understanding division of fractions - · pdf file18-01-2012 · understanding...

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Name © Pearson Education, Inc. 6 Reteaching 9-1 Reteaching 9-1 Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find 1 _ 8 4 4. Use a model to show 1 _ 8 . Divide each eighth into 4 equal parts. Each section shows 1 _____ (8 3 4) 5 1 __ 32 . 1 _ 8 4 4 5 1 __ 32 . Divide a fraction by a fraction. Find 3 _ 4 4 1 _ 4 . Use a number line. Use repeated subtraction. Count the number of 1 _ 4 s in 3 _ 4 . Subtract 1 _ 4 from 3 _ 4 until the difference is 0. 0 1 1 4 2 4 3 4 3 _ 4 2 1 _ 4 5 2 _ 4 2 _ 4 2 1 _ 4 5 1 _ 4 1 _ 4 2 1 _ 4 5 0 Count the number of times you There are three 1 _ 4 s. subtracted to find the quotient. 3 _ 4 4 1 _ 4 5 3 3 _ 4 4 1 _ 4 5 3 Solve each division sentence. Use a model if you wish. 1. 3 4 1 _ 3 5 2. 1 _ 5 4 4 5 Find each quotient. Simplify if possible. 3. 3 4 1 _ 2 5 4. 9 __ 10 4 1 __ 10 5 5. 1 _ 5 4 3 5 6. 3 __ 16 4 1 __ 16 5 7. 5 4 1 _ 3 5 8. 1 _ 2 4 6 5 9. 8 4 1 _ 4 5 10. 7 __ 12 4 1 __ 12 5 11. 6 _ 7 4 1 _ 7 5 12. Draw a Picture The square dancing club meets for 3 hours. Every 3 _ 4 hour, the dancers change partners. How many different partners will each dancer have in one meeting? Draw a picture to show your solution. 13. Writing to Explain Explain why the quotient of two fractions less than 1 is always greater than either fraction. 9 1 __ 20 6 9 1 __ 15 3 15 1 __ 12 32 7 6 3 4 3 _ 4 5 4; 4 partners. Check students’ drawings. The quotient is always greater than the fractions because you are dividing by a number less than one. 26 Topic 9

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Page 1: Understanding Division of Fractions - · PDF file18-01-2012 · Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find _1

Name

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-1Reteaching

9-1Understanding Division of FractionsDivide a fraction by a whole number.

18

18

18

18

18

18

18

18

132

Find 1 _ 8 4 4.Use a model to show 1 _ 8 .Divide each eighth into 4 equal parts. Each section shows 1 _____ (8 3 4) 5 1 __ 32 . 1 _ 8 4 4 5 1 __ 32 .

Divide a fraction by a fraction.Find 3 _ 4 4 1 _ 4 . Use a number line. Use repeated subtraction.Count the number of 1 _ 4 s in 3 _ 4 . Subtract 1 _ 4 from 3 _ 4 until the difference is 0.

0 114

24

34

3 _ 4 2 1 _ 4 5 2 _ 4

2 _ 4 2 1 _ 4 5 1 _ 4

1 _ 4 2 1 _ 4 5 0 Count the number of times you There are three 1 _ 4 s. subtracted to find the quotient. 3 _ 4 4 1 _ 4 5 3 3 _ 4 4 1 _ 4 5 3

Solve each division sentence. Use a model if you wish.

1. 3 4 1 _ 3 5 2. 1 _ 5 4 4 5

Find each quotient. Simplify if possible.

3. 3 4 1 _ 2 5 4. 9 __ 10 4 1 __ 10 5 5. 1 _ 5 4 3 5

6. 3 __ 16 4 1 __ 16 5 7. 5 4 1 _ 3 5 8. 1 _ 2 4 6 5

9. 8 4 1 _ 4 5 10. 7 __ 12 4 1 __ 12 5 11. 6 _ 7 4 1 _ 7 5

12. Draw a Picture The square dancing club meets for 3 hours. Every 3 _ 4 hour, the dancers change partners. How many different partners will each dancer have in one meeting? Draw a picture to show your solution.

13. Writing to Explain Explain why the quotient of two fractions less than 1 is always greater than either fraction.

9 1 __ 20

6 9 1 __ 15 3 15 1 __ 12

32 7 6

3 4 3 _ 4 5 4; 4 partners. Check students’ drawings.

The quotient is always greater than the fractions because you are dividing by a number less than one.

26    Topic 9

30429_T09_023-028.indd 26 1/18/12 8:48 PM

Page 2: Understanding Division of Fractions - · PDF file18-01-2012 · Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find _1

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Topic 9 27

Practice 9-1

Practice

9-1Understanding Division of Fractions Solve each division sentence using the models provided.

1. 3 � 1 _ 3 = 2. 1 _

4 � 6 � 3. 5 _

6 � 1 _

6 �

0 1 2 31

41

6

2

6

3

6

4

6

5

60

Find each quotient. Simplify if possible.

4. 8 � 1 _ 4 � 5. 1 _

7 � 4 � 6. 5 � 1 _

2 �

7. 7 _ 8 � 1 _

8 � 8. 11 __

12 � 1 __

12 � 9. 1 __

12 � 3 �

10. 6 � 2 _ 3 � 11. 7 � 1 _

3 � 12. 15

__ 16

� 1 __ 16

13. Draw a Picture Olivia has a piece of ribbon 1 _ 2 yard long. If she cuts it

into 6 equal pieces, what will be the length of each piece, in yards?

14. Geometry A regular polygon has a perimeter of 12 units. If each side measures 3 _

4 unit, how many sides does the polygon have?

15. Which division expression is shown by this model?

1 2 3 4 5 6 7 8 9

9

100

A 9 __ 10

� 1 __ 10

B 1� 1 __ 10

C 9 __ 10

� 1 D 10 � 9 __ 10

16. Writing to Explain When you divide a whole number by a fraction less than 1, will the quotient be greater than or less than the whole number? Explain, and give an example.

9 1 __ 24 5

32 1 __ 28 10

7 11 1 __ 36

9 21 15

Check students pictures: 1 __ 12 yard

16

The quotient will be greater because you are

dividing by a number less than 1. 4 ÷ 1 _ 3 = 12:

There are more fraction parts than wholes.

1

1 2 3

2 3 4 5 6 7 8 9 10 11 12

Each section 1

3

Page 3: Understanding Division of Fractions - · PDF file18-01-2012 · Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find _1

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32 Topic 9

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Reteaching

9-2Dividing a Whole Number by a Fraction

To divide a whole number by a fraction, you can multiply the whole number by the reciprocal of the fraction. The reciprocal of a number has the numerator and the denominator reversed. The product of a number and its reciprocal is 1.

Number � Reciprocal � Product

3 � 1 _ 3 � 1

1 _ 8 � 8 _

1 � 1

2 _ 3 � 3 _

2 � 1

Find 14 � 4 _ 7 .

Step 1

Rewrite the division as multiplication using the reciprocal of the divisor.

The reciprocal of 4 _ 7 is 7 _ 4 .

14 ÷ 4 _ 7 = 14 × 7 _ 4

Step 2

Divide out common factors if possible. Then multiply.

7

14 __ 1 � 7 _ 4 � 49 __ 2 2

Step 3

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

49 __ 2 � 24 1 _ 2

Find the reciprocal of each fraction or whole number.

1. 5 _ 7 2. 11 3. 9 _

2

Find each quotient. Simplify if possible.

4. 12 � 4 _ 5 5. 2 ÷ 1 _

4 6. 16 ÷ 8 __

10

7. 24 � 3 _ 4 8. 18 � 8 _

9 9. 25 ÷ 10

__ 11

10. 36 � 8 _ 9 11. 42 � 7 _

8 12. 40 ÷ 4 _

5

13. Karolyn makes rolls for a friend’s dinner party. She uses 3 lb of butter. Each stick of butter weighs 1 _

4 lb. How many sticks of butter

does Karolyn need to make her rolls?

7 _ 5 1 __ 11 2 _ 9

15 8 20 32 20 1 _ 4 27 1 _ 2 40 1 _ 2 48 50

12 sticks

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Topic 9 33

Practice 9-2

Practice

9-2Dividing a Whole Number by a FractionFind the reciprocal of each fraction or whole number.

1. 5 _ 9 2. 8 3. 7 _ 3

Find each quotient. Simplify if possible.

4. 8 4 2 _ 5 5 5. 4 4 1 _ 6 5 6. 18 4 3 _ 8 5

7. 12 4 1 _ 2 5 8. 42 4 7 _ 9 5 9. 10 4 5 _ 6 5

10. 20 4 3 _ 4 5 11. 22 4 5 _ 6 5 12. 7 4 2 _ 3 5

13. 9 4 1 _ 8 5 14. 15 4 1 _ 3 5 15. 6 4 1 _ 5 5

16. Writing to Explain Will the quotient of 5 4 7 _ 8 be greater than or less than 5? Explain.

17. Reasoning How many times will you need to fill a 1 _ 2 cup measuring cup to measure 4 cups of flour?

18. Geometry The distance around a circular flower bed is 36 feet. Jasper wants to put stakes every 8 inches ( 2 _ 3 of a foot) around the bed. How many stakes does he need?

19. Algebra Which expression is equal to 9 3 3 _ 2 ?

A 2 4 3 _ 9

B 3 4 9 _ 2

C 9 4 2 _ 3

D 9 4 3 _ 2

9_5 1_8 3_7

20 24 48 24 54 12 262_3 262_5 101_2 72 45 30

Thequotientwillbegreaterthan5becauseyouaredividingbyanumberthatislessthan1.

8

54stakes

30429_T09_29-34.indd 33 21/01/11 9:06 PM

Page 5: Understanding Division of Fractions - · PDF file18-01-2012 · Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find _1

4 2 4 _ 7 21 __ 8 1 1 _ 2

2 _ 3 3 _ 5 25 __ 28 1 _ 7 3 _ 8 6 2 _ 3 1 1 _ 9 4 _ 5 9 __ 10 1 1 _ 8 8 _ 9 16 __ 21

4 2 _ 3 servings

Sample answer: Each piece is 1 __ 12 of the cake.

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38 Topic 9

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Reteaching

9-3Dividing Fractions

To divide by a fraction, you can multiply by its reciprocal. The reciprocal of a number has the numerator and the denominator reversed.

Find 4 _ 5 ÷ 3 __

10 .

Step 1

Rewrite the division as multiplication using the reciprocal of the divisor.

The reciprocal of 3 __ 10

is 10 __

3 .

4 _ 5 � 3 __

10 � 4 _

5 × 10

__ 3

Step 2

Divide out common factors if possible. Then multiply.

2

4 _ 5 � 10

___ 3 � 8 _

3

1

Step 3

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

8 _ 3 � 2 2 _

3

Find each quotient. Simplify if possible.

1. 1 _ 2 � 1 _

4 � 1 _

2 � � 2. 4 _

7 � 8 __

21 = � �

Reciprocal of 1 _ 4 Reciprocal of 8 __

21

3. 1 _ 3 � 1 _

2 4. 2 _

5 ÷ 2 _

3 5. 5 _

8 ÷ 7 __

10

6. 3 _ 7 � 3 7. 1 _

3 � 8 _

9 8. 5 _

6 ÷ 1 _

8

9. 5 _ 9 � 1 _

2 10. 3 _

5 � 3 _

4 11. 3 _

4 ÷ 5 _

6

12. 9 __

10 � 4 _

5 13. 1 _

3 � 3 _

8 14. 4 _

7 ÷ 3 _

4

15. Aaron has 7 _ 8 gallon of bottled water. How many 3 __

16 -gallon servings

can he pour?

16. Draw a Picture Show how Rebecca can divide 3 _ 4 of a cake into

9 pieces. What fraction of the whole cake will each piece be?

Page 6: Understanding Division of Fractions - · PDF file18-01-2012 · Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find _1

2 _ 5 3 _ 4 1 1 _ 2 1 _ 9 1 1 _ 7 8 _ 9 1 2 _ 3 5 _ 8 2 1 _ 3 4 _ 9 3 __ 32 1 1 _ 2

No, Serena needs to use the reciprocal of the divisor, not the dividend, when she rewrites the expression as a multiplication expression: 4 __ 10 � 5 _ 8 � 4 __ 10 � 8 _ 5 , which can be simplified as 4 _ 5 � 4 _ 5 � 16 __ 25 .

10 boards

2 11 __ 12 servings

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Topic 9 39

Practice 9-3

Practice

9-3Dividing FractionsFind each quotient. Simplify if possible.

1. 1 _ 3 � 5 _ 6 � 2. 3 _ 8 � 1 _ 2 � 3. 7 _ 8 � 7 __ 12 �

4. 5 _ 9 � 5 � 5. 6 _ 7 � 3 _ 4 � 6. 2 _ 3 � 3 _ 4 �

7. 1 _ 2 � 3 __ 10 � 8. 5 __ 12 � 2 _ 3 � 9. 14 __ 15 � 2 _ 5 �

10. 1 _ 3 � 3 _ 4 � 11. 3 _ 8 � 4 � 12. 9 __ 10 � 3 _ 5 �

13. Writing to Explain Serena said that by looking for common factors and simplifying the expression, she found that 4 __ 10 � 5 _ 8 � 1 9 __ 16 . Do you agree with Serena? Why or why not?

5

10 __ 4 � 5 _ 8 � 25 __ 16 � 1 9 __ 16 4

14. A 5 _ 6 -yard piece of fencing is made of boards that are 1 __ 12 yard wide. How many boards make up the fence?

15. Nathan has 7 _ 8 lb of hummus. How many 3 __ 10 -lb servings does he have?

16. Algebra Which equation can you use to find the number of 1 _ 4 −inch pieces that can be cut from a piece of metal 5 _ 8 of an inch long?

A 5 _ 8 � 1 _ 4 � n

B 1 _ 4 � 5 _ 8 � n

C 5 _ 8 � 1 _ 4 � n

D 1 _ 4 � 8 _ 5 � n

Page 7: Understanding Division of Fractions - · PDF file18-01-2012 · Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find _1

Sample answers are shown. 2 5 6 6 7 9 8 9 3 4 7 10

Sample answer: Hayden’s estimate is closer because he used numbers closer to the original numbers.

Sample answer: About 8 pieces

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Reteaching

9-4Estimating Quotients

When you are working with fractions and mixed numbers, you can

estimate using rounding and compatible numbers.

Estimate 23 5 _ 6 � 8 3 _

7 . Estimate 31 1 _

6 � 4 5 _

8 .

23 5 _ 6 � 8 3 _

7

24 � 8 � 3

23 5 _ 6 � 8 3 _

7 � 3

Round each mixed number to the nearest whole number.

Divide.

31 1 _ 6 � 4 5 _

8

30 � 5 � 6

31 1 _ 6 � 4 5 _

8 � 6

Change 31 1 _ 6 and

4 5 _ 8 to the nearest

compatible whole numbers.

Think: 31 1 _ 6 and 4 5 _

8

are close to 30 and 5.

Estimate each quotient.

1. 11 1 _ 2 � 6 1 _

4 2. 19 1 _

3 � 3 2 _

3 3. 41 7 _

9 � 7 1 _

5

4. 35 1 _ 8 � 5 4 _

5 5. 61 3 _

8 � 8 5 _

9 6. 72 2 _

9 � 7 7 _

8

7. 86 3 _ 4 � 10 5 _

6 8. 26 9 __

10 � 2 5 _

8 9. 11 2 _

7 � 3 3 _

5

10. 7 9 __ 10

� 2 3 __ 10

11. 47 6 __ 10

� 7 1 __ 12

12. 60 5 __ 12

� 5 4 _ 9

13. Critical Thinking Which of these two estimates is closer to the actual quotient? How do you know?

Lisa’s estimate: 55 1 _ 2 � 6 3 _

4 � 54 � 6 � 9

Hayden’s estimate: 55 1 _ 2 � 6 3 _

4 � 56 � 7 � 8

14. Patrick uses wire to make wreaths. He has 31 1 _ 2 feet of wire left on

a spool. Estimate how many 3 3 _ 4 pieces can he cut from the longer

piece of wire.

Page 8: Understanding Division of Fractions - · PDF file18-01-2012 · Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find _1

Sample answers are shown. 6 4 7

2 9 310 8 59 4 1013 6 3

Sample answer: 6 is an overestimate because Tran rounded the width of the paper up and the width of the strips down.

Sample answer: Divide 22 by 2 7 _ 8 . Use compatible numbers to estimate: 21 � 3 � 7, so she has enough yarn to make at least 7 gift baskets.

Sample answer: I used compatible numbers: 250 � 10 � 25.

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Topic 9 45

Practice 9-4

Practice

9-4Estimating QuotientsEstimate each product.

1. 37 1 _ 3 � 5 7 _

8 � 2. 25 1 _

2 � 6 1 _

4 � 3. 49 4 _

5 � 6 1 _

2 �

4. 12 3 _ 4 � 5 5 _

9 � 5. 43 2 _

3 � 5 2 _

5 � 6. 8 1 _

3 � 2 9 __

10 �

7. 67 1 _ 5 � 7 2 _

7 � 8. 55 5 _

9 � 7 1 _

6 � 9. 19 6 _

7 � 4 1 _

8 �

10. 71 4 _ 5 � 7 8 _

9 � 11. 15 7 __

10 � 3 4 _

9 � 12. 79 4 _

7 � 8 5 _

8 �

13. 26 1 _ 4 � 2 3 _

8 � 14. 40 8 _

9 � 7 3 _

5 � 15. 58 1 _

3 � 19 5 _

6 �

16. Number Sense Tran wants to cut strips of paper that are 2 1 _ 4 in. wide.

His sheet of paper is 11 1 _ 2 in. wide. He estimates that 11 1 _

2 � 2 1 _

4 � 6,

so he can cut 6 strips from each sheet of paper. Is his estimate an overestimate or an underestimate? Explain.

17. Writing to Explain Eliza uses 2 7 _ 8 feet of yarn in each gift basket

she makes. Explain how to estimate how many baskets Eliza can make if she has 22 feet of yarn.

18. Geometry The area of this rectangle is 257 1 _ 4 sq in. What is the

best estimate of side length w?

257 sq in.1

4

1

210 in.

w

A 66,000 in.

B 50 in.

C 25 in.

D 5 in.

19. Critical Thinking What estimation method did you use to find the length of side w in Problem 18?

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-5Reteaching

9-5Dividing Mixed NumbersYou can follow these steps to find 5 1 _ 3 4 1 1 _ 3 and 21 4 2 1 _ 3 .

30422_T10_10_6RT-1

Step 1First estimate.Then write each number as an improper fraction.

Step 2Look for common factors. Simplify, then multiply.

Find the reciprocal of the divisor. Rewrite as a multiplication problem.

Step 3

1

1113

135 1

13

13Find 5 1 .

Estimate 5 1 5.

Estimate 21 2 10 .

43

163

73

211

34

163

73

211

37

211

43

163

37

211

37

211

34

163

34

441

163

3

1

4

91 91

321 2

13

12

Find 21 2 .

4 is close to 5,so the answer is reasonable.

129 is close to 10 ,

so the answer is reasonable.

Find each quotient. Simplify if possible.

1. 2 2 _ 3 4 3 1 _ 4 5 2. 1 3 _ 4 4 4 1 _ 8 5

3. 2 1 _ 5 4 2 1 _ 3 5 4. 5 1 _ 4 4 3 5

5. 10 4 3 1 _ 4 5 6. 7 1 _ 4 4 2 1 _ 8 5

7. Writing to Explain Paper needs to be cut for voting ballots. Each piece of paper is 10 1 _ 2 in. long. Each ballot should be 1 3 _ 4 in. long. How many ballots can be cut from one piece of paper?

32 __ 39 14 __ 33 33 __ 35 1 3 _ 4

3 1 __13 3 7 __

17

Six ballots can be cut.

30429_T09_47-52.indd 50 21/01/11 9:09 PM

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Topic 9 51

Practice 9-5

Practice

9-5Dividing Mixed NumbersFind each quotient. Simplify if possible.

1. 1 1 _ 2 � 2 1 _

3 � 2. 4 1 _

4 � 3 1 _

8 � 3. 2 1 _

4 � 5 1 _

2 �

4. 3 1 _ 2 � 2 1 _

4 � 5. 3 3 _

4 � 2 � 6. 1 1 _

2 � 2 1 _

4 �

7. 8 � 2 3 _ 4 � 8. 2 1 _

2 � 1 3 _

8 � 9. 4 2 _

3 � 1 3 _

4 �

10. Reasoning Is it possible to divide 15 by a mixed number and get a quotient that is greater than 15? Explain.

Room Gallons of Paint

Kitchen 2 1 _ 2

Bedroom 3 3 _ 4

Living room 4 1 _ 3

Max is painting the inside of an apartment complex. The table shows how many gallons of paint are needed to paint each type of room.

11. How many kitchens can Max paint with 20 gal?

12. How many living rooms can Max paint with 26 gal?

13. How many bedrooms can Max paint with 60 gal?

14. Find 4 1 _ 2 � 2 1 _

4 .

A 1

B 2

C 3

D 4

15. Writing to Explain Explain how you would find 4 1 _ 5 � 2 1 _

3 .

9 __ 14 1 9 __25 9 __ 22

1 5 _ 9 1 7 _ 8 2 _ 3

2 10 __11 1 9 __

11 2 2 _ 3

No, because you are dividing by a

number that is greater than 1.

8 kitchens

6 living rooms

16 bedrooms

Sample answer:

Write each mixed number as an improper

fraction. Find the reciprocal of the second

term and multiply the two fractions. Simplify the answer. 1 4 _ 5

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9-6Solving Equations

Here is how to solve addition, subtraction, multiplication, and division equations with fractions.

9

35

8

2 2

Solve 2 6 .

2 6

x 6 2

x 8

y

y

Solve a 3 .

a 3

a 3

y

2

a

y

a ( ) ( )

Addition Subtraction

Solve each equation and check your answer.

1. z � 2 1 _ 3 � 3 1 _

6 2. 6n � 3 _

4

3. x � 1 � 4 2 _ 3 4. y � 1 _

2 � 2 1 _

8

5. 3 _

8 � n � 10 6. 2 2 _

9 � 5 � x

7. Algebra The rainfall total for this month is 4 9 __ 10

in. Yesterday it rained 2 1 __

10 in. Use the equation n � 2 1 __

10 � 4 9 __

10 to calculate how

much rainfall was received before yesterday.

5 _ 6 1 _ 8

5 2 _ 3 1 1 __ 16

9 5 _ 8 4 _ 9

2 4 _ 5 in.

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Topic 9 57

Practice 9-6Practice

9-6Solving EquationsSolve each equation and check your answer.

1. y 1 1 1 _ 4 5 2 3 _ 8

2. w 2 2 5 3 1 _ 2

3. z 4 3 _ 4 5 4 1 _ 4

4. 1 _ 3 5 7 _ 8 q

5. 6 1 _ 2 5 5 _ 6 b

6. 2 1 _ 4 5 p 2 3 _ 8

7. 2 1 _ 4 5 x 4 1 _ 2

8. Number Sense Is the solution of m 4 2 _ 3 5 9 greater than or less than the solution of m 4 1 _ 4 5 9? Explain.

9. The bakery used 42 1 _ 3 c of flour. There were 10 1 _ 3 c left in the flour bin. Use the equation x 2 42 1 _ 3 5 10 1 _ 3 to find out how many cups of flour the bakery had to start with.

10. Alex had a ball of string. He cut the string into 26 equal pieces. Each piece measured 3 1 _ 4 in. Use the equation m 4 26 5 3 1 _ 4 to find the length of the ball of string.

11. Solve 12y 5 2 1 _ 4 .

A 1 1 _ 2 B 1 1 _ 8 C 7 __ 36 D 3 __ 16

12. Writing to Explain Write the steps you would use to solve the equation z 1 3 1 _ 5 5 6 3 _ 5 . Solve.

y 5 9 _ 8 or 1 1 _ 8 w 5 11 __ 2 or 5 1 _ 2 z 5 3 3 __ 16 q 5 8 __ 21 b 5 7 4 _ 5 p 5 2 5 _ 8 x 5 9 _ 8 or 1 1 _ 8

Sample answer: m 4 2 _ 3 5 9 is greater because you are dividing by a larger number.

52 2 _ 3 c

84 1 _ 2 in.

Change the mixed numbers to improperfractions and write z 1 16 __ 5 5 33 __ 5 . Then subtract 16 __ 5 from both sides of the equation. So,z 5 17 __ 5 , or 3 2 _ 5 .

30429_T09_53-58.indd 57 2/10/11 1:15 AM

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9-7Problem Solving: Look for a PatternSometimes you can solve a problem by identifying a pattern. Here are two types of patterns.

Patterns in sets of numbers

15 __ 4 , 13 __ 4 , 11 __ 4 , 9 _ 4 , 7 _ 4 , 5 _ 4 , 3 _ 4

Patterns in groups of figures

30422_T10_10-8RT-1

Ask yourself: Are the numbers increasing? Are they decreasing? Do they change by the same amount each time? Do you add, subtract, multiply, or divide to find the next number?

Ask yourself: How is the first figure modified to make the second figure? How is the second figure modified to make the third?

Remember: Once you have identified a possible number pattern, check at least two other consecutive numbers to make sure that the pattern is true for all of the numbers.

Find the missing numbers. Describe the pattern.

1. 3 _ 4 , 1, 1 1 _ 4 , 1 1 _ 2 , , , , , 2 3 _ 4

2. 89, 78, 67, , , , , 12, 1

3. 1 _ 5 , 4 _ 5 , 7 _ 5 , 10 __ 5 , , , , , 25 __ 5

4. Draw the next figure in the pattern below.

30422_T10_10-8RT-2

5. Number Sense The table below shows the number of cells in a culture. How many cells will there be at 4:30?

Time 1:00 1:30 2:00 2:30 3:00

Number of Cells 1 2 4 8 16

1 3 _ 4 2 2 1 _ 4 2 1 _ 2 Add 1 _ 4 56 45 34 23 Subtract 11 13 __ 5 16 __ 5 19 __ 5 22 __ 5 Add 3 _ 5

128 cells

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Page 14: Understanding Division of Fractions - · PDF file18-01-2012 · Understanding Division of Fractions Divide a fraction by a whole number. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 32 Find _1

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Topic 9 63

Practice 9-7

Practice

9-7Problem Solving: Look for a Pattern Find the missing numbers. Describe the pattern.

1. 1 _ 8 , 6 _

8 , 11 __

8 , 16

__ 8 , , , , , 41 __

8

2. 1 _ 4 , 1 _

2 , 1, , , , , 32, 64

3. 1.1, 1.1, 2.2, 6.6, , , ,

4. 14 1 _ 2 , 12 3 _

4 , 11, , , , , 2 1 _

4 , 1 _

2

5. 27, 9, 3, 1, 1 _ 3 , , , ,

6. 3, 5, 9, 15, , , , , 75

7. Number Sense In the figure, the sum of each row 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5

forms a pattern. What is the sum of the seventh row?

8. Which figure completes this pattern?

A

B

C

D

9. Writing to Explain How can you find the answer to 7 without finding the sum of the numbers in a row?

21 __ 8 26 __ 8 31 __ 8 36 __ 8 Add 5 _ 8

2 4 8 16 Multiply by 2

26.4 132 792 5,544 Multiply by 1, then by 2, then by 3, etc.

9 1 _ 4 7 1 _ 2 5 3 _ 4 4 Subtract 1 3 _ 4 1 _ 9 1 __ 27 1 __ 81 1 ___ 243 Multiply by 1 _ 3 or

23 33 45 59 Add 2, add 4, add 6, etc.

49

Sample answer: You can square the number of the row: row 2 � 22 or 4, row 3 � 32 or 9, row 7 � 72 or 49, and so on.

divide by 3