1

CHAPTER 1: INTRODUCTION TO FRACTIONS SKILLS FOR TRADES

Construction related measurements and calculations require knowledge of fractions. Fractions are a way of expressing a part of something. If you cut a piece of wood, you are using a fraction of the whole board. If you go home from work at lunch, you are only working a fraction of the day. There are two parts to every fraction, the numerator and the denominator.

The Numerator

(top number) refers to a specific part of the whole.

The Denominator

(bottom number) identifies the number of equal parts into which the whole has been divided.

Example: The fraction 83 tells you what part of the figure below is shaded. Three of the

eight equal parts are shaded. Therefore, the shaded portion represents 83 of an inch.

Darken or write fractions that represent the part of each inch that is indicated:

1) ________ 2) 8"5

3) 8"7

2

FRACTIONS IN WRITTEN FORM

Write the correct fraction for each of these. _______ 1) A yard contains 36 inches. 13 inches is what fraction of a yard? _______ 2) There are 12 inches in a foot. 7 inches is what fraction of the whole? _______ 3) 59 cents is what fraction of a dollar? _______ 4) There are 16 ounces in a pound. What fraction of a pound is 7 ounces? _______ 5) Alicia worked 11 months of the year. What fraction of the year did Alicia

work? _______6) Laura has to cut 150 2 x 4’s. This morning she cut 103. What fraction

of the job has she completed? _______7) There are 4 quarts in a gallon of paint. Tabitha used 1 quart to paint the

entryway. What fraction of the gallon did she use? _______8) 2,000 people applied to the pipefitters’ apprenticeship program. The

program accepted 147 people. What fraction of the applicants were accepted?

3

EQUIVALENT FORMS OF FRACTIONS-VOCABULARY

PROPER FRACTION- The numerator is less than the denominator.

Examples: 53

74

1615

A proper fraction is less than all the parts into which the whole is divided. The value of a proper fraction is always less than one. IMPROPER FRACTION- The numerator is equal to or larger than the denominator.

Examples: 58

23

99

An improper fraction may include all the parts into which a whole has been divided ( 8

8),

or it may include more than the total parts of the whole. The value of an improper fraction is either equal to or more than one. UNREDUCED FRACTION-The numerator is less than the denominator, but because it could be reduced, it isn’t a proper fraction.

Examples: 62

129

4832

MIXED NUMBER- A whole number is written next to a proper fraction

Examples: 833

1095

6512

4

REDUCING/SIMPLIFYING FRACTIONS Reducing or simplifying a fraction means writing it in its lowest term. Anyone who uses money is familiar with the concept of reducing fractions. A quarter is 25 out of 100

pennies 10025

or one of four quarters 41 . 10 dimes equals 1 dollar, therefore one dime

equals 101 of a dollar. 10 out of 100 pennies has the same value, 10 cents, and is

written not as 10010 but in its simplified form as

101 .

Below are examples of simplifying fractions.

Example 1: Simplify 10025

Step 1) First find a number that goes evenly into both the numerator and the denominator, in this case 25 is the only number that will work.

41

251002525 =

÷÷

Step 2) Next, see if another number will go evenly into the now reduced fraction. If no further reduction is possible, then the fraction is said to be at its lowest terms.

Example 2: Reduce 8154

Step 1) Find a number that goes evenly into the numerator and denominator. In this example, both numbers can be divided evenly by 9.

96

981954 =

÷÷

Step 2) Now that the fraction has been reduced once, check to see if it is possible to reduce it further. In this case, both numbers can be further divided by 3.

32

3936 =

÷÷

Step 3) Now check again to determine if the fraction can be reduced

further. If not, the fraction is at its lowest terms. 32 can not be reduced: it

is the answer.

Remember that no matter how much the numerator and denominator are reduced, the value of the fraction remains equal to the original just as 10 out of 100 pennies is equal to 1 out of 10 dimes.

5

Short cut: You can shorten the process if the numerator and denominator both end in zero, by crossing out the last zero in the numerator and the last zero in the denominator. This does not change the value, but does reduce the fraction.

Example 3: Simplify 10070

Step 1) Cross out a 0 at the end of each number.

Since you can only cross out one zero in the numerator, you can only cross out one zero in the denominator.

107

10070 =

Step 2) Check to see whether another number goes evenly into both the numerator and denominator of the fraction. Since no other number goes evenly into both 7 and 10 (besides 1), the fraction is reduced as far as it will go.

Example 4: Reduce 20080

Step 1) Cross out ending zeros.

208

20080 =

Step 2) Reduce further by dividing both the numerator and the denominator by 4.

52

44

208 =÷

Step 3) Check to determine if the fraction can be reduced further. It cannot.

Simplify each fraction to the lowest term.

1) 16

8 = ____ 2) 560

480= ____ 3) 400

8= ____ 4) 70

45= ____ 5) 32

4= ____

6)

8118

= ____

7) 7733

=____ 8) 3228

= ____ 9) 10035

=____ 10) 2510

= ____

11)

4928

= _____ 12) 6448

= ____ 13)12030

= _____ 14) 40040

= ____ 15) 21090

= _____

6

REVIEW

• Fractional parts show equal divisions of a whole

• Fractions with the same number on the top and bottom equal a whole

• Answers to fraction problems should always be in lowest terms.

ADDING AND SUBTRACTING FRACTIONS WITH THE SAME DENOMINATOR

With this much information you can easily add or subtract fractions with the same denominators (bottom numbers) by adding or subtracting the numerators (top numbers) and keeping the denominator the same. To work problems with mixed numbers (whole numbers and fractions), add or subtract the fractions first and then move to the whole numbers. Remember to reduce to their lowest terms all the fractions in your answers.

Example 1: Adding two fractions with the same denominator: 91

91 +

Step 1) Add the numerators: 1 + 1 = 2

Step 2) Write the sum from Step 1 over the denominator: 92

Example 2: Adding several fractions with a common denominator and reducing the

answer to lowest terms: 117

2214

22275

222

227

225 ===++ ++

Example 3: Adding mixed numbers to fractions with the same denominator:

31

315 +

Step 1) Add the fractions: 32

31

31 =+

Step 2) Place the whole number before the fraction: 325

(When adding two or more sets of mixed numbers, first add the fractions and then add the whole numbers separately). Example 4: Subtracting one fraction from another with a common denominator:

52

53 −

Step 1) Subtract the numerators: 123 =−

Step 2) Write the difference from Step 1 over the common denominator: 51

7

Example 5: 1012

1096 −

Step 1) Subtract the fractions and simplify/reduce: 54

108

101

109 ==−

Step 2) Subtract the whole numbers: 426 =−

Step 3) Put the whole number to the left of the fraction: 544

Work these problems. Simplify/reduce if possible:

1) =+83

82

_____ 2) =+171

173

_____ 3) =+42

41

_____

4) =++162

165

161 _____ 5) =−

6420

6442

_____ 6) =−162

167

_____

7) =+913

917 _____ 8) =−

317

3115

_____ 9) =−811

8110

_____

10) =++3612

364

368

_____ 11) =−4832

4846

_____ 12) =−653

6548

_____

13) =+195

199

_____ 14) =+3010

30102 _____ 15) =+

1328

1328 _____

16) =−2123

2193 _____ 17) =−

4056

40158 _____ 18) =−

102

1095 ______

19) 3261 + =

321015 _____ 20) =+

1879

18553 _____ 21) =−

913

9782 ______

22) =+16519

16562 _____ 23) =+

49154

49133 _____ 24) =+

99233

991011 _____

25) =−837

8513 _____ 26) =−

513

5421 _____ 27) =−

600303

6008041 _____

Extra Effort

28)

____49187

4942

49614 =++

29)

=−−5484

5453

544022 ______

30)

____8872

8834

888713 =−−

8

Some problems contain a whole number and a mixed number (or a fraction) where there is nothing to be added or subtracted from the fraction. In this case, deal with the fraction first by transferring it directly to your answer, then move to the whole numbers.

Example 1: 9 +72

=

Step 1) There is no fraction to be added to72

9 write 2/7 in the answer space

+ 2/7

Step 2) There is no whole number to be added to 9. 9 write 9 in the answer space answer: 9 2/7

+__2/7_

Example 2: 1+5112 =

Step 1) There is no fraction to be added to 1/5. 1 write 1/5 in the answer space

+12 1/5

Step 2) There is a whole number, 1, to be added to 12 1 write 13 in the answer space answer: 13 1/5

+12 1/5

Example 3: =− 69516

Step 1) There is no fraction to be subtracted from 5/9. 16 5/9 write 5/9 in the answer space

- 6____

Step 2) There is a whole number, 6, to be subtracted from 16 16 5/9 write 10 in the answer space answer: 10 5/9

- 6____

9

Work these problems:

1) =+327 _____ 2) =+ 11

43

_____ 3) =+169115 _____

4) =+ 3171117 _____

5) 14 + 4 +291510 = _____

6) =− 6857 _____

7) =− 2654 _____

8) =− 992199 _____

9) =− 33787782 _____

10) =+3263 _____

11) =− 815240 _____

12) =− 3836 _____

13) =+314 _____

14) =+838 _____

15) =+ 9174

_____

16) =− 9837777 ______

17) =+714955 _____

18) =− 9292100 _____

19) =− 7881431 _____

20) =−− 2216582 _____

You can only add or subtract fractions that have the same denominators. To this point all the fractions have had the same denominators, but that isn’t always the case. Turn the page to find out how to change problems with different denominators so that you can add or subtract them./

10

RAISING FRACTIONS

To add and subtract fractions with different denominators you will also need to learn how to raise fractions to higher terms. (This creates improper fractions. That’s okay – leave them alone for now) There are 2 ways to do this: one is by dividing the old denominator into the new.

Example 1: Raise 54

to 20ths. 20?

54 =

Step 1) In this problem 20 becomes the new denominator. To find the new numerator, divide the old denominator Into the new one: 20 ÷ 5 = 4

Step 2) Then multiply the numerator by that number, (4): 201644 =×

Step 3) Check your answer by dividing the raised fraction by 4.

54

420416 =

÷÷

The second way to raise fractions is by multiplying. Multiply the old denominator by whatever it takes to get to the new one, then multiply the original numerator by the same number. What you do to the bottom number, you must do to the top.

Example 2: Raise 97

to 27ths. 27?

97 =

Step 1) In this problem 27 becomes the new denominator. Multiply 9 X 3 to get 27

Step 2) Then multiply the numerator by that number, (3):

2721

3937 =

××

Step 3) Check your answer by dividing the raised fraction by 3.

97

327321 =

÷÷

Raise each fraction to higher terms.

1) 255

3 = 2) 3612

9 = 3) 328

7 = 4) 244

3 =

5) 279

7 = 6) 10010

5 = 7) 497

2 = 8) 486

3 =

9) 162

1 = 10) 183

2 = 11) 728

3 = 12) 153

2 =

13) 568

7 = 14) 484

1 = 15) 306

5 = 16) 505

4 =

11

CREATING MIXED/WHOLE NUMBERS FROM FRACTIONS

When adding fractions, the answer may be an improper fraction, such as 59

, a fraction that is greater than the whole. To finish solving a problem resulting in an improper fraction, you must convert it to a mixed number by dividing the denominator into the numerator and writing the remainder, if any, over the original denominator.

Example 1: Change 426

to a mixed/whole number.

Step 1) Divide the denominator into the numerator: 2

24

6264

Step 2) Write the 6 as a whole number. (There is a remainder of 2) Step 3) Write the remainder as a fraction over the original denominator: 4

26

Step 4) Reduce answer: 426 =

216

Example 2: Change 756

to a whole number

Step 1) Divide the denominator into the numerator 8

567

Step 2) Write 8 as a whole number (There is no remainder)

Step 3) No reducing necessary. The answer is 8. Practice changing improper fractions into mixed/whole numbers

1) 1122

= _____ 2) 935

= _____ 3) 5

100=_____

4) ____5

50 = 5) _____742 = 6) _____

241 =

7) 5

13= _____ 8)

90104

= _____ 9) 1212

= _____

10) _____3

77 = 11) _____8

345 = 12) _____27691 =

13) _____17447 = 14) _____

100000,1 = 15) _____

31563 =

12

HOW TO FIND THE COMMON DENOMINATOR A common denominator is that number which can be divided evenly by all the

denominators in a problem. The smallest number that can be divided evenly by all

denominators is called the lowest common denominator (LCD). First try using the largest denominator in the original problem as the LCD.

Example: =+41

81

Since 4 can go into 8 evenly, 8 can be used as the common denominator.

82

41

81

81 == and Rewritten, the problem becomes =+

82

81

Answer: 83

Solve these problems and reduce if necessary.

1) =+21

41

_____

2) =−31

65

_____

3) =+43

85

_____

4) =−41

127

_____

5) =++53

107

21

_____

6) =+253

54

____

7) =+5412

91

_____

8) =−114

7732

_____

9) =−83

3215

_____

10) =−72

217

_____

11) =++65

31

124

_____

12) =++2013

85

409

_____

13) =−73

5632

____

14) =++54

31

154

_____

15) =−21

3025

_____

16) =++8815

114

83

_____

17) =++65

31

124

_____

18) =−31

2411

_____

19) =−95

3627

_____

20) =++89

21

6421

_____

21) =++25

281

74

_____

22) =++53

209

107

_____ 23) =++3817

21

193

____ 24) =−−61

62

180110

_____

13

When none of the denominators will work as the (LCD), try:

• Going through the multiplication tables of the denominators to see if there

are common multiples

• Multiplying 2 small denominators together, then checking that number

against any other denominator (sometimes ½ of this will work—try it just to

see)

• Multiply all the denominators together ( Use this if all else fails: it always

works, but results in a VERY large number which means you will have a

lot of reducing/simplifying to do at the end)

Example 1: =++41

65

31

• Go through the multiplication table of the 3’s.

• 3 X 2 = 6 which cannot be divided by 4.

• 3 X 3 = 9 which cannot be divided by either 6 or 4.

• 3 X 4 = 12 which can

• Raise each fraction to 12ths.

be divided by 3, 6 and 4. Use 12 as the common

denominator.

,124

31 =

,1210

65 = ==

123

41

• Rewrite and add: 1217

123

1210

124 =++

• Answer (reduced): 1251

If there are whole numbers in the problem, deal with them separately

Example 2: 2 =+54

43

• Multiply the denominators to find the LCD: 4 x 5 = 20

• Raise each fraction to 20ths. 2016

54

2015

43 == and

• Rewrite and add: 2 20312

2016

2015 =+

• Answer: 32011

14

Solve these programs and simplify/reduce.

1) 3 83

1615 − = _____

2). 4 5107 1+ =_____

3) 5 165

43 + =_____

4) 7 61

32 + = _____

5) 7 210 114 − =____

6) 2 3

198 − = ____

7) 14 52

151 + =______

8) 8 32

2119 − = _____

9) 321

+77

= _____

10) 9 124

83 + =_____

11) 4

1662 + =_____

12) 3

25 1797 + =_____

13) 9 =− 41

53

______

14) =− 32

1615

_____ 15) 81

107 − =______

16) =− 72

43

____

17) =+ 32

217 _____

18) =− 9

4 552 ____

19) =+ 43

1048 _____

20) =− 83

659 _____

21) =+ 6

5434 ____

Extra effort

22) 32

+92

+1255 = ______

23) 52

+103

+1537 = _____

24) 735 +

1159 +

2112 = ______

15

BORROWING: A MULTI-STEP PROCESS

In some cases, to solve a subtraction problem you will have to borrow 1 from a whole number to create a needed fraction. This involves several critical steps:

1) LCD2)

(Identify/create the Lowest Common Denominator) Borrow

3) 1 from the larger whole number (never the smaller one)

Change• you may NOT change the value of the number, but you may state it differently

the borrowed 1 into the fraction you need it to be

($1 can be a single bill or 10 dimes or 4 quarters or 100 pennies) • fractions with the same top and bottom numbers equal 1:

11616;1

77;1

44 ===

4) Merge5)

the fraction from the borrowed 1 with that given in the problem (rewrite)

Subtract

Example 1: This is the easiest, involving only steps 1, 2, 3 & 5 (step 4 is not needed)

=−3268 Step 1) LCD

There is nothing above the

(you have only one: use 3rds as the LCD)

32

, yet you have to subtract it from

something.

Step 2) Borrow

1 from the 8, replacing the 8 with a 7 (+1) )1(78 +=

733 Step 3) Change

33 (You need more 3rds) Change the (+1) to

You now have a workable subtraction problem.

Step 4) is not needed

311

326

337 =− Step 5) Subtract

16

Example 2: The first fraction is smaller than the second fraction. Make it larger.

=−547

5113 Step 1) LCD

(you have only one: use 5ths as the LCD

51)1(1213 +→ Step 2) Borrow

1 from the 13, replacing it with 12 (+1)

)551( =+

Step 3) Change

the (+1) into 5/5

5612

55

5112 =+ Step 4) Merge and rewrite

the borrowed fraction with the problem

525

547

5612 =− Step 5)

Subtract

Practice borrowing by solving the following problems. Reduce if possible.

1) =−76

214 _____

2) =−546 _______

3) =−21

1031 ______

4) =−414 ______

5) =−107

529 ______

6) =−43

327 ______

7) =−157

3131 ______

8) =−326

3115 _____

9) =−3269 _______

10) =−5421 ______

11) =−

533

10313 ______ 12) =−

329

12719 ______

13) =−87142 _______

14) =−12712

4151 _____

15) =−215

11293 ______

16) =−11726

2152 ____

17) =−65263 _____

18) =−547

739 ______

19) =−97247 _____

20) =−

753

317 _____ 21) =−

432

8113 ______

22) =−1094

5318 _____ 23) =−

85

511 _____ 24) =−

75621 _____

17

25) =−433

12511 _____ 26) =−

1032 ______ 27) =−

972

5310 ____

WORD PROBLEMS: ADDITION AND SUBTRACTION Many of the trades use word problems on their qualifying tests. Look for key words that tell you whether you need to add or subtract. Words such as sum or total, or any problem that asks you to combine more than two numbers can be solved through addition. Difference, left, & remaining are some key words that tell you this is a subtraction problem. Figuring out an increase or decrease also may involve subtraction, as does the concept of something getting smaller (cut, shrunk, eaten or lost). In setting up subtraction problems the larger number always goes above the smaller one. To be correct, each answer must include numbers and words or symbols.

1) Vanessa cut off 833 inches from an 8” inch brick. How long was the remaining

piece?

2) Annette weighs 174 pounds. When she puts on her 4317 pound tool belt and her 5

lb. steel toed boots, how much does she weigh?

3) LaShawn cut a length of PVC into three pieces measuring, 16142 ”,

437 ” and

8113 ”.

How much PVC did she use?

18

4) Sue only used her truck for work. Monday morning the odometer on the truck showed 60,345

103 miles. Back at home Friday afternoon it showed 60,830

107 miles.

How far did she travel to work that week?

5) From an 80 pound bag of cement, Mary used 33 85 pounds to make concrete. How

much was left in the bag?

6) Kelly used up a box of nails to compete two projects. She used 24 41 lbs of nails for

the first project and 25 43 nails for the second one. How heavy was the box originally?

7) Before break, Crystal made up 3211 sets of nuts, bolts and washers. Before lunch,

she made up 9 more sets and after lunch, she finished another 5210 sets. How many

sets did she make?

8) If you lower the speed of a cement mixer from 12 rpms to 315 rpms, how much

slower does it spin?

9) Rosa’s 432 pound work boots weighed

8310 pounds after she worked in the mud

all day. How much was the increase?

19

PERIMETER

Perimeter is a measurement of the distance around an object. You will need to know how to measure perimeters to build a fence or to do other construction jobs that involve perimeters. To determine the perimeter of a shape, you must measure the sides, then add the sides together. Example: What is the perimeter of the following shape?

2 1/2'

16 1/12'

Add 2 1/2' + 2 1/2' (for each end of the figure) and 16 1/12' + 16 1/12' (for each side). The perimeter is then 37 1/6'. Use the symbol ' for feet and “ for inches. Note: Answers to perimeter problems must always include a unit of measurement. Determine the perimeter of the following shapes. 1) 2) P = _____ P = _____ 3) 4) 5) P = _____ P = _____ P = _____ 6) 7) 8) P = _____ P = _____ P = _____

413 '

2116 ’

216 '' 3

11 yd

317

ft

328 ft

8112 “ 6

16 yd 219

yd

311 yd

1612 “

8112 ''

318 yard

1016

yard

8110 miles

517

miles

20

PRACTICE PROBLEMS: ADDITION AND SUBTRACTION

1) =+61

31

_____

2) =+329

83

_____

3) =−43

87

_____

4) =−21

313 _____

5) =−95

951 _____

6) =+521

1077 _____

7) =−872

8110 _____

8) =++ 312511

21

_____

9) =−322

16713 _____

10) =−21

97

_____

11) =++ 11

22110

11216 ____ 12) =+

326

326 _____

13) =++1278

1278

1278 ___ 14) =−

643119

32131 __ 15) =++

217

539

324 ____

16) =++513

876

201313 ___ 17) =++

765

5117

352392 ___ 18) =++

5476

6133

302955 __

19) =−1054

10314 ____ 20) =−−

611

12116

9523 ____ 21) =−

1543

30715 _____

22) If a yard has 5 sides: 7’,2116 ’,

4113 ’,

2118 ’ and

316 ’, what is its perimeter?

Extra Effort

23) If the perimeter of a park is 41399 ’, and 3 sides are

21143 ’,

41142 ’, and

3155 ’,

what length is the 4th side?

24) The perimeter of a triangle is 96 yards. If one side is 3229 yards and another side

is 3130 yards, how long is the remaining side?

21

MULTIPLICATION OF FRACTIONS When multiplying fractions, you simply multiply the numerators together and the denominators together: no need for common denominators! Reduce/simplify if possible.

Example: =× 53

52

Step 1) Multiply the numerators. 2 x 3 = 6

Step 2) Multiply the denominators 5 x 5 = 25 Answer: 256

Multiply and reduce.

1) =×85

73

_____

2) =×75

32

_____

3) =×74

53

_____

4) =×95

31

_____

5) =×32

137

_____

6) =×21

65

_____

7) =×43

43

_____

8) =×87

103

_____

9) =×85

113

_____

10) =×41

21

_____

11) =×32

52

______

12) =×76

76

______

13) =×32

83

______

14) =×51

63

_______

15) =×32

21

______

Extra effort

16) =×25

74

______

17) =×1510

32

______

18) =×15

62

______

19) =×32

37

______

20) =×98

1113

______

21) =×49

3231

______

22

Shortcut

Cross canceling is a shortcut in multiplication of fractions. It is similar to simplifying or reducing. To cross cancel, divide any numerator and any denominator by a number that goes evenly into both. This process, though not required, makes the numbers smaller and, therefore, easier to multiply.

Example: 2112

87 ×

Step 1) Cancel 8 and 12 by 4. 8 ÷4 = 2 and 12 ÷ 4 = 3. Cross out the 8 and the 12 and replace them with 2 and 3.

213

27

21312

287 ×=→×→

Step 2) Cancel 7 and 21 by 7. Cross out the 7 and the 21 and replace them with 1 and 3.

3213

217

213

27

→×→=× Step 3) Multiply across by the new numbers.

Step 4) Simplify/ reduce.

21

63 =

Cross cancel and multiply.

1) =×2012

85

_____ 2) =×65

54

_____ 3) =×128

43

_____

4) =×97

73

_____ 5) =×103

125

_____ 6) =×31

1512

_____

7) =×109

65

_____ 8) =×2812

247

_____ 9) =×94

83

_____

10) =×103

115

_____ 11) =×154

125

_____ 12) =×1310

21

_____

13) =×207

354

_____ 14) =×93

153

_____ 15) =×169

278

_____

23

When there are three fractions in an equation, check to see if you can cross cancel to simplify the problem. Multiply the first two numerators together then multiply that answer by the third numerator. Do the same for the denominators. Simplify/reduce.

Example: =×× →→

482

53

714

22

53

71 ××

Multiply the first two numerators then multiply the answer times the last numerator.

331 =× 623 =× Multiply the first two denominators then multiply the answer times the last denominator.

3557 =× 70235 =×

Simplify/ reduce. 353

706 =

Multiply and simplify/reduce the following:

1) =××2

1

4

3

3

2 _____ 2) =××

3

1

8

5

9

4 _____ 3) =××

6

5

4

3

3

2 _____

4) =××8

7

6

5

5

3 _____ 5) =××

2

1

7

5

5

2 _____ 6) =××

7

6

6

5

4

3 _____

7) =××8

1

9

2

7

3 _____ 8) =××

4

1

3

1

2

1 _____ 9) =××

3

1

9

8

8

7 _____

10) =××5

2

10

3

12

5_____ 11) =××

9

7

8

5

8

3_____ 12) =××

5

3

8

7

14

9______

13) =××11

5

21

9

18

7_____ 14) =××

15

11

9

5

22

9 ____ 15) =××

8

3

12

1

7

4_____

16) =××12

5

4

3

20

1_____

17) =××

5

12

3

2

7

2_____ 18) =××

11

9

2

1

81

77______

24

CREATING IMPROPER FRACTIONS FROM WHOLE AND MIXED NUMBERS You cannot multiply fractions by whole or mixed numbers: you can only multiply a fraction by another fraction. To change a whole number into a fraction, use the given number as the numerator and put it over a 1 as its denominator. To change a mixed number into an improper fraction multiply the denominator by the whole number and add the numerator. Place the total over the original denominator. This changes the whole/mixed number into an improper fraction without changing its value. Example 1: Change 5 to an improper fraction. Place a 1 under the 5, creating

15 , which

is an improper fraction. Example 2: Change

433 to an improper fraction.

Step 1) Multiply the denominator by the whole number: 1243 =× Step 2) Add the result to the numerator: 12 + 3 = 15 Step 3) Place the total over the original denominator:

415

Example 3: Change 3

17 to an improper fraction. Step 1) 2137 =× Step 2) 21 + 1 = 22 Step 3)

322

Create improper fractions from the whole and mixed numbers below.

1) =414 _____

2) =3 _____

3) =327 _____

4) =748 _____

5) =542 _____

6) =1039 _____

7) =835 _____

8) 11 =_____

9) =4312 _____

10) =524 _____

11) =3220 _____

12) =5313 _____

13) =1132 _____

14) =854 _____

15) 14 =_____

16) =12532 ____

17) =

6123 _____ 18) =

7618 _____ 19) =

10943 ____ 20) =

9214 _____

25

Note: on this page all of the answers are left in improper fraction form!

MULTIPLYING WITH MIXED NUMBERS

Once you have converted whole or mixed numbers to improper fractions, multiply as shown in the following example:

Example 1: 20175 ×

Step 1) Write 5 as an improper fraction. 155 =

Step 2) Cross cancel 5 and 20 by 5. 42017

115

→×→

Step 3) Multiply the numerators and denominators.4

174

1711 =×

Step 4) Simplify/reduce 414

417 =

Example 2: =× 169

326

Step 1) Change 326 to an improper fraction. 3

20326 =

Step 2) Cross cancel 20 and 16 by 4. 49

35

4169

3520 ×=→×→

Cross cancel 9 and 3 by 3. 439

135 →→ ×

Step 3) Multiply the numerators and denominators. 415

4135=×=×

Step 4) Simplify/reduce. 4334

15 =

26

Multiply and reduce.

1) =× 31214

134 _____ 2) =× 7465

3 _____ 3) =× 10

75 _____

4) =× 7558

34 _____ 5) =× 41

872 _____ 6) =× 5

11312 _____

7) =× 9749 ______ 8) =× 21

1071024 _____ 9) =× 7

3312312 _____

10) =× 1211312 _____ 11) =× 149

5 921 _____ 12) =× 5466

59 _____

13) =× 32420

126 _____

14) =× 11514

310 _____

15) =× 116214

52 _____

16) =×× 43149

97 _____

17) =× 371516288 _____

18) =× 9

48721 _____

19) =×× 5339

8431 ____ 20) =×× 4

1215352

17 _____ 21) =×× 87

214

152

____

22) =×× 74327

219 ____ 23) =×× 13516

12756 _____ 24) =×× 6

59545

17 ____

27

WORD PROBLEMS: MULTIPLICATION

Use multiplication to solve word problems that ask you to: 1. Find a part of something 2. Use the information given about one thing to determine the weight/size/cost of

several things. Remember than fractional answers must be in simplified terms and include a word or symbol.

1) From home to work Lynn has to drive 75 miles. She usually stops for breakfast 32

of the way there. How far has she already driven when she stops for breakfast?

2) If one gallon of fresh water weighs 218 pounds, what is the weight of 2

11 gallons?

_____

3) For an entertainment center Kris plans to build, she wants four shelves that are

each 323 feet long. What is the total length of the four shelves Kris needs?

4) Mary gets paid $48/hour when she works overtime. If she worked 433 hours

overtime, how much would she be paid for the overtime work? ______________ Extra effort ( Hint: eliminate all but the essential numbers) 5) While on a 5 day vacation in Cancun, Aimee wanted to be outside 8 hours a day.

She spent 53

of her days lounging at the beach. How many days did Aimee spend lounging at the beach?

__________

6) If one fifty foot roll of 81

inch copper tubing costs $59.98, how much will 213 rolls

cost?

______

7) Nancy makes $14 an hour for regular time. When she works overtime she gets paid

time and a half. If she works 6 21

hours overtime in one week, how much extra

28

pay will she get for the overtime worked? _________________ DIVISION OF FRACTIONS

Now that you have mastered the processes required to multiply fractions, you are just one step away from division. To solve any division of fractions problem, remember:

1. You can only divide fractions by fractions, so change any whole/mixed numbers to improper fractions.

2. Invert (turn upside down) the fraction to the right of the division sign. 3. Change the division sign to a multiplication sign. 4. Multiply Example 1: 8

543 ÷

Step 1) is not needed

58

85 → Step 2) Invert the fraction on the right ( 8

5 ) to 58

58

43 × Step 3) change to x :

52

13 × Cancel 4 and 8 by 4.

511

56

52

13 ==x Step 4) Multiply and simplify/reduce

Example 2: 7

68 ÷

18 Step 1) Write 8 as a fraction.

67

76 → Step 2) Invert 7

6

Step 3) Change ÷ sign to x. 6

718

76

18 ×=÷ Cancel 8 and 6 by 2

. 3

19328

37

14 ==× Step 4) Multiply and simplify/reduce.

Example 3: 4

1312 ÷

41

37 ÷ Step 1) Change 3

12 to an improper fraction: 37

14

41 ÷ Step 2) Invert 4

1

14

37

14

37 ×=÷ Step 3) Change ÷ sign to x.

29

319

328

14

37 ==x Step 4) Multiply and simplify/reduce

Divide and simplify/reduce.

1) 21

32 ÷ = _____ 2)

161

161 ÷ = ____ 3)

1001

101 ÷ =____ 4) =÷

944 ____

5) 31615 ÷ =____ 6) 4

85 ÷ = _____ 7)

7618 ÷ = ____ 8)

611

1514 ÷ = ____

9) 324

97 ÷ = ____ 10)

87

127 ÷ = ____ 11) 5

2512 ÷ = ____ 12) 2

713 ÷ =____

13) 3112 ÷ = ____ 14)

2019

1092 ÷ = ____ 15)

229

1177 ÷ = ___ 16)

43

1271 ÷ = ___

17) 1271

924 ÷ = ____ 18)

1612

817 ÷ = ____ 19)

651

979 ÷ = ____ 20)

1211

322 ÷ =___

21) 323

833 ÷ _____ 22)

751

733 ÷ = _____ 23)

314

64402 ÷ = _____

Extra effort

24) 43

65

52 ÷÷ =___

25) 6454

722 ÷÷ ___

26) 43

87

641 ×÷ = ____

30

Word Problems: Division

To solve division of fractions word problems, first read the problem and determine-- “What is the entire thing to be evenly divided?” Write that number first, followed by the ÷ sign. The last number represents how many will be sharing or made from the first. The concept is: The whole divided/cut/packaged into how many

equal parts.

Remember to invert the number to the right of the ÷ sign. Always simplify your answer and include a word or symbol. 1) How many 7 2

1 inches pieces can be cut from 90 inches of rebar?

2) Kelly’s goal was to run 212 miles. She wanted to mark each 4

1mile with balloons.

How many would she need?

3) If Olivia needs 811 yards of denim to make 1 nail apron, how many can she make

from 433 yards of material?

4) How many 322 pound candles can be made from 3

210 pounds of candle wax?

5) How many 21

pound boxes of nails can be filled from 2116 pounds of nails?

6) Sheet metal 52 feet long is to be cut into 314 foot pieces. How many can be cut?

7) Leslie needs 412 gallons to paint a room. If all the rooms she’s painting are the

same size, how many can she paint with 9 gallons?

8) Marisol and Agnes are on a diet. They want to split 41

pound of chocolate evenly between them. How much would each get?

31

32

PRACTICE PROBLEMS: MULTIPLICATION AND DIVISION

1) 61

31 × = _____ 2)

32921× = _____ 3)

43

877 ÷ =_____

4) 315

213 ÷ = _____ 5)

98

2113 × = _____ 6)

521

1077 ÷ = _____

7) 273

973 ÷ = ______ 8)

4316

83 ×× = _____ 9)

313

957 ÷ =_____

10) 402524 ÷ = _____ 11)

15131

953 ÷ = _____ 12)

85

526

326 ×× = _____

13) 2116

3716 ÷ = _____ 14)

158

115

1211 ×× =_____ 15)

71

2193 ÷÷ = _____

16) 543× =_____ 17) 20

8011 × =_____ 18)

1094× = _____

Extra Effort

19) 517

623

30255 ×× =_____ 20)

755

71

838 ×÷ = _____ 21)

73

1511

3223 ×× = _____

22) 952

462310 × = _____ 23)

71

12511

167 ×× = _____ 24)

322

161113 ÷ = _____

33

Fractions packet answer key

Page 1

1) 81

2) 3) Page 2

1) 3613

2) 127

3) 10059

4) 167

5) 1211

6) 150103

7) 41

8) 2000147

Page 5

1) 21

2) 76

3) 501

4) 149

5) 81

6) 92

7) 73

Page 5 cont.

8) 87

9) 207

10) 52

11) 74

12) 43

13) 41

14) 101

15) 73

Page 7

1) 85

2) 174

3) 43

4) 21

5) 3211

6) 165

7) 10 92

8) 318

9) 91

10) 32

11) 247

Page 7 cont.

12) 139

13) 1914

14) 2 32

15) 16 134

16) 31

17) 2 41

18) 5 107

19) 16 21

20) 62 32

21) 79 32

22) 81 85

23) 7 74

24) 14 31

25) 6 41

26) 18 53

27) 38 121

28) 23 74

29) 15 21

30) 7 87

Page 9

1) 7 32

2) 11 43

3) 16 169

4) 20 1711

5) 28 2915

6) 1 85

7) 2 65

8) 21

9) 49 7877

10) 9 32

11) 32 152

12) 3 83

13) 4 31

14) 8 83

15) 9 174

16) 68 8377

17) 104 71

18) 8 92

19) 353 81

20) 59 65

Page 10

1) 2515

2) 3627

3) 3228

4) 2418

5) 2721

6) 10050

7) 4914

8) 4824

9) 168

10) 1812

11) 7227

12) 1510

13) 5649

14) 4812

15) 3025

16) 5040

34

Answer Key Continued

Answer Key Continued

Page 11 1) 2

2) 3 98

3) 20 4) 10 5) 6

6) 20 21

7) 2 53

8) 1 457

9) 1

10) 25 32

11) 43 81

12) 25 2716

13) 26 175

14) 10

15) 18 315

Page 12

1) 43

2) 21

3) 1 83

4) 31

5) 1 54

Page 12 cont.

6) 2523

7) 31

8) 774

9) 323

10) 211

11) 1 21

12) 1 21

13) 71

14) 1 52

15) 31

16) 1110

17) 1 21

18) 81

19) 367

20) 1 6461

21) 3 283

22) 1 43

23) 1 192

24) 91

Page 14

1) 3 169

2) 4 109

3) 6 161

4) 7 65

5) 7 143

6) 2 95

7) 14 157

8) 8 215

9) 4 21

10) 9 2417

11) 6 127

12) 25 92

13) 9 207

14) 4813

15) 4023

16) 2813

17) 8 61

18) 2 4511

19) 9 203

20) 9 2411

21) 5 127

Page 14 cont.

22) 6 3611

23) 7 109

24) 27 15459

Page 16

1) 3 149

2) 5 51

3) 54

4) 3 43

5) 8 107

6) 6 1211

7) 30 1513

8) 8 32

9) 2 31

10) 20 51

11) 9 107

12) 9 1211

13) 40 81

14) 38 32

15) 87 2215

16) 25 2219

17) 60 61

Page 16 & 17

18) 1 3522

19) 44 92

20) 3 2113

21) 10 83

22) 13 107

23) 4023

24) 14 72

25) 7 32

26) 1 107

27) 7 4537

Page 17 & 18

1) 4 85

inch 2)

196 43

pounds

3) 62 1615

inch

4) 485 52

miles

5) 46 83

lbs. 6) 50 lbs.

7) 31 151

sets

8) 6 32

rpms

9) 7 85

lbs.

35

Answer Key Continued

Page 19

1) 39 21

feet 2) 32 feet

3) 37 41

in. 4) 15 yds.

5) 21 32

yds.

6) 28 83

in.

7) 28 1513

yards

8) 34 2013

miles Page 20

1) 21

2) 3221

3) 81

4) 2 65

5) 1

6) 9 101

7) 7 41

8) 14 1211

9) 10 4837

10) 185

11) 37 225

12) 13 31

13) 25 43

14) 11 6435

15) 21 3023

16) 23 4029

17) 115 75

18) 165 1514

19) 9 54

20) 15 3617

21) 11 3029

22) 61 127

’

23) 58 61

’ 24) 36 yds. Page 21

1) 5615

2) 2110

3) 3512

4) 275

5) 3914

6) 125

7) 169

8) 8021

9) 8815

10) 81

Page 21 cont.

11) 154

12) 4936

13) 41

14) 101

15) 31

16) 1 73

17) 94

18) 1 32

19) 1 95

20) 1 995

21) 2 12823

Page 22

1) 83

2) 32

3) 21

4) 31

5) 81

6) 154

7) 43

8) 81

9) 61

Page 22 cont.

10) 223

11) 91

12) 135

13) 251

14) 151

15) 61

Page 23

1) 41

2) 545

3) 125

4) 167

5) 71

6) 2815

7) 841

8) 241

9) 277

10) 201

11) 19235

12) 8027

13) 665

14) 61

Page 23 cont.

15) 561

16) 641

17) 3516

18) 187

Page 24

1) 417

2) 13

3) 323

4) 760

5) 514

6) 1093

7) 843

8) 111

9) 451

10) 522

11) 362

12) 568

13) 1125

14) 837

15) 114

16) 12389

36

Page 24 cont.

17) 6139

18) 7132

19) 10439

20) 9128

Page 26

1) 11 21

2) 3 3533

3) 3 21

4) 25

5) 3223

6) 2 54

7) 43

8) 31 52

9) 42 10) 47

11) 4 91

12) 66 1513

13) 30 54

14) 15 117

15) 6

16) 2 41

17) 20 3728

Page 25/26 cont.

18) 10 76

19) 5 53

20) 87 43

21) 451

22) 25

23) 20 71

24) 27 31

Page 27 1) 50 miles

2) 12 43

lbs.

3) 14 32

feet 4) $180 5) 3 days 6) $209.93 7) $136.50 Page 29

1) 1 31

2) 1 3) 10 4) 9

5) 165

6) 325

Page 29 cont. 7) 21

8) 54

9) 61

10) 32

11) 12512

12) 1 74

13) 1 21

14) 3 191

15) 18 32

16) 2 91

17) 2 32

18) 3 115

19) 5 31

20) 2 1110

21) 36 22) 2

23) 10463

24) 2516

25) 89655

26) 2243

Page 30 1) 12 pieces 2) 10 markers

3) 3 31

aprons 4) 4 candles 5) 33 boxes 6) 12 pieces 7) 4 rooms

8) 81

pound Page 31

1) 181

2) 5 3229

3) 10 21

4) 3221

5) 12

6) 5 21

7) 34

8) 4 21

9) 2 154

10) 1253

11) 1 2119

12) 26 32

Page 31 cont.

13) 3721

14) 92

15) 49

16) 2 52

17) 2 43

18) 3 53

19) 140 20) 335

21) 8077

22) 26 65

23) 192137

24) 5 12817