Download 8-1 Multiplying a Fraction and a Whole Number

Name
Reteaching
8-1
Find 12 3 _14 .
Find _35 of 15, or _35 3 15.
12 3 _14 is the same as dividing
15 4 5 5 3, so _15 3 15 5 3
12 by 4.
Since _35 is 3 times _15,
12 4 4 5 3
3
_
5
3
_
5
12 3 _14 5 3
3 15 5 3 3 __15 3 15+ 5 3 3 3 5 9.
3 15 5 9
Find each product.
1. _45 3 20 5
16
4. _25 of 15 5
6
2. _67 of 14 5
5. 400 3 _38 5
12
150
3. 24 3 _34 5
18
7 of 80 5
6. __
10
56
7. Reasoning Can you use division and mental math to find _23 of 24?
Why or why not?
Yes, because 3 is the greatest common factor
of 3 and 24. _23 of 24 5 16
The chart shows the average high temperatures for different months in Phoenix, Arizona.
Phoenix Weather
Month
Average High
February
70°F
May
93°F
July
105°F
8. What is _45 the average temperature in July?
84°F
9. What is _12 the average temperature in February?
35°F
10. What is _23 the average temperature in May?
18 Topic 8
62°F
© Pearson Education, Inc. 6
Reteaching 8-1
Multiplying a Fraction
and a Whole Number
Name
Practice
8-1
Multiplying a Fraction
and a Whole Number
Find each product.
12
4. _18 of 72 9
2. _56 30 25
600
5. 900 _23 3. 42 _56 35
13 of 100 6. __
20
65
Practice 8-1
1. _34 16 9 of 81. Explain.
7. Reasoning Without multiplying, tell which is greater, _56 of 81 or __
10
__
5.
9 of 81 because __
9 is greater than _
10
10
6
Departure City
Pittsfield, Massachusetts
Reno, Nevada
Driving Distances
Destination City
Providence, Rhode Island
Wendover, Utah
Distance
132 mi
400 mi
8. Mike drove _13 of the distance between
Pittsfield, Massachusetts, and Providence,
Rhode Island. How far did he drive?
44 mi
9. Bimal drove _35 of the distance between
Reno, Nevada, and Wendover, Utah.
How far did he drive?
240 mi
10. Estimation How many more miles does Bimal
have to drive to get to Wendover, Utah?
160 mi
11. There are 25 students in Mr. Fitch’s sixth-grade class. If _35 of the
students are girls, how many girls are in Mr. Fitch’s class?
A 5 girls
B 10 girls
C 15 girls
D 20 girls
© Pearson Education, Inc. 6
12. Writing to Explain Explain how you would find the product of 36 and _23.
Write the problem as 36 ⴛ _23 .
36 ⴜ 3 ⴝ 12, so _13 ⴛ 36 ⴝ 12.
_
2 ⴛ 36 ⴝ 2 ⴛ (_
1 ⴛ 36) ⴝ 2 ⴛ 12 ⴝ 24.
3
3
Topic 8
19
Name
Reteaching
8-2
Estimating Products
When you are working with fractions and mixed numbers, you can estimate using
rounding, compatible numbers, or compatible benchmark fractions.
3 21 ≈ 6
__
10
Think: 20 ÷ 10 2.
3 12 using a compatible
Estimate __
10
benchmark fraction.
3 to a compatible
3 12
__
Round __
10
10
3 is
benchmark fraction. Since __
10
‚
‚
1
_
close to 4 and 4 is a factor of
1 12 3
_
twelve, use _14 .
4
3 12 ≈ 3
__
10
Think: 12 4 3.
1 3 3.
3 2 6.
Sample answers are shown.
Estimate each product by using compatible numbers or benchmark fractions.
1. _16 19 4. 31 _25 7. 43 _27 3
12
12
2.
4 10 _
7
7 18 5. __
12
5 8. 35 __
12
5
9
15
3. _58 23 9 90 6. __
16
9. 16 _49 15
45
8
Estimate each product by rounding each factor to the nearest whole number.
10. 6_23 5_18 ‡ Round 6_23:
11. 10_29 4_56 50
7
Round 5_18:
12. 2_78 3_34 =
12
5
Multiply:
35
4 13. 4_15 2__
10
8
14. Reasonableness Carlotta estimated that _37 20 is about _37 14 6. Is her estimate
reasonable? Why or why not?
No, a better
choice for a number compatible to
__
3
20 is 21: 7 21 9.
6 18 shown below different? Is one
15. Critical Thinking Why are the estimates of __
10
estimate better than the other?
6 18 ≈ __
6 × 20 12
__
10
10
6 × 18 ≈ _
1 18 9.
__
10
2
Sample answer: The first estimate uses a
compatible whole number to estimate and the
second uses a compatible benchmark fraction.
Both are reasonable ways to estimate, and in this
case, one method is not better than the other.
24
Topic 8
© Pearson Education, Inc. 6
Reteaching 8-2
3 21 using a whole number
Estimate __
10
that is compatible with the denominator.
3 21
__
Change 21 to the nearest
10
whole number that is
‚
‚
compatible with 10.
3 20 6
__
10
Name
Practice
Estimating Products
8-2
Sample answers are shown.
6 ⫻ 5_
3 ⫽
2. 3 ⫻ 2_15 ⫽
3. __
2
6
10
4
5. 6_12 ⫻ 2_13 ⫽
6. _78 ⫻ 2_38 ⫽
12
14
8. _14 ⫻ 17 ⫽
9. _35 ⫻ 51 ⫽
15
4
12 ⫻ 8 ⫽
12. 11 ⫻ _12 ⫽
42 11. __
4
25
5 ⫻ 13 ⫽
15. __
6 14. 7__17 ⫻ 2_23 ⫽ 21
12
Estimate each product.
1. 4_58 ⫻ _13 ⫽
4. 2_79 ⫻ 4_25 ⫽
7. 38 ⫻ _38 ⫽
10. 7_49 ⫻ 5_67 ⫽
16. Show three ways to estimate _35 ⫻ 5_34. Identify each method you use.
Sample answer: Compatible numbers: _35 ⴛ 5 ⴝ 3;
Rounding: 1 ⴛ 6 ⴝ 6; Compatible benchmark: _12 ⴛ 6 ⴝ 3.
3 miles from his office. He estimates that he
17. Explain It Mr. Simpson lives 11__
10
commutes 11 ⫻ 2 ⫻ 5, or 110 miles each week. Is his estimate an overestimate or an
underestimate? Explain.
Sample answer: 110 is an underestimate because
Mr. Simpson rounded the number of miles for each
trip to the lower whole number before he multiplied.
_
1
7
__
18. Which benchmark fraction could you use to estimate the product of 38 ⫻
12
?_______
2
19. Geometry Which is the best estimate for the area of a square with sides equal to 3_15
inches?
A
B
C
D
3 sq in.
6 sq in.
9 sq in.
16 sq in.
© Pearson Education, Inc. 6
9
20. Joyce and Marianne have money jars. Joyce has 54 dimes in her jar. Marianne has __
10
as many dimes as Joyce. Estimate the number of dimes that Marianne has in her jar.
A
B
C
D
60 dimes
45 dimes
6 dimes
5 dimes
Topic 8
25
Practice 8-2
4 ⫽
13. _89 ⫻ 6__
10
3
2
30
6
5
Name
Reteaching
8-3
Multiplying Fractions
Find _34 ⫻ _25.
Draw a picture.
Shade the squares.
There are 20 squares in all.
6 squares have overlapping shading.
3 ⫻_
6.
2 ⫽ __
_
5
4
20
Use the
denominators to
determine the
number of
squares: 5 tall and
4 wide.
6 ⫽ __
3
Simplify: __
20
10
Multiply the numerators and the
denominators. Simplify if possible.
(3 ⫻ 2)
3 ⫻_
2 ⫽ _____
_
5
4
(4 ⫻ 5)
Reteaching 8-3
6
⫽ __
20
Simplify first. Divide a numerator and a
denominator by their GCF. Then multiply.
3
_
4
(2 ⫼ 2)
3
⫻ _25 ⫽ _____
⫻ _____
5
(4 ⫼ 2)
⫽ _32 ⫻ _15
3
⫽ __
10
3
⫽ __
10
Draw a picture to solve.
1. _13 ⫻ _45 =
__
4
15
2. _34 ⫻ _39 =
_
1
3. _12 ⫻ _78 =
4
__
7
16
Write an equation for each picture.
5.
4.
_
3 ⴛ_
2 ⴝ_
1
4
3
2
Find each product. Simplify if possible.
3 =
6. _57 ⫻ __
10
9. _56 ⫻ _38 =
__
3
14
__
5
16
6 =
7. _12 ⫻ __
15
5 =
10. _67 ⫻ __
12
_
1
5
__
5
14
8. _47 ⫻ _12 =
11. 8 ⫻ _34 =
12. Number Sense Can you simplify before multiplying 14 ⫻ _47 ? Explain.
30
_
2
7
6
Yes. Rewrite 14 as a fraction with 1 in the denominator.
Then simplify the 14 and the 7 using their
8 ⴝ 8.
14 ⴛ _
4ⴝ_
2ⴛ_
4ⴝ_
GCF of 7: __
7
1
1
1
1
Topic 8
© Pearson Education, Inc. 6
_
1 ⴛ_
1 ⴝ_
1
3
2
6
Name
Practice
8-3
Multiplying Fractions
Write an equation for each picture.
1.
2.
_
3 3_
3
4 5 __
12 5 __
7
14
8
56
_
2 3_
1 5 __
2
9
3
27
Find each product. Simplify if possible.
__
13
30422_T10_10-1PR-1
13 5
7 3 __
3. __
10
14
6. _34 3 16 5
12
3 5
7. _25 3 __
10
17 5
9. _35 3 __
21
35
__
17
13 3 100 5
12. __
20
65
10. _18 3 72 5
13. _38 3 _49 5
__
7 30422_T10_10-1PR-2
3
4
_
_
10
__
3
25
9
_
1
6
5.
7
3
9
5
8. _56 3 42 5
15 3 __
24 5
11. __
9
25
13 5
1 3 __
14. __
2
16
Pamela spent _23 of an hour doing homework. She solved math problems
for _25 of that time and read her science book for _35 of that time. What
fraction of one hour did Pamela spend:
15. solving math problems?
__
4
15
16. reading her science book?
__
4
21
35
_
8 5 1_
3
5
5
__
13
32
_
2
5
7 live within a mile of school.
17. Of the students in Mr. Moore’s room, __
13
4
_
Of those students, 7 live within half a mile of school. What fraction
of all students in Mr. Moore’s class live within half a mile of school?
3
A __
13
4
B __
13
3
C __
15
4
D __
15
© Pearson Education, Inc. 6
18. Writing to Explain Without multiplying, tell which is greater:
55 3 81 or __
9 3 81. Explain.
__
6
10
__
55 3 81 is greater because __
55 is more than 9
6
6
9 is less than 1. Eighty-one times 9 is
and __
10
greater than 81 3 1.
Topic 8
31
Practice 8-3
20
4. _45 3 _78 5
Name
Reteaching
8-4
Multiplying Mixed Numbers
How to find the product of two mixed numbers: Find 3_23 ⫻ 4_12.
Step 1
Round 3_23 to 4 and 4_12 to 5:
4 ⫻ 5 ⫽ 20
Estimate the product by rounding.
Step 2
9
11 and 4_
1 ⫽_
3_23 ⫽ __
3
2
2
Write each mixed number as an improper
fraction.
3
9
3
11 × _
11 × _
/ ⫽ __
3_23 ⫻ 4_12 ⫽ __
3/
2
1
2
Look for common factors and simplify.
1
Step 3
3 ⫽ __
33
11 ⫻ _
__
1
2
2
33 ⫽ 16_
1
__
Multiply the numerators and denominators.
Write the product as a mixed number.
2
2
Find each product. Simplify if possible.
1. 2_34 ⫻ 3_12
4. 1_25 ⫻ 3_14
9_58
11
4__
20
2. 2_15 ⫻ 2_23
5. 4_12 ⫻ 16
Evaluate each expression for K ⫽ 2_13 .
7. 12K
28
8. 1_34 K
13
5__
15
72
1
4__
12
6. 1_38 ⫻ 2_12
19_12
7
3__
16
9. 2_23K
6_29
3. 6 ⫻ 3_14
10. Reasonableness What is a reasonable estimate for 7_34 ⫻ 2_23 ?
_
3
Explain.
24, since 74 rounds to 8 and
2_23 rounds to 3 and 8 ⴛ 3 ⴝ 24.
11. The cups of mushrooms in a recipe is 2_12 times the cups of onions.
The cups of onions is 1_12. Solve c ⫽ 1_12 ⫻ 2_12 to find c, the cups of
mushrooms.
3_34 c of mushrooms
© Pearson Education, Inc. 6
Reteaching 8-4
16_12 is close to 20, so the answer is reasonable.
36
Topic 8
Name
Practice
8-4
Multiplying Mixed Numbers
Find each product. Simplify if possible.
1. 3_12 ⫻ 1_23
4. 2_16 ⫻ 1_15
7. 1_23 ⫻ 2_14
5_56
2_35
3_34
2. 1_18 ⫻ 2_13
5. 3_16 ⫻ 18
8. 10 ⫻ 1_13
Evaluate each expression for S ⫽ 1_45 .
10. 2_13S
4_15
11. 3_34S
2_58
57
13_13
6_34
3. 7 ⫻ 1_14
6. 1_18 ⫻ 2_12
9. 2_45 ⫻ 3_13
12. 5_16S
8_34
13
2__
16
9_13
3
9__
10
Use the table to answer the questions.
13. If Berkeley receives 1_14 times its average January
rainfall, how much rain will it receive?
14. How much rain will Berkeley receive if it is 2_13 times
the October average?
3_12 in.
January
7 in.
3__
10
April
1_45 in.
October
1_12 in.
Practice 8-4
4_58 in.
Average Rainfall in
Berkeley, California
15. Which month has about twice the rainfall as April?
January
16. Jessie stacked photographs of 6 zoo animals on top of each
other to create a display. Each photo is 14_14 in. high. How high is
the display?
A 84_23 in.
B 85_12 in.
C 86_34 in.
D 87 in.
© Pearson Education, Inc. 6
17. Writing to Explain Explain how you would find 2 ⫻ 2_13 using the
Distributive Property.
Multiply 2 ⴛ 2 and 2 ⴛ _13 , and then add the
products to get 4 ⴙ _23 or 4_23.
Topic 8
37
Name
Reteaching
Problem Solving:
Multiple-Step Problems
8-5
Some word problems have hidden questions that must be answered before you can
solve the problem.
A paved trail is 8 miles long. Rita runs _38 of the length of the trail and walks the rest of the
way. How many miles of the trail does Rita walk?
What do you know?
What are you asked to find?
How can you find the distance that Rita
walks?
What is the hidden question? The hidden
question will help you find data you need to
solve the problem.
Rita runs _38 of an 8-mile trail.
How many miles of the trail that Rita
walks.
Subtract the distance Rita ran from the
length of the trail.
How many miles did Rita run?
To answer, find _38 3 8 5 3.
Write and answer the hidden question(s) in each problem. Then solve the problem.
1.
April surfed for _13 of the 6 hours she was at the beach. She spent the rest of the time
building a sand castle. How many hours did she spend building the castle?
Hidden question:
Solution:
2.
How many hours did April surf? _13 3 6 5 2
6 2 2 5 4; 4 hours
Bill put gasoline in 2 of his 5-gallon cans and 4 of his 2-gallon cans. He filled all the
cans to the exact capacity. How many gallons of gasoline did he buy?
How many gallons are in two 5-gallon cans?
Hidden question 1:2 3 5 5 10
How many gallons are in four 2-gallon cans?
Hidden question 2:4 3 2 5 8
10 1 8 5 18; 18 gallons of gas
Solution:
3.
It costs Le Stor $20 to buy a shirt. The store sells the shirt for 2_12 times its cost. What
is the profit for 100 of these shirts? Hint: Profit equals sales minus cost.
What
is the selling price of the shirt?
_
1
3
22 $20 5 $50
Hidden question 2:What is the profit for one shirt?
$50 2 $20 5 $30
Solution:
100 3 $30 5 $3,000
Hidden question 1:
42 Topic 8
© Pearson Education, Inc. 6
Reteaching 8-5
Use the data to solve: 8 2 3 5 5, so Rita walked 5 of the 8 miles.
Name
Practice
Problem Solving:
Multiple-Step Problems
8-5
Write and answer the hidden question(s) in each problem. Then solve the problem.
1.
Tiwa spent 1_12 hours setting up her computer. It took her 3 times as long to install the
software. How long did it take Tiwa to set up the computer and install software?
Hidden question(s):
Solution:
2.
How many
hours did it take to install software?
3 ⴛ 1_12 ⴝ 4_12
1_12 ⴙ 4_12 ⴝ 6; 6 hours
Lon bought 40 ounces of sliced ham. He used _34 of the ham to make sandwiches for
his friends and _15 of the ham in an omelet. How many ounces of ham were left?
How many ounces of ham were used to make
Hidden question(s): sandwiches? 3/4 × 40 = 30; How many ounces
of ham were used in the omelet? 40 ⴛ _15 ⴝ 8
Solution:
40 ⴚ 30 ⴚ 8 ⴝ 2; 2 ounces
Lionel cut off _16 of a 48-inch piece of rope. Marsha cut off _14 of a 36-inch piece of
rope. They compared their cut pieces. Whose piece is longer? How much longer?
_1 ⴛ 48 ⴝ 8;
How
long
is
Lionel’s
cut
piece?
Hidden question(s):
6
How long is Marsha’s cut piece? _14 ⴛ 36 ⴝ 9
Solution:
Marsha’s rope is 1 inch longer: 9 ⴚ 8 ⴝ 1.
4.
Melanie bought 3 CDs. The country music CD cost $15. The rock music CD cost _23
as much as the country music CD. The platinum edition CD cost twice as much as
the rock CD. What was the cost of the three CDs?
Practice 8-5
3.
How much did the rock CD cost? _23 ⴛ $15 ⴝ $10;
Hidden question(s): How much did the platinum CD cost?
2 ⴛ $10 ⴝ $20
Solution:
$15 ⴙ $10 ⴙ $20 ⴝ $45
© Pearson Education, Inc. 6
5. Writing to Explain Choose one of the problems above. Explain how you determined
the hidden question and why it was necessary to answer that question in order to
solve the problem.
Sample answer: In Problem 1, the question asks for the time
spent on two tasks, so I needed to add the two times to
answer the question. Only the time for one task was given
in the problem, so the hidden question had to be to find the
other time.
Topic 8
43