Transcript
CHAPTE R
6
Multiply and Divide Decimals
connectED.mcgraw-hill.com
The
BIG Idea
Investigate
How does using estimation help me to multiply and divide decimals accurately?
Animations Vocabulary Math Songs Multilingual eGlossary
Learn Personal Tutor Virtual Manipulatives
Make this Foldable to help you organize your notes.
d s an l ply ma lti eci u M eD vid Di
Audio Foldables
Review Vocabulary
Practice Self-Check Practice
r that has a digit in the decimal decimal A numbe ce, and beyond. tenths place, hundredths pla
0.32
eGames Worksheets
tenths
hundredths
Assessment
Review Vocabulary English compatible numbers decimal point rounding 238
Español
números compatibles punto decimal redondear
When Will I Use This?
Your Turn! You will solve thhiis teerrr. problem in the chap
Multiply and Divide Decimals 239
Are You Ready for the Chapter?
Text Option
You have two options for checking Prerequisite Skills for this chapter.
Take the Quick Check below.
Multiply. 1. 16 × 30
2. 12 × 28
3. 320 × 10
4. 34 × 21
5. 56 × 50
6. 109 × 34
7. How much will it cost for Gary to reserve 13 lanes?
Bowling Reservations 1 lane
$12
8. Mr. Irwin bought 4 concert tickets for his family. If each ticket costs $24, how much did Mr. Irwin spend?
Divide. 9. 114 ÷ 6 12. 1,500 ÷ 10
10. 84 ÷ 4
11. 64 ÷ 4
13. 512 ÷ 8
14. 5,000 ÷ 5
15. Hugo spent $216 on 4 sweaters. If each sweater cost the same amount, find the cost of one sweater.
Round each decimal to the nearest whole number. 16. 2.7
17. 0.7
18. 18.2
19. 6.34
20. 9.8
21. 9.4
Online Option 240
Take the Online Readiness Quiz.
Multiply and Divide Decimals
Multi-Part Lesson
1
PART
Multiply Decimals by Whole Numbers A
Main Idea I will estimate products of whole numbers and decimals.
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, and decimals. GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution.
B
C
Estimate Products of Whole Numbers and Decimals Previously, you used rounding and compatible numbers to estimate products with whole numbers. You can use the same methods to estimate products of whole numbers and decimals.
Use Rounding G GASOLINE Mrs. James buys about 15.8 gallons of gas each week for her car. About how many gallons of gas does she w buy in 5 weeks? Estimate the product of 15.8 and 5. 15.8 × 5 16 × 5 16 × 5 = 80
Round 15.8 to 16 because 15.8 is closer to 16 than 15. Multiply.
So, 15.8 × 5 is about 80. Mrs. James buys about 80 gallons of gas in 5 weeks.
SHOPPING A gel pen at a craft store costs $1.49. Lucinda wants to buy three gel pens, each in a different color. She has $6 to spend. Does Lucinda have enough money to buy three gel pens? Explain. 3 × $1 = $3 Estimate $1.49 as $1. 3 × $2 = $6 Estimate $1.49 as $2. The actual cost of the pens is between $3 and $6. Lucinda has enough money to buy three gel pens.
Lesson 1A Multiply Decimals by Whole Numbers 241
Use Compatible Numbers Estimate the product of 19 and 11.25. E 19 × 11.25 Compatible numbers are numbers that are close to the actual numbers and easy to multiply. 92 × 4.2 −−
−−
100 4 ×− −− 400
20 × 10 20 and 10 are compatible numbers. 20 × 10 = 200 So, 19 × 11.25 is about 200.
Use Compatible Numbers M MEASUREMENT A worker loads 16 packages onto a truck. Each package weighs 58.5 pounds. About how many pounds are loaded onto the truck? Judge the reasonableness of your answer. 58.5 × 16 Estimate the product of 58.5 and 16. 60 × 20
60 and 20 are compatible numbers.
So, 58.5 × 16 is about 1,200. The truck is holding about 1,200 pounds. Since both numbers were rounded up, the estimate is higher than the actual product.
Estimate each product product. See Examples 11–44 1. 2 × $4.10
2. 5 × $12.65
3. 18.4 × 10
4. 24.7 × 3
5. A group of five friends is going to see a movie. If one ticket costs $8.50, is $45 a reasonable estimate for the cost of 5 tickets? Explain.
6. About 57 people rent skates at the ice rink each day. It costs $4.25 to rent skates. About how much does the rink make in skate rentals in one day?
7. A cantaloupe weighs 1.8 pounds. About how much do 3 cantaloupes weigh?
8.
242
Multiply and Divide Decimals
E
TALK MATH Explain how to round
18.9 to the nearest whole number.
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Estimate i each h product. d See Examples 1–4 9. 4 × $4.62 13. 7.2 × 5
10. 3 × $23.07
11. $15.50 × 6
12. $16.85 × 9
14. 14.5 × 3
15. 8 × 19.7
16. 10 × 26.2
17. Measurement A turkey costs $0.89 per pound and weighs 14 pounds. Is $15 a reasonable estimate for the cost of the turkey? Explain.
18. A dragonfly can fly at a speed of 17.8 miles per hour. At that speed, about how far could a dragonfly fly in 5 hours?
Art
Artsonia is the world’s largest kids’ online art museum, where students have their own art gallery online. Friends and family can order items like the ones shown below that display students’ artwork.
Estimate the cost of each order. 19. 4 keychains
20. 2 plush bears
21. 3 coffee mugs and 2 keychains
22. 5 mouse pads and a plush bear
23. If you have $50, how many coffee mugs can you buy? Use the guess, check, and revise strategy.
24. OPEN ENDED Write a real-world multiplication problem involving whole numbers and decimals in which you would use estimation to solve. Describe the steps that you used to solve the problem. 25.
E
WRITE MATH Explain why it is important to use estimation
when solving multiplication problems involving decimals.
Lesson 1A Multiply Decimals by Whole Numbers 243
Multi-Part Lesson
1
Multiply Decimals
PART
A
B
C
Main Idea
Multiply Decimals by Whole Numbers
I will explore multiplying a decimal by a whole number.
You can use decimal models to multiply a decimal and a whole number. Each 10 by 10 square represents 1 whole.
Materials colored pencils
Model 0.4 × 2 using decimal models.
grid paper
0.4
Shade four rows of each decimal model to represent 0.4.
2
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals. Also addresses GLE 0506.1.4.
Combine both shaded parts onto one model.
So, 0.4 × 2 = 0.8.
and Apply It Use decimal models to solve. 1. 0.5 × 2
2. 0.2 × 3
3. 2 × 0.6
4. The product of a multiplication problem is represented by the decimal model shown. Write two possible factors for the model. 5.
244
E TALK MATH How could you solve Exercises 1–3 without using decimal models?
Multiply and Divide Decimals
Multi-Part Lesson
1
Multiply Decimals
PART
A
Main Idea I will multiply a decimal by a whole number.
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals. Also addresses GLE 0506.1.7.
B
C
Multiply Decimals by Whole Numbers Multiplying a decimal by a whole number is similar to multiplying whole numbers.
Find 3 × 0.96.
One Way:
Use repeated addition.
3 × 0.96 means 0.96 + 0.96 + 0.96. 21 0.96 0.96 + 0.96 −−−− 2.88
Another Way: 1
0.96 ×3 −−− 2.88
Count decimal places. There are two places to the right of the decimal point. Count two decimal places from right to left in the product.
MEASUREMENT Find the area of a board that is 4 feet by 3.62 feet. Estimate 4 × 3.62 2 3.62 × −−−−4 14.48
4 × 4 or 16
There are two places to the right of the decimal point. Count two decimal places from right to left.
The area of the bulletin board is 14.48 square feet.
Lesson 1C Multiply Decimals by Whole Numbers
245
Multiply. See Examples 1 and 2 Multiply 1. 6 × 0.5
2. 4 × 2.6
3. 7 × 0.89
4. 3 × 2.49
5. 52 × 2.1
6. 3.4 × 7
7. A recipe for a cake calls for 3.5 cups of sugar. How many cups of sugar are needed for 4 cakes? 8.
E
TALK MATH Is the product of 2.8 and 2 greater than 6 or
less than 6? How do you know?
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Multiply. M lti l See Examples l 1 and d2 9. 2 × 1.3
10. 3 × 0.5
11. 1.8 × 9
12. 2.4 × 8
13. 4 × 0.02
14. 0.66 × 5
15. 0.7 × 4
16. 1.59 × 7
17. 0.48 × 3
Algebra Evaluate each expression if x = 3, y = 0.2, and z = 4.5. 18. (7 × 2) × y
19. xz
20. (9 - x) × y
Use the information to solve the problem.
21. Did Sonia or Tom make more money mowing lawns? Explain.
246
Multiply and Divide Decimals
22. OPEN ENDED Write a multiplication problem in which the product has two decimal places. 23. NUMBER SENSE Place the decimal point in the answer below to make it correct. Explain your reasoning.
498 × 8.32 = 414336 24.
E
WRITE MATH Write a real-world problem that can be solved using
multiplication. One factor should be a decimal.
Test Practice 25. One pound off tomatoes costss $1.59. How much do 5 pounds of tomatoes cost? A. $5.59
C. $7.95
B. $6.59
D. $9.79
26. Mr. Thomas’ car can travel 17.6 miles per gallon of gas. If his gas tank holds 11 gallons, how far can he travel on a full tank of gas? F. 176 miles
H. 286.6 miles
G. 193.6 miles
I. 352 miles
27.
SHORT RESPONSE The Marino family bought 4 tickets to the circus. What was the total cost of the tickets? ets?
28.
GRIDDED RESPONSE Vanessa is making bracelets for a fundraiser. Each bracelet uses 28.5 centimeters of string. How many centimeters of string will she need to make 20 bracelets?
Estimate each product. (Lesson 1A) 29. 2.3 × 6
30. 1.35 × 4
31. 8.7 × 17
32. Mrs. Walsh needs to buy 1.75 pounds of clay for each student in her class. If she has 22 students, about how many pounds of clay does she need to buy? (Lesson 1A)
Lesson 1C Multiply Decimals by Whole Numbers
247
Multi-Part Lesson
2
PART
Multiply Decimals A
B
C
D
Main Idea
Use Models to Multiply Decimals
I will explore using decimal models to multiply decimals.
You can also use decimal models to multiply a decimal by a decimal.
Materials colored pencils
Find 0.3 × 0.7 using a decimal model.
Step 1
Draw a 10-by-10 decimal model.
grid paper
Each small square represents 0.01.
Get ConnectED
Step 2
Shade a rectangle that is 0.3 units wide and 0.7 units long.
GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals. Also addresses GLE 0506.1.4.
0.3 0.7
There are twenty-one hundredths in the region that are shaded. So, 0.3 × 0.7 = 0.21.
248
Multiply and Divide Decimals
Find 0.4 × 2.4 using decimal models.
Step 1
Draw three 10-by-10 decimal models.
Step 2
Shade a rectangle that is 0.4 units wide and 2.4 units long.
0.4 2.4
Step 3
Shade the same amount of squares on a 10-by-10 grid. There are ninety-six hundredths in the shaded region.
So, 0.4 × 2.4 = 0.96.
and Apply It Use decimal models to find each product. 1. 0.4 × 0.8
2. 0.5 × 0.6
3. 0.2 × 0.9
4. 0.3 × 1.8
5. 0.6 × 2.2
6. 1.4 × 0.8
7. Diana’s house has a rectangular window on the door with a height of 1.5 feet and a width of 0.8 feet. What is the area of the window? 8.
E
WRITE MATH How many 10-by-10 decimal models do
you need to find 0.5 × 3.2?
Lesson 2A Multiply Decimals 249
Multi-Part Lesson
2
Multiply Decimals
PART
A
Main Idea I will multiply a decimal by a decimal.
Get ConnectED
B
C
D
E
Multiply Decimals To multiply a decimal by a decimal, multiply as with whole numbers. To decide where to place the decimal point, count the number of decimal places in each factor. The product has the same number of decimal places.
GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals.
Find 1.8 × 3.9. Estimate 1.8 × 3.9
2 × 4 or 8
1.8 × 3.9 −−−− 162 + 540 −−−− 702
one decimal place one decimal place
7.02
Count two decimal places from the right and place the decimal point.
So, 1.8 × 3.9 = 7.02. Check 8 ≈ 7.02
PEANUTS Nina is buying peanuts in bulk. The peanuts cost $0.75 for each pound. She purchased 2.1 pounds of peanuts. How much will Nina pay? Round to the nearest cent. Find $0.75 × 2.1. Estimate $0.75 × 2.1 0.75 × 2.1 −−−− 75 + 1500 −−−−− 1575 1.575
$1 × 2 or $2
two decimal places one decimal place
Count three decimal places from the right and place the decimal point.
$1.58 Round to the nearest cent.
Nina will need to pay $1.58. Check $2 ≈ $1.58
250
Multiply and Divide Decimals
Annex Zeros GUITARS Demarcus is buying a new guitar pick that costs $0.50. The sales tax is found by multiplying the cost of the guitar pick by 0.06. What is the cost of the sales tax for the pick? Find 0.50 × 0.06. 0.50 × 0.06 −−−−− 300
two decimal places two decimal places
0.0300 or 0.03 Count four decimal places from the right. Annex zeros if needed.
The sales tax is $0.03.
Multiply. See Examples 11–33 Multiply 1. 0.6 × 0.8
2. 1.7 × 2.4
3. 0.9 × 3.8
4. 0.96 × 7.1
5. 2.6 × 3.65
6. 5.2 × 0.07
7. 0.8 × 2.78
8. 0.35 × 0.15
9. 3.24 × 6.4
10. Measurement Jocelyn’s swimming pool is shown. Find the area of the swimming pool. 11. A nutrition label says that one serving out of a bag of chips has 3.7 grams of fat. How many grams of fat are there in 2.5 servings? 12. Apples cost $0.98 per pound. How much would it cost to purchase 5.5 pounds of apples? 13.
E
TALK MATH Compare and contrast multiplying decimals
and multiplying whole numbers.
Lesson 2B Multiply Decimals 251
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Multiply. l i l See Examples 1–3 14. 0.9 × 0.3
15. 1.6 × 3.2
16. 0.5 × 6.7
17. 0.81 × 7.3
18. 5.6 × 3.9
19. 1.2 × 0.05
20. 0.28 × 0.03
21. 0.73 × 0.21
22. 5.14 × 0.4
23. 0.98 × 0.23
24. 0.11 × 0.91
25. 6.34 × 0.7
26. It costs $4.25 for one pound of roast beef. How much will it cost to purchase 2.5 pounds of roast beef? Round to the nearest cent. 27. A certain car travels 22.6 miles on one gallon of gasoline. How many miles can the car travel on 3.6 gallons of gas? Find the area of each figure. 28.
29. 1.3 m 6.4 cm
1.3 m 1.8 cm
Algebra Evaluate each expression if x = 0.3, y = 6.2, and z = 0.5. 30. xy + z
31. y × 0.4 - z
32. 3.5z + y
33. y(0.6 + 0.2)
34. yz
35. 5 - (z - x)
According to a recent U.S. Census, the population of Virginia is about 7.07 million people. 36. About twenty-four hundredths, or 0.24, of all the people living in the state of Virginia are under the age of 18. How many people in Virginia are under the age of 18? 37. If forty-nine hundredths, or 0.49, of all the people living in Virginia are female, how many females live in Virginia?
252
Multiply and Divide Decimals
Virginia
38. NUMBER SENSE Place the decimal point in the answer to make it correct. Explain your reasoning. 0.7839 × 2.569 = 20138391 39. OPEN ENDED Write a multiplication problem using two decimals that has an answer that is greater than 0.1 and less than 0.2. 40.
E
WRITE MATH Write a multiplication problem in which
the product is between 0.05 and 1.
Test Practice 41. Mikayla is buying deli meat and cheese for sandwiches for a family reunion. How much will she spend on 4.5 pounds of ham and 3.5 pounds of cheese?
43. Mr. Garner’s gas tank holds 17.5 gallons of gasoline. How much will it cost him to fill up his gas tank if gasoline costs $2.48 per gallon? F. $34 G. $35 H. $42.16 I. $43.40
42.
A. $7.39
C. $20.66
B. $13.27
D. $33.93
SHORT RESPONSE To find the amount of tax added to a purchase, the total amount is multiplied by 0.06. If Morgan buys a pair of shoes for $38.95, how much tax is added to the purchase price? Round to the nearest cent.
44.
GRIDDED RESPONSE The average honeybee can travel 10.5 feet per second. How many feet can the honeybee travel in 3.7 seconds?
Multiply. (Lesson 1C) 45. 12.3 × 6
46. 4.56 × 9
47. 72.3 × 22
48. A group of 9 students go rollerskating. It costs $6.75 for each person to rollerskate. Is $54 a reasonable estimate for the cost of 9 students to rollerskate? Explain. (Lesson 1A)
Lesson 2B Multiply Decimals 253
Multi-Part Lesson
2
Multiply Decimals
PART
A
Main Idea I will multiply a decimal by powers of ten.
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals.
B
C
D
E
Multiply Decimals by Powers of Ten Multiplying a decimal by a power of 10 moves the decimal point to the right the same number of places as the number of zeros in the power of 10.
Decimal
Power of Ten
Product
4.26
×1
= 4.26
4.26
× 10
= 42.6
4.26
× 100
= 426
4.26
× 1,000
= 4,260
SKIMBOARD Josh will need to make 10 payments of $32.25 to purchase a new skimboard. What is the total cost of the skimboard? Find $32.25 × 10 mentally. $32.25 × 10 = $32.25 10 has one zero. Move the decimal one place to the right.
= $322.50 Add a zero at the end to make fifty cents. The total cost is $322.50.
Multiply by a Power of Ten Greater than One Find 0.5 × 100. 0.5 × 100 = 0.50 100 has two zeros. Move the decimal point two places to the right. Annex zeros as needed.
= 50
Since 50 is a whole number, remove the leading zero and decimal point.
So, 0.5 × 100 is 50.
254
Multiply and Divide Decimals
A pattern is also found when multiplying a decimal by powers of 10 like 0.1 and 0.01 that are less than one. Multiplying a decimal by a power of 10 that is less than one moves the decimal point to the left the same number of places as there are after the decimal point. Decimal
Power of 10
Product
61.3
× 0.1
= 6.13
61.3
× 0.01
= 0.613
61.3
× 0.001
= 0.0613
Multiply by a Power of Ten Less than One Find 3.4 × 0.1. 3.4 × 0.1 = 3.4 x You may need to anne zeros to complete your multiplication.
= 0.34
There is one place after the decimal point, so move the decimal point in 3.4 one place to the left.
Find 8.6 × 0.01. 8.6 × 0.01 = 08.6 = 0.086
There are two places after the decimal point, so move the decimal point in 8.6 two places to the left.
Find each product. product Examples 11–44 1. 0.54 × 10
2. 8.32 × 100
3. 7.46 × 1,000
4. 0.98 × 100
5. 1.63 × 10
6. 0.853 × 1,000
7. 0.72 × 0.1
8. 0.9 × 0.01
9. 67.2 × 0.001
10. 76.5 × 0.001
11. 0.81 × 0.1
12. 1.23 × 0.01
13. A music store has 10 flutes. Each flute costs $325.50. What is the cost of all 10 flutes? 14.
E
TALK MATH Suppose you want to find the cost of
10 text messages that cost $0.25 each. Explain how you can mentally find the cost.
Lesson 2C Multiply Decimals 255
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Find i d each h product. d Examples 1–4 15. 0.397 × 100
16. 1.76 × 100
17. 0.78 × 10
18. 0.13 × 10
19. 1.4 × 1,000
20. 4.81 × 1,000
21. 0.48 × 0.1
22. 0.93 × 0.01
23. 0.22 × 0.001
24. 1.5 × 0.01
25. 9.75 × 0.001
26. 48.5 × 0.1
27. One carton of milk costs $0.99. What is the total cost of 100 cartons of milk? 28. Measurement Mike made a model of a 300-foot building. His model was one hundredth, or 0.01, the size of the actual building. How tall was his model? 29. The school purchased 100 trophies to give to the honor roll students. If one trophy costs $4.32, how much would it cost for all of the trophies? 30. Titus and Cam’s hourly charge for yard work is shown. Suppose Titus and Cam each worked 10 hours. How much money did they earn together? 31. The length of Jenny’s cat is 10 times the length of her hamster. What is the length of her cat?
Titus
Cam
$8.25/h
$5.85/h
32. What is one tenth, or 0.1, the cost of the paintbrush?
$3
.20
The table shows a school store’s prices. 33. How much will it cost for 10 pencils and 1 notebook? 34. How much would it cost to buy 10 pens from the school store?
256
Multiply and Divide Decimals
Item
Price
Notebook
$1.25
Pencil
$0.50
Binder
$2.15
Pen
$0.80
35. CHALLENGE Find 0.346 × 102. 36. FIND THE ERROR Rico is finding 7.5 × 100. Find and correct his mistake.
7.5 × 100 07.5 = 0.075
37.
E
WRITE MATH Write a real-world problem that involves
multiplying a decimal by a power of 10.
Test Practice 38. Elyse is cutting strips of paper for a scrapbook page. Each strip is 1.5 centimeters wide. How wide will 10 strips of paper be? A. 1.5 cm
39. Neal is saving $4.25 a week for 100 weeks. How much will he have saved altogether? F. $4.25 G. $42.50
B. 15 cm
H. $425.00
C. 150 cm
I. $4,250.00
D. 1,500 cm
Multiply. (Lesson 2B) 40. 0.4 × 0.3
41. 0.3 × 2.8
42. 1.6 × 2.5
43. 0.26 × 2.5
44. 3.6 × 7.55
45. 4.3 × 0.04
46. Sophie and two of her friends each purchased a ticket to ride the rollercoaster. If it costs $1.45 for 1 ticket, how much did Sophie and her friends spend in all? (Lesson 1C)
Lesson 2C Multiply Decimals 257
Multi-Part Lesson
2
Multiply Decimals
PART
A
B
C
D
Problem-Solving Strategy:
Look for a Pattern
Main Idea I will solve problems by looking for a pattern. Marvin is shopping for new M cclothes. He receives a member discount on his purchases. The table shows the amount Marvin pays for his clothes and the amount of his discount.
Cost
Discount Amount
Marvin Pays
Shirt
$34.70
$3.47
$31.23
Pants
$48.50
$4.85
$43.65
Jacket
$62.30
$6.23
$56.07
Boots
$68.80
Item
Based on the pattern, how much will Marvin pay, and how much will the discount be for boots that cost $68.80?
Understand What facts do you know? • Marvin receives a discount on his purchased items. • We know the cost of the boots. What do you need to find? • We need to find the amount Marvin will pay for the boots and the amount of the discount.
Plan
One way to solve the problem is by looking for a pattern. Then, extend the pattern to find the amount paid.
Solve
Use your plan to solve the problem. If you multiply the cost of the item by 0.1, you can find the discount. Subtract the discount from the total cost to find how much Marvin pays. $68.80 × 0.1 = $6.88 $68.80 - $6.88 = $61.92
Check
The discount is $6.88. Marvin pays $61.92.
The answer fits the pattern. So, it is correct.
GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals. GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution. Also addresses GLE 0506.1.5.
258 2
Multiply and Divide Decimals
Refer to the problem on the previous page. 1. What happens to the decimal point when the cost of the item is multiplied by 0.1?
3. Marvin only had $50 left to spend. With his discount, would he be able to purchase another shirt that costs $54.70? Explain.
2. How much would the discount be if Marvin bought a hat that cost $16.50?
4. Why was finding the pattern a helpful strategy to find the cost of Marvin’s discount?
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Solve. Use the look for a pattern strategy. 5. Every year, Victoria receives $30 for her birthday, plus $2 for each year of her age. Lacey receives $20 for her birthday and $4 for each year of her age. In 2010, Victoria is 10, and Lacey is 6. In what year will they both receive the same amount of money?
6. Describe the pattern below. Then find the next three numbers. 0.03, 0.3, 3, 30, , ,
7. Draw the next two figures in the pattern.
9. Algebra Copy and complete the table. What is the pattern? Input (x)
Output (y)
5
9
10
19
15
20
39
49
10. Adrienne did 13 sit-ups the first day, 20 sit-ups the second day, and 27 sit-ups the third day. If the pattern continues, how many sit-ups will she do on the sixth day? 11. Describe the pattern below. Then find the next two numbers. 784.5, 78.45, 7.845, ,
8. Samuel will arrive at the airport on the first plane after 10 A.M. Airplanes arrive every 50 minutes beginning at 6 A.M. When will Samuel’s plane arrive?
12.
E
WRITE MATH Write a real-world
problem involving a pattern with decimals. Give it to a classmate and see if he or she can continue it.
Lesson 2D Multiply Decimals 259
Mid-Chapter Check Estimate each product. (Lesson 1A) 1. 4 × $3.14
2. 7 × $5.89
3. 3 × 6.3
4. 8 × 2.7
13. The dimensions of a swimming pool are shown. Find the area of the pool. (Lesson 1C)
5. The store sells newspapers for $1.75 each. Mrs. Horace wants to buy 24 for her classroom. Estimate the total cost. (Lesson 1A)
6. Renee ran 1.3 miles every day for one week. About how many miles did she run in one week? (Lesson 1A)
Multiply. (Lesson 1C) 7. 5 × 2.4
8. 9 × 6.7
9. 2 × 9.5
10. 7 × 4.8
11. The table shows a list of coins and the average mass for that coin. Coin
2.67
Nickel
5.01
Dime
2.28
Quarter
5.66
15. 7.5 × 3.6
16. 0.6 × 8.2
17. 0.06 × 7.4
18. Dale bought 10 apples at the store for $0.49 each. How much did he pay for all 10 apples? (Lesson 2C) Find each product. (Lesson 2C) 19. 6.2 × 100
20. 45.7 × 0.1
21. 0.12 × 10
22. 92.3 × 0.01
23. Describe the pattern below. Then find the next number. (Lesson 2D)
If Jeremy had 3 dimes and 2 quarters in his pocket, what would be the total mass of those coins? (Lesson 1C) 12. Find the area of the parallelogram in square meters.
260
14. 1.3 × 4.1
Mass (g)
Penny
(Lesson 2B)
Multiply. (Lesson 2B)
0.9 m 1.8 m
Multiply and Divide Decimals
0.732, 7.32, 73.2, 24.
E
WRITE MATH A stadium has
3,720 seats and one tenth, or 0.1, of the seats are reserved for students. Explain how to find how many seats are reserved for students. Then solve. (Lesson 2C)
Multi-Part Lesson
3
PART
Divide Decimals by Whole Numbers A
Main Idea I will estimate quotients of decimals and whole numbers.
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, and decimals. GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution.
B
C
D
Estimate Quotients of Decimals and Whole Numbers To estimate quotients of decimals and whole numbers, you can use rounding and compatible numbers.
ANIMALS A live sponge is an animal that lives in the ocean. Sponges filter water to get food. One sponge can filter 26.4 liters of water in 4 hours. About how many liters of water can a sponge filter in one hour? 26.4 ÷ 4 28 ÷ 4 28 ÷ 4 = 7
Change 26.4 to 28 because 28 and 4 are compatible numbers. Divide mentally.
So, a sponge can filter about 7 liters of water in one hour.
Use Compatible Numbers MONEY Ms. Glover buys 2 kites for a total of $15.18. If each kite costs the same, about how much did each kite cost? Explain why your answer is reasonable. $15.18 ÷ 2 $16 ÷ 2 $16 ÷ 2 = $8
Change $15.18 to $16 because 16 and 2 are compatible numbers. Divide mentally.
So, $15.18 ÷ 2 is about $8. Since 2 × 8 = 16 and 16 ≈ 15.18, your answer is reasonable.
Lesson 3A Divide Decimals by Whole Numbers 261
Round Money
Rounding a decimal is the same as rounding a whole number.
To round $32.57 to the nearest dollar, look at the number to the right. Since it is 5, round $32.57 to $33.
RESTAURANT Three friends went out to dinner. The total cost of their meal was $32.57. If the friends split the bill evenly, about how much will each person pay? Explain why your answer is reasonable. $32.57 ÷ 3 $33 ÷ 3 $33 ÷ 3 = $11
Round $32.57 to the nearest whole dollar amount. Divide mentally.
Each friend would pay about $11. Since 3 × 11 = 33 and 33 ≈ 32.57, your answer is reasonable.
Use Rounding Estimate 78.74 ÷ 42. 78.74 ÷ 42 80 ÷ 40 80 ÷ 40 = 2
Round 42 to 40. Replace 78.74 with 80 because 80 and 40 are compatible numbers. Divide mentally.
So, 78.74 ÷ 42 is about 2.
Estimate each quotient quotient. See Examples 11–44 1. $19.50 ÷ 5
2. $47.25 ÷ 25
3. 16.8 ÷ 4
4. 87.3 ÷ 11
5. 44.7 ÷ 5
6. 195.8 ÷ 12
7. It costs $158.75 to purchase 15 tickets to the state fair. About how much does one ticket cost? 8. Jake bought 3 drawing pens for $8.07. Each pen costs the same amount. About how much did he spend for 1 pen? 9.
E
TALK MATH Describe the steps for estimating dividing
money by a whole number.
262
Multiply and Divide Decimals
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Estimate E ti t each h quotient. ti t See Examples 1–4 10. $28.20 ÷ 6
11. $7.92 ÷ 6
12. 88.3 ÷ 9
13. 128.9 ÷ 12
14. 576.4 ÷ 62
15. $15.47 ÷ 7
16. 56.3 ÷ 18
17. 32.17 ÷ 7
18. $175.32 ÷ 3
19. A canoe rental company offers two trips along the river. Neil and Gabriela choose the longer trip. If they split the cost of the rental, about how much will each person pay?
Canoe Rental Trip
Distance (miles)
Cost
A
5.75
$12.98
B
8.5
$16.32
20. It costs $88.50 for 6 adults to see an exhibit on Egyptian mummies. About how much does it cost 1 adult to see the exhibit? 21. Measurement There are 74.4 grams of sugar in 3 servings of grapes. About how many grams of sugar are in a single serving?
Measurement
The table shows the density of each object. Density describes how tightly the particles in an object are packed together. You can find density by dividing an object’s mass by its volume.
Substance
Mass (g)
Volume (cm3)
Aluminum
13.5
5
Gold
56.7
3
121.5
9
Mercury
22. Estimate the density of aluminum. 23. Is the density of gold greater than the density of mercury? Explain. 24. The density of gold is about how many times greater than the density of aluminum?
25. OPEN ENDED Write a division problem involving money. Describe the steps that you used to estimate the quotient. 26.
E
WRITE MATH Explain how to use compatible numbers
to estimate 39.12 ÷ 21.
Lesson 3A Divide Decimals by Whole Numbers 263
Multi-Part Lesson
3
PART
Divide Decimals by Whole Numbers A
Main Idea I will explore dividing a decimal by a whole number.
B
C
D
E
Divide Decimals by Whole Numbers You can use place-value models to divide a decimal by a whole number.
Materials base-ten blocks
Find 3.6 ÷ 3 using models.
Step 1
Model 3.6 as 3 wholes and 6 tenths.
Step 2
Separate the blocks into three equal groups.
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals. Also addresses GLE 0506.1.4.
There is one whole and two tenths in each group. So, 3.6 ÷ 3 = 1.2.
and Apply It Use models to find each quotient. 1. 3.4 ÷ 2
2. 6.3 ÷ 3
3. 5.6 ÷ 4
4. 2.7 ÷ 3
5. 4.8 ÷ 2
6. 2.8 ÷ 4
7.
E
TALK MATH How could you solve Exercises 1–6 without
using models?
264
Multiply and Divide Decimals
Multi-Part Lesson
3
PART
Divide Decimals A
Main Idea I will divide a decimal by a whole number.
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals. Also addresses GLE 0506.1.7.
B
C
D
E
Divide Decimals by Whole Numbers As shown in the previous lesson, dividing a decimal by a whole number is similar to dividing whole numbers.
RIBBON There are 8.4 meters left on a roll of ribbon. Nancy wants to cut the ribbon in half. What will be the size of each piece of ribbon? Find 8.4 ÷ 2. Estimate 8 ÷ 2 = 4 4.2 2 8.4 8 −−− 04 4 −−− 0
Place the decimal point directly above the decimal point in the dividend.
So, 8.4 ÷ 2 = 4.2. Each piece of ribbon will be 4.2 meters long. Check 4.2 ≈ 4
Divide a Decimal by a Two-Digit Number Find 6.54 ÷ 12. 0.545 12 6.540 - 60 54 48 −−−− 60 60 −−−− 0
Place the decimal point.
Annex a zero and continue dividing.
6.54 ÷ 12 = 0.545
Lesson 3C Divide Decimals by Whole Numbers 265
When dividing decimals, sometimes the answer does not come out evenly. You need to round the quotient to a specified place-value position.
Round a Quotient MONEY Corey bought 3 books at the book fair for $49.90. If each book costs the same amount, what is the price of each book? Round to the nearest cent. x You may need to anne 0s when dividing a decimal by a whole number.
Find $49.90 ÷ 3. 16.633 49.900 3 -3 19 18 −−−− 19 -18 10 9 −−− 10 9 −−− 1
Place the decimal point.
Divide until you place a digit in the thousandths place, so you can round to the nearest cent.
The cost of each book is $16.63.
Divide. Round to the nearest tenth if necessary Divide necessary. Examples 11–33 1. 3 48.33
2. 2 8.8
3. 4 8.43
4. 145.8 ÷ 12
5. 22.11 ÷ 11
6. 65.9 ÷ 14
7. 18 38.4
8. 14 3.66
9. 15 36.16
10. Mandy and 5 of her friends bought a package of bottled water for $4.98. How much will each friend pay to the nearest cent, if the cost is divided equally? 11.
E
TALK MATH Is the quotient 9.3 ÷ 10 greater than one
or less than one? Explain.
266
Multiply and Divide Decimals
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Divide. Di id R Round d tto th the nearestt ttenth th if necessary. Examples 1–3 12. 12.4 ÷ 9
13. 14.14 ÷ 14
14. 55.44 ÷ 11
15. 7 7.21
16. 4 6.28
17. 6 276.2
18. 380.5 ÷ 30
19. 75.2 ÷ 24
20. 4.68 ÷ 22
21. 42 223.6
22. 20 8.14
23. 12 361.5
24. Four girls swam the 4-by-200 meter freestyle relay in a total of 540.5 seconds. If each girl swam their part of the race in the same amount of time, what was the time for one girl? 25. The table shows the prices for two packs of batteries. Which pack is a better buy? Explain your reasoning. 26. The total cost of four movie tickets is $33.40 What is the cost of one ticket?
Batteries Pack
Number of Batteries
Price
A B
4 6
$2.44 $3.90
Use the information to solve the problem.
27. How much money do they each make working together if the money is split evenly?
28. OPEN ENDED Write a real-life problem that involves dividing a decimal by a whole number. 29.
E
WRITE MATH Why is rounding important when dividing
a decimal by a whole number?
Lesson 3C Divide Decimals by Whole Numbers 267
Multi-Part Lesson
3
Divide Decimals by Whole Numbers
PART
A
Main Idea I will divide a decimal by powers of ten.
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals.
B
C
D
E
Divide Decimals by Powers of Ten Dividing a decimal by a power of 10 moves the decimal point to the left the same number of places as the number of zeros in the power of 10. Decimal
Power of Ten
Quotient
53.7
÷1
= 53.7
53.7
÷ 10
= 5.37
53.7
÷ 100
= 0.537
53.7
÷ 1,000
= 0.0537
Laura will need to make 10 payments to pay for her new motorized scooter. How much will each payment be if the scooter cost $219.60? Find $219.60 ÷ 10. $219.60 ÷ 10 = 219.60 10 has one zero. Move the decimal = $21.96
point 1 place to the left.
So, each payment will be $21.96.
Divide by a Power of Ten 36.4 ÷ 1,000 36.4 ÷ 1,000 = 036.4 1,000 has three zeros. Move the decimal point 3 places to the left.
= 0.0364 Insert a leading zero. So, 36.4 ÷ 1,000 = 0.0364.
268
Multiply and Divide Decimals
The quotient of a number and a power of ten that is greater than 1 will always be less than the original number.
SCIENCE In the past 100 years, scientists measured the movement of the continents to be 190.5 centimeters. If the continents moved the same amount each year, how much did the continents move in one year? Divide the total distance moved by 100. 190.5 ÷ 100 = 1.905 100 has 2 zeros. So move the decimal point 2 places to the left.
= 1.905 The continents moved 1.905 centimeters each year.
Find each quotient quotient. Examples 11–33 1. 7.2 ÷ 10
2. 21.5 ÷ 100
3. 19.2 ÷ 1,000
4. 10 14.7
5. 100 5.62
6. 100 18.7
7. 6.3 ÷ 1,000
8. 0.05 ÷ 1
9. 0.012 ÷ 100
10. 100 8.72
12. 100 98.6
11. 10 2.46
13. Herbert bought red beans to make chili to sell at a fundraiser. The total price of the red beans was $13.90. How much did it cost for each can?
Items Purchased Item
Quantity
red beans shredded cheese
10 6
14. Sarah was trying to decide what kind of cat food to buy at the store. Smiley Cat costs $7.90 for 10 pounds. Purr Plus costs $0.85 per pound. Which brand is the better buy? Explain your reasoning.
15.
E
TALK MATH Suppose you want to find the number of
10-millimeter ribbons that can be cut from a ribbon that is 192.6 millimeters long. Explain how you can mentally find the number of ribbons that can be cut.
Lesson 3D Divide Decimals by Whole Numbers 269
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Find Fi d each h quotient. ti t Examples l 1–3 16. 8.3 ÷ 100
17. 208 ÷ 100
18. 5.71 ÷ 1
19. 1 0.07
20. 1,000 437
21. 1,000 32.7
22. 0.4 ÷ 100
23. 102.6 ÷ 100
24. 78.2 ÷ 10
25. 10 5.4
26. 100 2.1
27. 1,000 0.75
28. Christy purchased 6.75 pounds of licorice. How much licorice does she need to put in each bag if she divides the total amount into 10 equal-sized bags? For Exercises 29–31, use the table at the right. 29. Each row of the grocery store parking lot has 10 parking spots of equal width. What is the width for each spot? 30. How much space is given for each parking spot at the mall parking lot if each spot has an equal width?
Number of Parking Spots per Row
Combined Width of all Parking Spots (m)
Grocery Store
10
31.92
Hardware Store
10
31.96
Mall
10
31.30
Location
31. Which location gives the greatest space for each parking spot? 32. In Alaska, scientists are studying the melting of glaciers. They found that in the past 10 years, the glaciers receded about 59.05 feet. If the glaciers receded at the same rate, how much did they recede in one year?
33. WHICH ONE DOESN’T BELONG? Identify the expression that does not belong with the other three. Explain your reasoning.
6.7 ÷ 10
4 ÷ 1,000
0.2 ÷ 1
52.1 ÷ 100
34. CHALLENGE Tell what number you must divide 180 by to get 1.8. 35.
E
WRITE MATH Describe the relationship between the number
of places a decimal is moved to the left and the change in the value of the number.
270
Multiply and Divide Decimals
Estimation Roll Off Estimate Dividing a Decimal by a Whole Number
Get Ready! Players: 2, 3, or 4 players
Get Set! Each group will need two number cubes, paper, and pencil.
You will need: 2 number cubes, paper and pencil
The player with the most correct estimates is the winner for that round. Roll the cubes to continue play as time permits.
Go! Roll both number cubes. The sum of the cubes will be the quotient. Each player will have 5 minutes to write down as many division equations as possible with an estimate that equals the sum of the numbers rolled on the number cubes. At least one of the numbers in the equation must be a decimal. At the end of five minutes, share the equations with the rest of the group and tally the number of equations with correct estimates.
Game Time Estimation Roll Off
271
Multi-Part Lesson
4
PART
Divide Decimals A
Main Idea I will explore using models to divide a decimal by a decimal.
B
C
D
E
Divide Decimals Using Models You can use models to divide decimals by decimals.
Materials base-ten blocks
Find 2.4 ÷ 0.8.
Step 1
Model 2.4 using two wholes and four tenths.
Get ConnectED 1 GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals. Also addresses GLE 0506.1.4.
1
0.4
Step 2
Since you are dividing by tenths, replace both of the ones blocks with tenths.
Step 3
Separate the tenths into groups of eight tenths to show dividing by 0.8.
0.8
0.8
0.8
3 groups
There are three groups of eight tenths in 2.4. So, 2.4 ÷ 0.8 = 3.
272
Multiply and Divide Decimals
Model 0.3 ÷ 0.06.
Step 1
Model 0.3 using 3 tenths.
Step 2
Replace the tenths with hundredths since you are dividing by hundredths.
Step 3
Separate the hundredths into groups of 6 hundredths to show dividing by 0.06.
5 groups
There are 5 groups of 6 hundredths in 0.3. So, 0.3 ÷ 0.06 = 5.
and Apply It Use models to find each quotient. 1. 2.4 ÷ 0.3
2. 1.6 ÷ 0.4
3. 1.8 ÷ 0.9
4. 0.2 ÷ 0.04
5. 0.4 ÷ 0.05
6. 0.2 ÷ 0.02
7. Grace purchased some packages of gum for $1.50. If each package costs $0.50, how many packages did Grace purchase? 8. Tell why the quotient 0.9 ÷ 0.09 is a whole number. What does the quotient represent? 9.
E
WRITE MATH Refer to Activity 2. Explain why in Step 2 you
replaced the tenths with hundredths.
Lesson 4A Divide Decimals 273
Multi-Part Lesson
4
Divide Decimals
PART
A
Main Idea I will divide decimals by decimals.
Get ConnectED GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, decimals.
B
C
D
E
Divide Decimals When dividing by decimals, change the divisor into a whole number. To do this, multiply both the divisor and the dividend by the same power of 10. Then divide as with whole numbers.
TRAVEL The Prease family drove 213.9 miles to the beach. Their car used 9.3 gallons of gas. How many miles did they drive per gallon of gas? Find 213.9 ÷ 9.3. Multiply 9.3 by 10 to make a whole number.
23. 9.3 213.9 93 2,139. 1 86 −−−−− 279 Multiply 213.9 by the 279 same number, 10. −−−− 0 So, they drove 23 miles per gallon
Place the decimal point. Divide as with whole numbers.
of gas.
Divide Decimals Find 5.32 ÷ 1.4. Multiply 1.4 by 10 to make a whole number.
1.4 5.32 Multiply 5.32 by the same number, 10.
3.8 14 53.2 - 42 11 2 - 11 2 0
So, 5.32 ÷ 1.4 = 3.8. Check 3.8 × 1.4 = 5.32
274
Multiply and Divide Decimals
Place the decimal point. Divide as with whole numbers.
Sometimes it is necessary to annex a zero in the quotient and/or the dividend.
Zeros in the Quotient and Dividend Find 26.4 ÷ 0.22. 26.40 0.22
Multiply each by 100.
120. 22 2,640. - 22 44 44 −−−− 00
Place the decimal point. Divide. Write a zero in the ones place of the quotient because 0 ÷ 22 = 0.
So, 26.4 ÷ 0.22 = 120 Check 120 × 0.22 = 26.4 Find 0.06 ÷ 1.5. 0.06 1.5
Adding a zero to the end of a decimal does not change the value of the decimal. 0.6 = 0.60
Multiply each by 10.
0.04 15 0.60 - 0 60 60 −−−− 0
Place the decimal point. 15 does not go into 6, so write a 0 in the hundredths place. Annex a 0 to continue.
So, 0.06 ÷ 1.5 is 0.04. Check 0.04 × 1.5 = 0.06
Divide. Examples 11–44 Divide 1. 6.89 ÷ 1.3
2. 4.58 ÷ 0.5
3. 0.66 ÷ 0.3
4. 16.5 ÷ 0.03
5. 0.462 ÷ 0.2
6. 18.6 ÷ 0.12
7. 0.07 ÷ 1.4
8. 3.822 ÷ 0.04
9. 0.08 ÷ 1.6
10. Mrs. Chibas is making chocolate chip cookies for her daughter’s class. She bought a tub of chocolate chip cookie dough that contained 57.6 ounces of dough. If each cookie needs 1.2 ounces of dough, how many cookies can she make? 11.
E
TALK MATH Explain how to solve 0.808 ÷ 0.4.
Lesson 4B Divide Decimals 275
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
Divide. Di id Examples l 1–4 12. 14.4 ÷ 0.4
13. 13.2 ÷ 0.3
14. 3.51 ÷ 0.78
15. 84.78 ÷ 0.15
16. 0.92 ÷ 0.4
17. 12.42 ÷ 4.6
18. 0.242 ÷ 0.4
19. 1.404 ÷ 0.45
20. 0.4 ÷ 0.05
21. 29.12 ÷ 1.3
22. 7.98 ÷ 4.2
23. 2.07 ÷ 0.9
24. Measurement Kaya works for a T-shirt company and has 49.5 yards of fabric to make specialty T-shirts. If each T-shirt needs 4.5 yards of fabric, how many T-shirts can she make? 25. A pizza chef can make one pizza in 52.3 seconds. How many pizzas can the pizza chef make in 418.4 seconds? 26. Sei paid $5,787.28 in rent for a certain number of months. If the cost of rent for one month is $723.41, how many months has Sei already paid? 27. Kasey is 4.5 feet tall. Her brother, Jerome, is 6.75 feet tall. How many times taller is Jerome than Kasey?
28. OPEN ENDED Write a real-world problem that involves dividing a decimal by a decimal. 29. NUMBER SENSE Without solving, would 1.98 ÷ 0.51 be closer to 4 or 5? Explain. 30. FIND THE ERROR Sonia is finding 13.2 ÷ 2.2. Help find and correct her mistake.
13.2 2.2
31.
E
0.6 22 13.2 13 2 −−−−− 0
WRITE MATH Explain how you can check your answer when
dividing decimals.
276
Multiply and Divide Decimals
Test Practice 32. Lamar spent $16.25 on peanuts. If he bought 2.6 pounds of peanuts, how much does 1 pound of peanuts cost?
34. The area of the rectangle is 335.62 square inches. If the length is 34.6 inches, what is the width?
34.6 in.
A. $5.55
C. $6.55
B. $6.25
D. $7.25
33. Mason has 1.71 meters of string to decorate his locker. He cut the string into 0.19-meter lengths. How many pieces does he have? F. 6
H. 8
G. 7
I. 9
35.
A. 8.6 inches
C. 10.3 inches
B. 9.7 inches
D. 10.9 inches
SHORT RESPONSE Patricia has 3.75 pounds of hamburger. She is making hamburgers with 0.25 pound of hamburger each. How many hamburgers can she make?
Divide. (Lesson 3D) 36. 10 46.33
37. 100 48.3
38. 1,000 9,435.2
39. Mr. Ginobli pays the same amount each month on a loan. In one year, he paid a total of $1,276.20. How much did Mr. Ginobli pay each month for the bank loan? (Lesson 3C) Find each product. (Lesson 2C) 40. 52.8 × 100
41. 3.87 × 10
42. 0.26 × 1,000
43. 73.54 × 0.1
44. 6.02 × 0.01
45. 132.49 × 0.001
46. The flag at the right is the flag of Switzerland. What is the area of the flag? (Lesson 2B)
3.15 m 4.5 m
Lesson 4B Divide Decimals 277
Multi-Part Lesson
4
PART
Divide Decimals A
C
B
Problem-Solving Investigation Main Idea I will choose the best strategy to solve a problem.
Blake measured the amount of rain each . day for five days for his first experiment. Rain Experiment Day Amount (in.) Monday Tuesday Wednesday Thursday Friday
1.2 0.5 2.4 3.75 1.0
Then, Blake measured the amount of rain for another five days for a second experiment. It rained the same total amount as the first experiment, but this time it rained an equal amount each day.
YOUR MISSION: Determine how much it rained each day in the second experiment.
Understand
What facts do you know?
• the amount it rained each day for the first experiment • the total amount of rain for each experiment is the same What do you need to find? • how much it rained each day for the second experiment
Plan Solve
Solve by making a diagram. 1.2 + 0.5 + 2.4 + 3.75 + 1.0 = 8.85 total amount of rain 8.85
in Experiment 1 Think: What is 8.85 ÷ 5?
8.85 ÷ 5 = 1.77 So, it rained 1.77 inches each day of Experiment 2.
Check
Since 1.77 × 5 = 8.85, the answer is reasonable.
GLE 0506.2.5 Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, and decimals. GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution.
278
Multiply and Divide Decimals
EXTRA
% )# E # T4 IC !C 2A 0R P
Begins on page EP2.
• • • •
Determine reasonable answers. Look for a pattern. Draw a diagram. Choose an operation.
Use any strategy shown to solve each problem. 1. Musa wants to buy 3 pairs of pants for $39.54 each and 2 shirts for $17.99 each. Does she need to save $170, $160, or $150?
5. Kristen wants to buy a new skateboard. So far, she has saved $31.26 and $45.50 from her two lawn-mowing jobs. How much more money does she need to buy the skateboard if it costs $131.99?
6. Teresa can spend $10 on food. Which of the following combination of foods can she NOT buy? • hamburger and fruit salad • 2 fruit salads and ice cream • hamburger and ice cream
2. NUMBER SENSE Describe the pattern below. Then find the next three numbers.
Item
Price
2.54, 2.6, 2.66, 2.72, , ,
Ice cream Hamburger Fruit Salad
$3.29 $6.79 $2.25
3. Victor bought a small pizza at Pizza Palace. The price of an extra large pizza is equal to 2 times the price of a small pizza plus $3. How much did Victor pay for the pizza?
7. BAR DIAGRAM Brandon ran 21.6 laps over three days. If he ran the same amount each day, how many laps did he run each day?
8. Thomas noticed that for every 4 cars that went through a car wash, he saw 2 trucks, if 24 vehicles went through the car wash, how many were trucks?
4. Anne is setting up tables for a conference. She is putting 5 rectangular tables end-toend to make one large table. One rectangular table can seat one person at each end and 2 people on each side. How many people will be able to sit around the new large table?
9. Measurement The width of a swimming pool is 18.6 yards and divided into 6 equal rows. What is the width of each row?
10.
E
WRITE MATH What strategy did you
use to solve Exercise 4? Explain why your strategy is appropriate to solve the problem.
Lesson 4C Divide Decimals 279
The Sun is amazing. It is a star that is 4.5 billion years old. At its center, the temperature is 27 million degrees Fahrenheit. That’s about 67,000 times hotter than an oven! The Sun is not only incredibly hot, but also very large. If it was hollow, 1,000,000 Earths could fit inside of it. Because the Sun is so big, it has a lot of gravity. This gravity pulls on the 8 planets and keeps them in orbit.
Planet
280
Average Distance from Sun (miles)
Each planet’s orbit is a different shape. Orbital eccentricity describes the shape of a planet’s orbit. Scientists use decimals to measure orbital eccentricity. If a planet’s orbit is perfectly circular, its orbital eccentricity is 0.0. The more ovalshaped the planet’s orbit, the closer the decimal is to 1.0. Because each orbit is different, the planets are not always the same distance from the sun. Scientists describe these distances from the sun as averages.
Orbital Eccentricity
Gravitational Pull, ( meters )
_ second2
Mercury
35,983,093
0.205
3.59
Venus
67,237,910
0.007
8.87
Earth
92,955,820
0.017
9.81
Mars
141,633,330
0.055
3.77
Jupiter
483,682,810
0.094
25.95
Saturn
886,526,100
0.057
11.08
Uranus
1,783,935,996
0.046
10.67
Neptune
2,795,084,800
0.011
14.07
Multiply and Divide Decimals
Use the information on the previous page to solve each problem. 1. The gravitational pull of Neptune is
about how many times larger than the gravitational pull of Mercury? 2. The gravitational pull of Jupiter is
about how many times larger than the gravitational pull of Mars? 3. Which planet’s orbital eccentricity is
about 4 times the amount of Neptune’s orbital eccentricity?
4. The gravitational pull of Uranus is
about 1.25 times larger than what planet? 5. The gravitational pull of Earth is
about 2.5 times larger than what planet? 6. Based on the gravitational pull on
each planet, would you expect to weigh more or less on Mercury? Explain your answer.
If you drove 60 miles per hour, it would take you 176 years to get to the Sun.
Problem Solving in Science 281
Chapter Study Guide and Review Be sure the following Key Concepts are written in your Foldable.
Vocabulary powers of 10
Vocabulary Check M Div ultip ide ly a De nd cim als
Key Concepts Multiply Decimals (Lessons 1 and 2) • To multiply decimals, multiply as with whole numbers. The product has the same number of decimal places as the sum of the number of decimal places in each factor.
Choose the correct term or number to complete each sentence. 1. The product of 4.63 × (0.01, 100 ) is 463. 2. To mentally divide 8.3 by 1,000, move the decimal point three places to the (left, right). 3. When multiplying decimals and placing the decimal point, find the (sum, product) of the number of decimal places in each factor.
Divide Decimals (Lessons 3 and 4) • To divide decimals, change the divisor into a whole number. Place the decimal point directly above the decimal point in the dividend, then divide as with whole numbers. Powers of Ten (Lessons 2 and 3) To . . .
move the decimal point to the . . .
multiply by a power of 10 less than 1
left
multiply by a power of 10 greater than 1
right
divide by a power of 10 less than 1
right
divide by a power of 10 greater than 1
left
282
Multiply and Divide Decimals
4. The quotient of 45.8 ÷ 0.01 is (458, 4,580). 5. When dividing decimals, place the decimal point in the quotient directly above the decimal point in the (dividend, divisor). 6. A pattern using (powers, factors) of ten is used in dividing 4.19 ÷ 10. 7. The quotient of 3.6 ÷ 0.6 is the same as the quotient of 36 ÷ (0.06, 6 ). 8. Numbers that are easy to divide mentally are called (compatible, decimal) numbers.
Multi-Part Lesson Review Lesson 1
Multiply Decimals by Whole Numbers
Estimate Products of Whole Numbers and Decimals Estimate each product. 9. 6 × 6.1 11. 4 × $9.21
10. 8 × 0.8 12. 3 × $5.89
13. A bus travels 55 miles per hour for 2.75 hours. Estimate the number of miles driven.
EXAMPLE 1
Estimate the product of 25 × 4.7. 25 × 4.7 Round 4.7 to 5.
25 × 5 25 × 5 = 125
Multiply.
So, 25 × 4.7 is about 125.
Multiply Decimals by Whole Numbers Multiply.
(Lesson 1A)
(Lesson 1C)
EXAMPLE 2
14. 6.2 × 5
15. 3 × 4.2
Find 2.6 × 8.
16. 0.08 × 13
17. 2.09 × 20
2.6 × 8 −−−− 20.8
18. How much would five breakfast cereal bars cost?
one decimal place Count one decimal place from right to left.
So, 2.6 × 8 is 20.8.
Lesson 2
Multiply Decimals
Multiply Decimals
(Lesson 2B)
Multiply.
EXAMPLE 3
19. 0.3 × 2.5
20. 3.6 × 6.1
21. 0.08 × 3.4
22. 3.5 × 7.2
23. A rectangular patio measures 8.5 feet by 10.75 feet. What is the area of the patio in square feet?
Find 8.4 × 4.2. 8.4 × 4.2 −−−− 16 8 + 336 0 −−−−− 35.28
one decimal place one decimal place
two decimal places
Chapter Study Guide and Review 283
Chapter Study Guide and Review
Lesson 2
Multiply Decimals
(continued)
Multiply Decimals by Powers of Ten
(Lesson 2C)
Find each product.
EXAMPLE 4
24. 0.5 × 100
Find 5.8 × 1,000. 5.8 × 1,000 = 5800.
25. 6.3 × 0.1 = 5,800 26. 47.8 × 0.01
So, 5.8 × 1,000 = 5,800.
27. 0.87 × 1,000
EXAMPLE 5
1,000 has 3 zeros, so move the decimal point 3 places to the right.
Find 6.7 × 0.01. 28. The local sports store sold 100 boxes of golf balls in the past week for $7.49 each. What was the total cost? 29. A remote control car with a speed of 2.7 feet per second was recently modified to go 10 times faster. What is the new speed of the car?
6.7 × 0.01 = 0.067 = 0.067
So, 6.7 × 0.01 = 0.067.
Problem-Solving Strategy: Look for a Pattern
There are 2 places after the decimal point, so move the decimal point 2 places to the left.
(Lesson 2D)
Solve. Use the look for a pattern strategy.
EXAMPLE 6
30. Find the next three numbers.
Find the pattern in the table.
48.5, 47, 45.5, 44, , , 31. Draw the next figure in the pattern.
Input (x)
Output (y)
3
7.8
6
15.6
9
23.4
12
31.2
Study the relationship between each input and output. Each output is 2.6 times the input. So, the pattern is 2.6 times x, or 2.6x.
284
Multiply and Divide Decimals
Lesson 3
Divide Decimals by Whole Numbers
Estimate Quotients of Decimals and Whole Numbers Estimate each quotient.
(Lesson 3A)
EXAMPLE 7
32. 35.9 ÷ 9
33. 4.6 ÷ 4
Estimate the quotient of 37.1 ÷ 6.
34. 9.4 ÷ 3
35. 63.2 ÷ 7
37.1 ÷ 6
36. The amount of miles an athlete ran in a week was 27.3 miles. What was the approximate number of miles the athlete ran per day, if he ran the same amount each day?
36 ÷ 6 = 6
Divide mentally.
So, 37.1 ÷ 6 is about 6.
Divide Decimals by Whole Numbers Divide.
36 ÷ 6
Change 37.1 to 36 because 36 and 6 are compatible numbers.
(Lesson 3C)
EXAMPLE 8
56.4 37. 4
38. 8 4.52
Find the quotient of 12.6 ÷ 7.
39. 67.2 ÷ 5
40. 84.7 ÷ 7
1.8 12.6 7 7 −−− 56 56 −−−− 0
41. Marty spent $20.93 for baseball cards over the past week. If he bought one pack per day and each pack of cards is the same price, how much does each pack of cards cost?
Divide Decimals by Powers of Ten Find each quotient.
Place the decimal point.
So, 12.6 ÷ 7 = 1.8.
(Lesson 3D)
EXAMPLE 9
42. 71.6 ÷ 0.1
43. 5.14 ÷ 0.01
Find 46.8 ÷ 100.
44. 28.7 ÷ 1,000
45. 309 ÷ 100
46.8 ÷ 100 = 0.468
46. The smaller glass holds 10 times less than the larger glass. How many milliliters does the smaller glass hold?
= 0.468
100 has 2 zeros, so move the decimal point 2 places to the left.
So, 46.8 ÷ 100 = 0.468. V = 83.4 mL
V = ? mL
Chapter Study Guide and Review 285
Chapter Study Guide and Review
Lesson 4
Divide Decimals
Divide Decimals
(Lesson 4B)
Divide. 47. 3.96 ÷ 0.6
EXAMPLE 10
48. 22.26 ÷ 4.2
49. 0.525 ÷ 1.5
50. 4.2 ÷ 0.6
51. 48.38 ÷ 8.2
52. 0.063 ÷ 0.3
53. Stacey cut the board below into 12.6-inch pieces. How many boards did she cut?
Find 18.6 ÷ 0.2. 18.6 0.2
Multiply 0.2 by 10 to make a whole number. Multiply 18.6 by the same number, 10.
93. 2. 186. 18 −−−− 6 6 −−− 0
Place the decimal point. Divide as with whole numbers.
So, 18.6 ÷ 0.2 = 93. Check 93 × 0.2 = 18.6
Problem-Solving Investigation: Choose the Best Strategy Solve.
(Lesson 4C)
EXAMPLE 11
54. George, Francine, and Alisha spent $45.75 on a birthday gift for their dad. If they equally divided the price, how much did each person pay?
Jamal rode his bike 5.7 miles over 3 days. If he rode his bike the same amount each day, how far did he ride in one day? 5.7 miles
55. Randy bought 4 pencils and a pad of notebook paper. If each pencil cost $0.30, how much did he spend in all?
?
5.7 ÷ 3 = 1.9 So, Jamal rode his bike 1.9 miles each day.
$1
.49
286
Multiply and Divide Decimals
Practice Chapter Test Estimate each product. 1. 7 × 0.9
2. 12 × 1.7
3. 5 × $2.79
4. 3 × $21.35
5. What is the area of the rectangle in square centimeters? 4.8 cm
12. Melissa and her 3 friends went to the movies and spent $40.25 on tickets, popcorn, and a drink. About how much will each person pay if they divide the total price equally? 13. Toby and his 5 friends ordered a large pizza for $15.99, and 6 large drinks for $1.99 each. What is the total cost, not including tax?
15.5 cm
Divide. Multiply. 6. 7.2 × 0.5 8. 7.06 × 6.1
14. 5.7 ÷ 3
15. 0.2 ÷ 5
16. 4.8 ÷ 6
17. 10.5 ÷ 3
7. 2.8 × 3.1 9. 0.04 × 19.3
10. The table shows a list of recycling prices per pound for certain metals. Metal
Price per pound ($)
Aluminum
0.92
Copper
2.04
Lead
0.68
Tin
6.67
Bridget collected 100 pounds of aluminum cans to recycle. How much money would she earn from collecting the cans? 11. How many cubes will the next figure have if the pattern below continues?
18. Ten bananas contain 290.5 grams of carbohydrates. If each banana contains the same amount of carbohydrates, then how many grams of carbohydrates are in one banana?
Divide. 19. 5.98 ÷ 4.6
20. 2.36 ÷ 0.4
21. 88.4 ÷ 6.8
22. 18.45 ÷ 2.05
23. Tracy paid $157.50 for concert tickets. If each ticket costs $15.75, how many tickets did she buy? 24.
E
WRITE MATH Explain the advantages
and disadvantages of using an estimate when computing with decimals.
Practice Chapter Test
287
Test Practice
Mitch had $3.45 left after buying some supplies for school. If he originally had 10 times that amount, how much money in dollars did he have before he bought supplies? Read the Test Item
Be sure to count all of the decimal places in the numbers you are multiplying before placing the decimal in your answer.
You need to multiply the remaining amount of money by ten to determine how much money he started with. Solve the Test Item Multiply 3.45 by 10 to find the amount of money he started with. 3.45 × 10 −−−− 34.50
two decimal places two decimal places
Check your work. 34.50 ÷ 10 = 3.45 Fill in the grid.
34
.
50
Read each question. Then fill in the correct answer on the answer sheet provided by your teacher or on a sheet of paper. 1. Monica downloaded 8 new songs that cost $0.99 each. How much did Monica spend for the songs she downloaded?
2. Find the area of the parallelogram in square centimeters by multiplying base by height. 6.2 cm
A. $6.89 13.7 cm
B. $7.72 C. $7.92 D. $8.00
288
Multiply and Divide Decimals
F. 84.94 G. 78.92 H. 42.47 I. 19.9
3. The table shows Mr. Davidson’s monthly water bill for June through September. He splits the bill equally among 3 people. How much will each person have to pay for the month of June? Monthly Water Bill
7. About 12.4 million citizens in California speak Spanish as their first language. About 3 million citizens in Florida speak Spanish as their first language. How many more people speak Spanish in California than in Florida?
Bill ($)
A. 12.1 million
C. 9.4 million
June
32.76
B. 10.4 million
D. 8.6 million
July
24.58
August
24.89
September
26.43
Month
A. $5.46
C. $8.81
8. How many places are to the right of the decimal point in the product of the following problem?
B. $8.19
D. $10.92
45.929 × 1.7
4. The History Museum is selling tickets at the prices shown below. Tickets
Price
Adult
$16.50
Children
$11.50
If a family purchases two adult tickets and two children tickets, how much will they spend for the tickets?
5.
6.
F. $23.00
H. $56.00
G. $33.00
I. $66.00
SHORT RESPONSE A family is hiking a trail that is 627 yards long. What is this distance in feet? (Hint: 1 yard equals 3 feet.)
F. 1 place
H. 3 places
G. 2 places
I. 4 places
9. Mary bought one of each item shown below. What was the total price for all of the items bought? .59 $1
10.
GRIDDED RESPONSE A piece of ribbon 3.2 meters long is cut into 0.4-meter pieces. How many pieces will there be?
.49 $2
A. $5.07
C. $5.99
B. $5.79
D. $6.07
.99 $1
SHORT RESPONSE Mrs. Neal’s class found the mass of the class gerbil to be 96.3 grams. Each day, the gerbil eats 0.1 of its body weight in food. How much food does the gerbil eat?
NEED EXTRA HELP? If You Missed Question . . . Go to Chapter-Lesson . . . For help with . . .
1
2
3
4
5
6
7
8
9
10
6-1C
6-2B
6-3C
6-1C
2-2B
6-4B
5-3C
6-2B
5-2C
6-2C
GLE 2.5 GLE 2.5 GLE 2.5 GLE 2.5 GLE 2.5 GLE 2.5 GLE 2.5 GLE 2.5 SPI 2.5 GLE 2.5
Test Practice 289
