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Chapter 6Multiplying and Dividing Decimals and Fractions

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Chapter 6Multiplying and Dividing Decimals and Fractions

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66Multiplying and Dividing Decimals and Fractions

Lesson 6-1 Multiplying Decimals by Whole Numbers

Lesson 6-2 Multiplying Decimals

Lesson 6-3 Problem-Solving Strategy: Reasonable Answers

Lesson 6-4 Dividing Decimals by Whole Numbers

Lesson 6-5 Dividing by Decimals

Lesson 6-6 Problem-Solving Investigation: Choose the Best Strategy

Lesson 6-7 Estimating Products of Fractions

Lesson 6-8 Multiplying Fractions

Lesson 6-9 Multiplying Mixed Numbers

Lesson 6-10 Dividing Fractions

Lesson 6-11 Dividing Mixed Numbers

Five-Minute Check (over Chapter 5)

Main Idea and Vocabulary

California Standards

Example 1

Example 2

Example 3

Example 4

Example 5

6-16-1 Multiplying Decimals by Whole Numbers

• I will estimate and find the product of decimals and whole numbers.

• scientific notation

Standard 5NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of results.

Find 18.9 × 4.

One Way: Use estimation.

Round 18.9 to 19.18.9 × 4 19 × 4 or 76

18.9× 4

Since the estimate is 76, place the decimal point after the 5.

Another Way: Count decimal places.

18.9× 4

Find 12.7 × 5.

B. 63.5

C. 60.35

D. 63.35

Find 0.56 × 7.

One Way: Use estimation.

Round 0.56 to 1.0.56 × 7 1 × 7 or 7

0.56× 7

Since the estimate is 7, place the decimal point after the 3.

Another Way: Count decimal places.

0.56× 7

Find 0.47 × 8.

C. 4.76

D. 0.392

Find 3 × 0.016.

0.016× 3

Find 0.026 × 2.

A. 0.052

B. 0.52

C. 0.0052

D. 0.502

ALGEBRA Evaluate 5g if g = 0.0091.

0.0091× 5

5g = 5 × 0.0091 Replace g with 0.0091.

ALGEBRA Evaluate 3h if h = 0.0054.

A. 1.62

B. 0.162

C. 0.00162

D. 0.0162

The average distance from Earth to the Sun is 1.5 × 108 kilometers. Write the distance in standard form.

One Way: Use order of operations.

Evaluate 108 first.Then multiply.

1.5 × 108 = 1.5 × 10,000,000= 150,000,000 kilometers

Another Way: Use mental math.

Move the decimal point to the right the same number of places as the exponent of 10, or 8 places.

1.5 × 108 = 1.50000000

= 150,000,000

The average distance from the Sun to the planet Jupiter is 58.8 × 107 kilometers. Choose the answer showing the distance written in standard form.

A. 588,000,000 kilometers

B. 58,000,000 kilometers

C. 5,880,000,000 kilometers

D. 5,800,000 kilometers

Five-Minute Check (over Lesson 6-1)

Main Idea

Example 1

Example 2

Example 3

Example 4

6-26-2 Multiplying Decimals

• I will multiply decimals by decimals.

Estimate 8.3 × 2.9 8 × 3 or 24

×8.32.9747

+ 16624.07

Find 8.3 × 2.9.

one decimal placeone decimal place

two decimal places

Answer: So, the product is 24.07.

Check for Reasonableness

Compare 24.07 to the estimate. 24.07 is about 24.

Find 4.5 × 3.9.

A. 17.55

C. 18.44

D. 19.45

Find 0.12 × 5.3.

Estimate 0.12 × 5.3 0 × 5 or 0

×0.12

+ 600.636

two decimal placesone decimal place

three decimal places

Compare 0.636 to the estimate. 0.636 is about 0.

Find 0.14 × 3.3.

A. 0.636

B. 0.543

C. 0.462

D. 0.723

ALGEBRA Evaluate 1.8r if r = 0.029.

1.8r = 1.8 × 0.029 Replace r with 0.029.

×0.029

1.8232

+ 290.0522

one decimal place

Annex a zero to make four decimal places.

three decimal places

ALGEBRA Evaluate 2.7x if x = 0.038.

A. 2.738

B. 0.1026

C. 0.0126

D. 0.2106

Carmen earns $14.60 per hour as a painter’s helper. She worked a total of 15.75 hours one week. How much money did she earn?

×$14.60

15.757300

229.9500

two decimal placestwo decimal places

73001460+

Estimate 14.60 × 15.75 15 × 16 or 240.

Compare $229.95 to the estimate. $229.95 is about $240.

Answer: So, Carmen earned $229.95.

Alex went shopping for 6.5 hours and spent $32.50 per hour. How much did she spend?

A. $211.25

B. $225

C. $250.25

D. $211.50

Main Idea

Example 1: Problem-Solving Strategy

6-36-3 Problem-Solving Strategy: Reasonable Answers

• I will solve problems by determining reasonable answers.

Standard 5MR3.1 Evaluate the reasonableness of the solution in the context of the original situation.

For their science project, Stephanie and Angel need to know about how much more a blue whale weighs in pounds than a humpback whale.

They have learned that there are 2,000 pounds in one ton. While doing research, they found a table that shows the weights of whales in tons.

Understand

What facts do you know?

• There are 2,000 pounds in one ton.

• A blue whale weighs 151.0 tons.

• A humpback whale weighs 38.1 tons.

What do you need to find?

• A reasonable estimate of the difference in the weight of a blue whale and a humpback whale.

Estimate to find the weight of each whale in pounds and then subtract to find a reasonable estimate of the difference.

Blue whale:

Answer: A reasonable estimate for the difference in the weight of a blue whale and a humpback whale is 220,000 pounds.

Humpback whale:

2,000 × 151

2,000 × 38.1

2,000 × 150

2,000 × 40

300,000 80,000

300,000 – 80,000 = 220,000

Look back at the problem. A blue whale weighs about 150 – 40 or 110 more tons than a humpback whale. This is equal to 110 × 2,000 or 220,000 pounds. So the answer is reasonable.

Main Idea

Example 1

Example 2

Example 3

6-46-4 Dividing Decimals by Whole Numbers

• I will divide decimals by whole numbers.

Standard 5NS2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors.

Find 7.2 ÷ 3.

Estimate 7.2 ÷ 3 7 ÷ 3 or about 2.

3 7.22 4.

– 61 2

– 1 20 7.2 ÷ 3 = 2.4 Compared to the estimate,

the quotient is reasonable.

Find 6.4 ÷ 4.

C. 1.6

D. 0.8

Find 6.6 ÷ 15.

Estimate 6.6 ÷ 15 7 ÷ 15 or about 0.5.

15 6.60 4.

– 06 6

– 6 06

6.6 ÷ 15 = 0.44 Compared to the estimate, the quotient is reasonable.

0– 60

Find 8.8 ÷ 16.

A. 5.5

B. 0.55

C. 0.22

D. 2.2

During a science experiment, Nita measured the mass of four unknown samples. Her data is shown below.

Sample 1: 6.22 g

Sample 2: 6.00 g

Sample 3: 6.11 g

Sample 4: 6.11 g

First, add all the data together.

Answer: So, the mean mass of Nita’s samples is 6.11 grams.

6.226.006.116.11

Divide by the number of addends to find the mean mass.

4 24.446.11

Greta bought 4 pairs of socks for $25.36. If each pair of socks costs the same amount, how much was each pair?

A. $6.34

B. $6.00

C. $4.63

D. $3.64

Main Idea

Example 1

Example 2

Example 3

Example 4

6-56-5 Dividing by Decimals

Dividing Decimals

• I will divide decimals by decimals.

Standard 5NS2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors.

Find 21.44 ÷ 6.4.

Estimate 21 ÷ 7 = 3

64 214.43 3.

– 19222 4

3 2–

6.4 21.444

Divide as with whole numbers.

Annex a zero to continue.

Place the decimal point.

21.44 divided by 6.4 is 3.34.

Compare to the estimate.

3.34 × 6.4 = 21.44

Find 25.45 ÷ 5.5.

A. 5.9

B. 5.09

C. 5.90

D. 50.9

Find 72 ÷ 0.4.

4 720.18 .

– 432

0.4 72.00

Answer: So, 72 ÷ 0.4 = 180.

Check 180 × 0.4 = 72

Find 45 ÷ 0.9.

A. 0.50

D. 5.0

Find 0.024 ÷ 2.4.

24 0.240 0.

– 002

2.4 0.0241

Answer: So, 0.024 ÷ 2.4 = 0.01.

Check 0.01 × 2.4 = 0.024

24 does not go into 2, so write a 0 in the tenths place.

Find 0.036 ÷ 1.2.

A. 0.03

C. 0.3

D. 1.2

Ioviana bought a stock at $42.88 per share. If she spent $786.85, how many shares did she buy? Round to the nearest tenth.

42.88 786.85

Find 786.85 ÷ 42.88.

42.88 78685.008 3.

– 428835805

–1501 0

Answer: So, to the nearest tenth, 786.85 ÷ 42.88 = 18.4. So, Ioviana bought about 18.4 shares.

1286 4–2146 02144 0–

Oprah bought televisions for everyone in her audience at $245.75 per TV. If she spent $21,773.45 about how many people were in the audience? Round to the nearest whole number.

A. 88.6 people

B. 88 people

C. 89 people

D. 90 people

Main Idea

Example 1: Problem-Solving Investigation

6-66-6 Problem-Solving Investigation: Choose the Best Strategy

• I will choose the best strategy to solve a problem.

Standard 5MR1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.

Standard 5NS2.1 Add, subtract, multiply, and divide with decimals; . . . and verify the reasonableness of results.

MIGUEL: At the store, I saw the following items: a batting glove for $8.95, roller blades for $39.75, a can of tennis balls for $2.75, and weights for $5.50. I have $15 and I would like to buy more than one item.

YOUR MISSION: Find which items Miguel can buy and spend about $15.

Understand

What facts do you know?

• You know the cost of the items and that Miguel has $15 to spend.

What do you need to find?

• You need to find which items Miguel can buy.

Make an organized list to see the different possibilities and use estimation to be sure he spends about $15.

Since the roller blades cost more than $15, you can eliminate the roller blades. The batting glove is about $9, the weights are about $6, and the can of tennis balls is about $3.

Start with the batting glove:

• 1 glove + 1 weights ≈ $9 + $6 or $15

• 1 glove + 2 cans of tennis balls ≈ $9 + $6 or $15

List other combinations that contain the weights:

• 2 weights + 1 can of tennis balls ≈ $12 + $3 or $15

• 1 weights + 3 cans of tennis balls ≈ $6 + $9 or $15

List the remaining combinations that contain only tennis balls:

• 5 cans of tennis balls ≈ $15

Check the list to be sure that all of the possible combinations of sporting good items that total no more than $15 are included.

Example 1

Example 2

Example 3

6-76-7 Estimating Products of Fractions

• I will estimate products of fractions using compatible numbers and rounding.

• compatible numbers

Standard 5MR2.5 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.

Standard 5NS2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.

Find a multiple of 5 that is close to 16.

× 1615

× 16 means of 16. 15

Estimate × 16.15

× 1515

× 15 = 315

Answer: So, × 16 is about 3.15

15 and 5 are compatible numbers since 15 ÷ 5 = 3

15 ÷ 5 = 3

Estimate × 19.19

C. 212

D. 214

Find a multiple of 4 that is close to 23.

× 2314

Estimate × 23 first. 14

Estimate × 23.34

× 2414

× 24 = 614

Use 24 since 24 and 4 are compatible numbers.

24 ÷ 4 = 6

Answer: So, of 23 is about 18.34

If of 24 is 6, then of 24 is 6 × 3 or 18.34

Estimate × 29.35

A. 1725

1 × 16

Estimate × .45

1 × 16

Answer: So, × is about .45

Estimate × .56

A. 113

D. 213

Estimate the area of the rectangle.

Round each mixed number to the nearest whole number.

× 7 × 2 = 14

Answer: So, the area is about 14 square inches.

Estimate the area of a rectangle with a width of

9 in. and a length of 3 in.45

A. 30 sq. in.

B. 27 sq. in.

C. 40 sq. in.

D. 36 sq. in.

Main Idea

Key Concept: Multiply Fractions

Example 1

Example 2

Example 3

Example 4

6-86-8 Multiplying Fractions

Multiplying Fractions

• I will multiply fractions.

Find × .15

=1 × 1 5 × 6

Multiply the numerators.Multiply the denominators.

Simplify.

Find × .17

A. 214

Find × 7.58

=5 × 7 8 × 1

Multiply.

Simplify.

Estimate × 7 = 3 .12

Write 7 as . 71

or 438

4 is about 3 .38

Find × 9.78

B. 7 79

D. 8 19

Find × .37

× =3 × 2 7 × 9

Multiply.

Simplify.

Estimate × 0 = 0.12

is about 0.2

Find × .46

B. 1242

ALGEBRA Evaluate pq if p = and q = .34

=3 × 2 4 × 3

The GCF of 2 and 4 is 2. The GCF of 3 and 3 is 3. Divide the numerator and the denominator by 2 and 3.

pq = ×34

Replace p with and q with .34

Simplify.

Answer: So, × = .34

ALGEBRA Evaluate gh if g = and h = .25

C. 1050

Main Idea

Key Concept: Multiply Mixed Numbers

Example 1

Example 2

Example 3

6-96-9 Multiplying Mixed Numbers

• I will multiply mixed numbers.

Find × 3 .38

× 3 13

Estimate Use compatible numbers × 3 = 112

3 × 10 8 × 3

Write 3 as . 13

× 103

or 114

Divide 10 and 8 by their GCF, 2. Divide 3 and 3 by their GCF, 3.

Simplify. Compare to the estimate.

Find × 2 .45

A. 2 2

12D. 3

B. 2 12

Belinda lives 1 times farther from school than

Elena does. If Elena lives 4 miles from school,

how far from school does Belinda live?

Elena lives 4 miles from school. Multiply 4 × 1 .15

4 × 115

21 × 3 5 × 2

First, write mixed numbers as improper fractions.

Then, multiply the numerators and multiply the denominators.

Simplify.= or 66310

Answer: So, Belinda lives 6 miles from school.310

Mariah is making 4 times the recipe for crispy

treats. If the recipe calls for 1 cups of butter, how

much butter will she need?

A. 8516

B. 5 15

C. 5 5

D. 2516

ALGEBRA If y = 3 and w = 2 , what is the value

of wy?

=Divide the numerator and denominator by 2 and 5.

Simplify.

wy = 2 × 3 45

34 Replace w with 2 and y with 3 .

154 ×

= or 10 12

ALGEBRA If m = 4 and n = 2 , what is the value of mn?

C. 13 3

A. 18514

B. 18014

Key Concept: Divide Fractions

Example 1

Example 2

Example 3

Example 4

Example 5

6-106-10 Dividing Fractions

• I will divide fractions.

• reciprocal

Find the reciprocal of 8.

Since, 8 × = 1, the reciprocal of 8 is .18

Answer: 18

Find the reciprocal of 6.

B. 0.6

Answer: 53

Find the reciprocal of .35

Since, × = 1, the reciprocal of is .35

Find the reciprocal of .23

1 × 6 3 × 5

Multiply by the reciprocal .65

Divide 3 and 6 by the GCF, 3.

Multiply numerators. Multiply denominators.

Find ÷ .13

Find ÷ .23

B. 1221

5 ÷ 16 Multiply by the reciprocal .

Simplify.

Find 5 ÷ .16

= or 30301

Find 6 ÷ .18

A relay race is of a mile long. There are 4 runners

in the race. What portion of a mile will each racer

Divide into 4 equal parts.34

Answer: So, each runner ran of a mile.316

= Multiply.

Simplify. = 316

÷ 434 Multiply by the reciprocal.=

Three ladies decided to knit the world’s longest

scarf. It was of a mile long. If each lady knit the

same amount, what portion of a mile did each lady

Main Idea

Key Concept: Dividing by Mixed Numbers

Example 1

Example 2

Example 3

6-116-11 Dividing Mixed Numbers

• I will divide mixed numbers.

Find 6 ÷ 2 .14

6 ÷ 214

Estimate 6 ÷ 3 = 2

Write mixed numbers as improper fractions.=

Multiply by the reciprocal.=

Divide 2 and 4 by the GCF, 2, and 25 and 5 by the GCF, 5.=

Simplify. = or 2 1

Check for Reasonableness 2 is about 3.12

Find 3 ÷ 1 .34

34D. 2

B. 2 12

ALGEBRA Find a ÷ b if a = 2 and b = .58

a ÷ b

Write the mixed number as an improper fraction.=

= 2 ÷58

Multiply by the reciprocal.=218

Replace a with 2 and b with . 58

Simplify.=6316

or 31516

ALGEBRA Find f ÷ g if f = 3 and g = .23

D. 216

A. 535

C. 5 1315

B. 2 7

Estimate 180 ÷ 4 = 45

180 ÷ 3 34

Write mixed numbers as improper fractions.

A team took 3 days to complete 180 miles of an

adventure race consisting of hiking, biking, and river

rafting. How many miles did they average each day?

= ÷1801

Multiply by the reciprocal.= ×1801

Simplify.=72015

Compare to the estimate.

Answer: So, the team averaged 48 miles each day.

A cross country skier took 4 days to travel 240

miles. How many miles did he average each day?

D. 52 miles12

A. 51 miles37

B. 51 miles6

C. 50 miles 34

Five-Minute Checks

Dividing Decimals

Multiplying Fractions

Lesson 6-1 (over Chapter 5)

Lesson 6-2 (over Lesson 6-1)

(over Chapter 5)

Find 6 – 2 .14

C. 4 12

A. 3 12

B. 8 34

(over Chapter 5)

Find 5 – 3 .13

C. 8 69

A. 2 49

B. 1 79

D. 269

(over Chapter 5)

Find the value of n.

n + 1 = 835

A. 625

C. 4 35

(over Chapter 5)

Find the value of n.

D. 1 56

n + 10 = 1212

(over Lesson 6-1)

Find 3.8 × 2.

A. 5.8

B. 7.6

C. 5.6

Find 0.6 × 25.

(over Lesson 6-1)

Find 0.038 × 15.

A. 0.63

B. 1.35

C. 0.57

(over Lesson 6-1)

Find 0.0003 × 17.

A. 0.0021

B. 0.0034

C. 0.0051

D. 0.21

(over Lesson 6-1)

Mercury is approximately 3.6 × 107 miles from the Sun. How far is this?

A. 360 mi

B. 1,800,000 mi

C. 36,000,000 mi

D. 3,000,000 mi

(over Lesson 6-1)

(over Lesson 6-2)

Find 75.4 × 2.9.

A. 150.36

B. 77.36

C. 125

D. 218.66

(over Lesson 6-2)

Find 0.05 × 0.123.

A. 0.15

B. 0.00615

C. 0.00506

(over Lesson 6-2)

Evaluate 2.5y if y = 4.8.

A. 4.8

B. 10.48

C. 8.5

(over Lesson 6-2)

Selam makes $6.75 an hour. Last week, she worked 12.4 hours. How much did she earn?

A. $36.75

B. $48.55

C. $48.50

D. $83.70

(over Lesson 6-3)

Determine a reasonable answer. Mr. Nieto has 63.75 yards of fencing. How many feet of fencing is that?

A. 127.50 ft

B. 191.25 ft

C. 255 ft

D. 33.75 ft

Cafeteria workers made 23.5 gallons of punch for an awards banquet. They are serving the punch in 1-quart pitchers. How many containers do they need for all the punch? (1 gal = 4 qt)

A. 11.75 pitchers

B. 40 pitchers

C. 4 pitchers

D. 94 pitchers

(over Lesson 6-3)

(over Lesson 6-4)

Find 27.09 ÷ 9. Round to the nearest tenth if necessary.

A. 7.09

C. 3.01

(over Lesson 6-4)

A. 75.7

C. 35.5

D. 102

(over Lesson 6-4)

A. 35.4

C. 35.04

D. 23.5

(over Lesson 6-4)

Find the mean for the following set of data: 7.8, 9.02, 2.62.

A. 4.5

B. 4.45

C. 6.48

D. 5.55

(over Lesson 6-5)

Find 24.36 ÷ 4.2.

A. 5.8

B. 5.66

D. 6.18

(over Lesson 6-5)

Find 15.39 ÷ 0.05.

A. 128.5

D. 307.8

(over Lesson 6-5)

Find 0.648 ÷ 0.12.

A. 0.85

B. 1.48

C. 5.4

D. 5.6

(over Lesson 6-5)

Find 0.782 ÷ 3.4.

A. 0.023

B. 0.015

(over Lesson 6-6)

Choose the best strategy to solve the problem. The sum of three consecutive numbers is 42. What are the three numbers?

A. 12, 14, 16

B. 15, 12, 9

C. 13, 14, 15

C. 1 × 36 = 6

(over Lesson 6-7)

Estimate the product.

A. × 38 = 816

× 3816

B. × 36 = 616

(over Lesson 6-7)

A. 1 × 45 = 45

C. × 45 = 3023

× 4423

B. × 45 = 4523

(over Lesson 6-7)

B. × 45 = 3038

C. × 45 = 1712

A. × 1 = 12

(over Lesson 6-7)

B. 25 × 40 = 1,000 ft2

C. 26 × 40 = 1,040 ft2

A. 25 × 40 = 975 ft2

A pool is 25 feet wide and 39 feet long.

Estimate the area.

(over Lesson 6-8)

Multiply. Write in simplest form.

(over Lesson 6-8)

C. 113

(over Lesson 6-8)

Evaluate n if n = . Write each product in

simplest form.

(over Lesson 6-8)

Evaluate 10n if n = . Write each product in

simplest form.

B. 423

A. 7 12

(over Lesson 6-9)

× 1 23

B. 2 23

C. 1 12

A. 3 35

(over Lesson 6-9)

B. 24 12

C. 21 12

A. 10 15

5 × 4 56

D. 31 12

(over Lesson 6-9)

The length of a square sandbox is 4 feet. What is

the area of the sandbox?

B. 23 ft2 59

C. 21 ft2 59

A. 21 ft2 79

(over Lesson 6-9)

ALGEBRA If a = 4 and t = 1 , what is the value

of at?

C. 5 35

A. 5 5

(over Lesson 6-10)

Divide. Write in simplest form.

(over Lesson 6-10)

A. 3 12

(over Lesson 6-10)

6 ÷ 23

(over Lesson 6-10)

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