simply need to convert the wholenumber to a fraction first, and then use the multiplicationrules you just learned to multiply the two fractions.

Front 89

Operations with Fractions_Multiplying Fractions by Whole Numbers

Multiply the fraction 3⁄5 by 2.

Back 89

Front 90

Operations with Fractions_Multiplying Fractions by Whole Numbers

Suppose you’re going to place three spacers on astud bolt. Each spacer is 3⁄8 inch thick. How much space willthe three spacers take up?

Back 90

Front 91

Operations with Fractions _Using Cancellation When Multiplying

Describe cancellation when multiplying?

Back 91

When you’re multiplying fractions, you can sometimes usea process called cancellation as a shortcut to arrive at thecorrect answer. Remember, though, that the cancellationprocess only works with multiplication. You can never usecancellation when you’re adding or subtracting fractions.

Front 92

Operations with Fractions _Using Cancellation When Multiplying

Multiply the fraction 3⁄8 by the fraction 8⁄11. Use general multiplication method.

Back 92

Front 93

Operations with Fractions _Using Cancellation When Multiplying.

Multiply the fraction 3⁄8 by the fraction 8⁄11. Use cancellation method.

Back 93

Front 94

Operations with Fractions _Using Cancellation When Multiplying.

Multiply the fraction 15⁄16 by the fraction 1⁄3. Use general multiplication method.

Back 94

Front 95

Operations with Fractions _Using Cancellation When Multiplying.

Multiply the fraction 15⁄16 by the fraction 1⁄3. Use cancellation method.

Back 95

Front 96

Operations with Fractions _Using Cancellation When Multiplying.

What to remember for applying cancellation for multiplication?

Back 96

Remember that you can only use cancellationwhen you’re multiplying fractions. Also, when you use cancellation,you must always cancel in pairs. You need to cancelthe numerator from one fraction, and the denominator fromanother fraction. You can’t cancel two numerators, or twodenominators.

Front 97

Operations with Fractions _Using Cancellation When Multiplying.

When should use cancellation method in multiplication?

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Remember also that you’re not required to use cancellationat any time. The idea of using cancellation is to take a shortcutthat makes solving a fraction multiplication problemeasier, and to avoid having to reduce the answer at the end.

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Operations with Fractions_Using Cancellation When Multiplying.

Multiply the fraction 2⁄8 by the fraction 4⁄6. Use general multiplication method.

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Front 99

Operations with Fractions _Using Cancellation When Multiplying.

Multiply the fraction 2⁄8 by the fraction 4⁄6. Use cancellation method.

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Front 100

Operations with Fractions_Dividing Fractions

What is the process of dividing fractions?

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The process for dividing fractions is very similar to thatof multiplying. Follow these steps to divide one fractionby another:

1. Set up the division problem. Then, invert (turn over) thedivisor. This means that you’ll switch the numerator anddenominator of the second fraction.

2. Change the division sign to a multiplication sign, andmultiply the fractions together.

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Operations with Fractions_Dividing Fractions

Example: Divide the fraction 1⁄2 by the fraction 3⁄5.

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Operations with Fractions_Dividing Fractions

Example: Divide the fraction 3⁄4 by the fraction 7⁄8.

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Operations with Fractions_Dividing Fractions

Example: Determine how many 1⁄16-inch spacers you wouldneed to fill a 5⁄8-inch gap. To solve the problem, you’ll need todivide the size of the gap (5⁄8 inch) by the size of the spacers(1⁄16 inch).

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Operations with Fractions_Dividing Fractions by whole numbers

Example: Divide the fraction 2⁄5 by the number 6.

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Operations with Fractions_Dividing Fractions by whole numbers

Example: Suppose that you need to divide a measurementof 15⁄16 inch into three equal sections. What will be the lengthof each section?

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Operations with Fractions_Dividing Whole Numbers by Fractions

Example: Divide the whole number 9 by 1⁄3.

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Operations with Fractions_Dividing Whole Numbers by Fractions

Example: Divide the whole number 5 by 3⁄4.

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Operations with Fractions_Dividing Whole Numbers by Fractions

Example: Suppose you have 10 containers of oil. Each container holds 1 gallon. One of your jobs is to fill the machines in your department with this oil. Each machine holds 2⁄5 gallon of oil. How many machines can you fill?To solve this problem, you determine the number of times 2⁄5goes into 10.

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The answer is 25. You can fill 25 machines with the oil.