Name _____________________________________________________________________________________________________
Factors and Divisibility
One number is divisible by another
number when the quotient is a whole
number and the remainder is zero.
You can use the divisibility rules to
help you find factors of a number.
R 7-1
Divisibility Rules
Divisible by
Rule
2
The number is even.
3
The sum of the digits is divisible
by 3.
Is 9 a factor of 117?
4
The number formed by the tens and
ones digits is divisible by 4.
If 117 is divisible by 9, then 9 is a
factor of 117.
5
The ones digit is 0 or 5.
6
The number is divisible by 2 and 3.
Use the divisibility rule for 9: the sum
of the digits is 1 ! 1 ! 7 " 9. 9 is
divisible by 9, so 117 is divisible by 9.
9
The sum of the digits is divisible
by 9.
Example 1
10
The ones digit is 0.
Example 2
List all the factors of 32.
Try 1, then 2, then 3, and so on.
The factors of 32 are
1, 2, 4, 8, 16, and 32.
Try:
1
2
3
4
5
6
7
8
Is it a factor?
Yes,
Yes,
No.
Yes,
No.
No.
No.
Yes,
Factors:
1 # 32 " 32
2 # 16 " 32
1, 32
2, 16
4 # 8 " 32
4, 8
8 # 4 " 32
8, 4
Mental Math Tell if each number is divisible by 2, 3, 4, 5, 6, 9, or 10.
1. 39
2. 120
3. 56
4. 4,731
5. 356
6. 425
7. 1,240
8. 331
9. 744
© Scott Foresman, Gr. 5
10. 123
(209)
Use with Chapter 7, Lesson 1.
Name _____________________________________________________________________________________________________
Factors and Divisibility
H 7-1
Fill in the chart. Write if each number is divisible by 2, 3, 4, 5, 6, 9, or 10.
Divisible by
1.
24
2.
65
3.
79
4.
60
5.
45
6.
423
7.
5,170
8.
7,536
9.
9,000
10.
2,222
11.
31,770
12.
501,264
2
3
4
5
6
9
10
List all the factors of each number.
13. 33
14. 74
15. 53
16. 94
17. Math Reasoning If a number is divisible by 12, is it also divisible by 2, 3, 4, and 6?
Test Prep Circle the correct letter for each answer.
18. Which number is divisible by 2, 3, 4, 5, 6, 9 and 10?
F 1,422
© Scott Foresman, Gr. 5
G 9,002
(210)
H 45,360
J 64,860
Use with Chapter 7, Lesson 1.
Name _____________________________________________________________________________________________________
Prime Factorization
R 7-2
A whole number greater than 1 is either a prime number or a composite number.
A prime number has exactly two factors, 1 and the number itself.
A composite number has more than two factors.
Example 1
List the numbers from 50 to 60. Which numbers are prime?
50
51 52
! !
! 53 54
! 55
! 56
! 57 58
! 59 60
!
Cross out all the numbers that are divisible by 2 since they are not prime.
Cross out all the numbers divisible by 3 since they are not prime. Continue
until only prime numbers remain. The prime numbers from 50 through 60
are 53, 57, and 59.
Example 2
Use a factor tree to find the factorization of 24. Write 24 as a product
of two factors. Then write each factor that is not prime as a product.
Write 24 as a product.
2 is prime, 12 is not prime.
24
2
2
!
12
2
!
2
!
6
!
2
!
2
!
Write 12 as a product.
Write 6 as a product.
3
2 and 3 are prime.
If a factor appears more than once, exponents can be used.
24 " 2 ! 2 ! 2 ! 3 " 23 ! 3
Complete each factor tree. Then write the prime factorization using exponents if possible.
1.
9
" 32
!
2.
16
3.
!
!
15
" 15
!
© Scott Foresman, Gr. 5
(212)
!
!
!
x
Use with Chapter 7, Lesson 2.
" 24
Name _____________________________________________________________________________________________________
Prime Factorization
H 7-2
Complete each factor tree. Then write the prime factorization, using exponents if possible.
42
1.
6
56
2.
!
!
!
!
7
!
" 42
!
!
" 56
Write the prime factorization of each composite number, using exponents when possible.
If a number is prime, write prime.
3. 55
4. 29
5. 28
6. 120
7. 72
8. 64
9. 45
10. 36
11. 80
12. 49
13. 13
14. 83
15. 30
16. 31
17. 88
18. 77
19. 59
20. 99
21. 57
22. 60
Test Prep Circle the correct letter for each answer.
23. Which is the prime factorization of 72?
A 2!2!2!3
B 22 ! 3 ! 3
C 22 ! 33
D 23 ! 32
H 2 ! 32 ! 7
J 2 ! 3 ! 72
24. Which is the prime factorization of 294?
F 2 ! 147
© Scott Foresman, Gr. 5
G 2 ! 21 ! 7
(213)
Use with Chapter 7, Lesson 2.
Name _____________________________________________________________________________________________________
Common Factors and GCF
R 7-3
You can use what you know about divisibility and fact families to help
you find the greatest common factor.
Find the greatest common factor of 9 and 12.
List all of the factors of each number. Remember to use divisibility rules.
Factors of 9: 1, 3, 9
Factors of 12: 1, 2, 3, 4, 6, 12
Then list the common factors, those numbers that are in both lists.
Common factors: 1, 3
The largest number in this list is 3. The greatest common factor (GCF)
of 9 and 12 is 3. The greatest common factor is sometimes called the greatest
common divisor (GCD).
The common factors of 9 and 12 are 1 and 3.
The greatest common factor (GCF) of 9 and 12 is 3.
The greatest number that can be in each group is 3.
List the factors of each number. Then find the GCF for each set of numbers.
1. 15
10
factors:
factors:
2. 24
GCF:
18
factors:
factors:
GCF:
3. 45
27
factors:
factors:
GCF:
© Scott Foresman, Gr. 5
(215)
Use with Chapter 7, Lesson 3.
Name _____________________________________________________________________________________________________
Common Factors and GCF
H 7-3
List the factors of each number. Then find the GCF of each set of numbers
1. 25, 34
2. 16, 20
25 !
16 !
34 !
20 !
GCF !
GCF !
3. 16, 28, 48
4. 20, 35, 70
16 !
20 !
28 !
35 !
48 !
70 !
GCF !
GCF !
5.
Draw a Venn diagram to show the
factors of 24 and 30.
What are the common factors?
What is the greatest common factor?
6.
A toy shop has 40 red marbles and 56 yellow marbles lying loose in a bin. The shop
owner wants to package them so that there are the same number of each per each bag.
How many bags will there be?
How many red marbles will be in each bag?
How many yellow marbles?
Test Prep Circle the correct letter for each answer.
7.
Which is the GCF for 12, 20, and 36?
A 12
8.
B 4
C 36
D 9
H 13
J 1
Which is the GCF for 5, 13, and 26?
F 5
© Scott Foresman, Gr. 5
G 2
(216)
Use with Chapter 7, Lesson 3.
Name _____________________________________________________________________________________________________
Common Multiples and LCM
R 7-4
A multiple of a whole number is the product of that number and any other whole
number. 44 is a multiple of 4 because 4 " 11 ! 44. 44 is also a multiple of 11.
Common multiples of numbers are multiples that are the same for each number.
Multiples of 4: 4, 8, 16, 20, 24, 28, 32, 36, 40
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Common multiples: 20, 40, . . . LCM: 20
Some common multiples of 4 and 8 are 8, 16, 24, and 32. The least common
multiple (LCM) is 8.
List the first ten multiples of each number.
1. 2
2. 3
3. 5
4. 7
5. 9
Using the multiples above, list the common multiples shown for each set of numbers.
Then circle the LCM.
6. 2, 7
7.
2, 3
8. 9, 7
9.
3, 5
List the first ten multiples of each number. Then find the LCM of each number.
10. 5, 6
5!
6!
LCM !
11. 3, 5
3!
5!
© Scott Foresman, Gr. 5
LCM !
(218)
Use with Chapter 7, Lesson 4.
Name _____________________________________________________________________________________________________
Common Multiples and LCM
H 7-4
List the first ten multiples of each number. Then find the LCM of the numbers.
1. 4, 5
4!
5!
LCM !
2. 8, 9
8!
9!
LCM !
3. 3, 5, 6
3!
5!
6!
LCM !
4. Laurie set up the household task schedule at the right.
If she begins her schedule on the first day of each month,
on which days will she have two tasks to do?
Monthly Schedule
Task
5. On which days will she have 3 tasks?
Frequency
Take out trash
Every 2 days
Water plants
Every 4 days
Clean room
Every 7 days
Test Prep Circle the correct letter for each answer.
6. Which is the least common multiple of 12 and 18?
A 12
B 18
C 6
D 36
7. Which is the least common multiple of 6, 8, and 12?
F
12
© Scott Foresman, Gr. 5
G 24
(219)
H 36
J 48
Use with Chapter 7, Lesson 4.
Name _____________________________________________________________________________________________________
Problem-Solving Skill
R 7-5
Reasonable Answers
It is important to know whether an answer to a problem is reasonable. If you
estimate first, you can check to see if the answer is close to the estimate. If it is,
then the answer is reasonable.
The school secretary is arranging bus transportation
for a field trip for the students in the classrooms listed
in the chart, 5 teachers, and 6 volunteer aides. Would
it be reasonable to order 5 buses if each has 42
passenger seats?
Classroom
Number of
students
4
19
5
20
6
22
7
24
8
16
Round 42 to 40 and multiply 40 by 5 to get the total
number of seats on 5 buses, which would be 200.
The total number of people going on the field trip is
only 112, so it would not be reasonable to order
5 buses.
Students are thinking of having Leo’s Print Shop print up tickets for their school play.
Tickets cost $0.07 each to print. The school theater holds 490 people, and there will be
two performances.
1. Which sentence tells how to estimate the cost of printing tickets for the play?
a. Multiply the printing cost by 490.
b. Multiply the printing cost by 490, then multiply that answer by 2.
c. Neither of the above.
2. If tickets cost $0.07 each to print, what is a reasonable price for printing tickets for
one performance?
a. $0.35
b. $3.50
c. $35.00
3. Each ticket is sold for $4.00. If 475 tickets are sold for the first performance and
485 tickets are sold for the second performance, what is a reasonable estimate for
the amount received from ticket sales?
a. $400
© Scott Foresman, Gr. 5
b. $1,000
(221)
c. $4,000
d. $10,000
Use with Chapter 7, Lesson 5.
Name _____________________________________________________________________________________________________
Problem-Solving Skill
H 7-5
Reasonable Answers
The school gym holds 670 people. The teachers
will play the students in a basketball game to raise
money for a new floor. The gym seats will be divided
equally into two sections—one for students and one
for senior citizens and general admission ticketholders.
Ticket Prices
Students
$2.25
Senior Citizens
$1.50
General Admission
$3.00
1. How would students determine whether it is
reasonable to print 500 student tickets?
a. Divide the total number of tickets by 2 and compare that number with 500.
b. Multiply the total number of tickets by 2 and compare that number with 500.
c. Neither of the above.
2. Which expression would you use to estimate the amount of money expected
from selling tickets to students?
a. (700 ! 2) " $3.00 # P
b. (700 ! 2) " $2.00 # P
c. (670 ! 2) " $1.00 # P
3. What is a reasonable estimate of the cost of having team rosters printed for each
attendee if the cost of printing one is $0.03?
a. $2.10
b. $210.00
c. $21.00
4. So far, 57 people have called to reserve 2 general admission tickets each.
Which of the following should you do to estimate the amount of money the
school would earn from these sales?
a. Add 60 to the ticket price and multiply by 2.
b. Multiply 60 by the ticket price.
c. Multiply 60 by the ticket price and double that product.
5. Math Reasoning A week before the game, 88 general admission tickets and 97
student tickets have been sold. Is it reasonable to say that the school has earned
more from student ticket sales so far?
© Scott Foresman, Gr. 5
(222)
Use with Chapter 7, Lesson 5.
Name _____________________________________________________________________________________________________
Relating Fractions and Decimals
R 7- 6
A fraction can be used to represent a part of a region or part of a set.
A fraction has a numerator above the fraction bar
numerator
!!
and a denominator below the fraction bar.
denominator
Write a fraction and a
decimal for the part
that is shaded.
numerator: number of shaded parts
!!!!!
denominator: number of equal parts
2
!!
10
Write a fraction and a
decimal for the part
that is shaded.
0.2
# 2 " 10 #10 !"
2"
.0
20
0
numerator: number of shaded parts
!!!!!
denominator: number of equal parts
3!
!5
0.6
= 3 " 5 # 5 !"
3"
.0
30
0
Write a fraction and a decimal for each shaded part.
1.
2.
3.
4.
5.
■■■■
✚✚✚✚
6.
© Scott Foresman, Gr. 5
(224)
Use with Chapter 7, Lesson 6.
Name _____________________________________________________________________________________________________
Relating Fractions and Decimals
H 7-6
Write a fraction and a decimal for each shaded part.
1.
4.
2.
▲▲▲▲
▲▲▲▲
▲
5.
3.
★
★
6.
Write the following as a fraction and as a decimal.
7. 15 hundreths
8. 8 tenths
9. 20 hundreths
10. 79 hundreths
11.
➨➨ ➨➨➨➨➨➨➨➨➨ ➨ ➨ ➨ ➨ ➨ ➨ ➨ ➩ ➩
Test Prep Circle the correct letter for each answer.
12. There are 7 posters for the Talent Show. Five posters are on yellow poster board.
The others are on white. What fraction of the posters are on white?
7
A !2!
2
B !7!
7
C !5!
2
D !5!
13. Sixty-five of the 100 chorus members are girls. Which is the decimal
describing the part of the chorus made up of girls?
F 1.65
© Scott Foresman, Gr. 5
G 0.135
(225)
H 0.35
J 0.65
Use with Chapter 7, Lesson 6.
Name _____________________________________________________________________________________________________
Equivalent Fractions
R 7-7
Equivalent fractions are fractions that
name the same amount.
2
!3!
4
!!
6
and
2
!3!
are equivalent fractions.
4
!!
6
"
You can find equivalent fractions by multiplying or dividing both the numerator
and the denominator of a fraction by the same nonzero number because
multiplying or dividing by 1 does not change the number.
Example 1
Find an equivalent fraction for !23! by multiplying
the numerator and the denominator by the
same number.
Two fractions equivalent to
2
!!
3
are
4
6
!! and !!.
6
9
Example 2
2
!!
3
"
2#2
!!
3#2
"
2
!!
3
"
2#3
!!
3#3
"
Find a fraction equivalent to
denominator of 10.
32
!!
40
32
!!
40
"
!!
5
What do you divide 40 by to get 5?
Divide 10 by the same number.
32
!!
40
"
32 $ 8
!!
40 $ 8
"
4
!!
6
6
!!
9
with a
4
!!
5
Find each equivalent fraction.
4
2
#
! " !!
1. !
5
#
2
1
3
10
3
4
4. !! " !!
3
$
10 $
15 $
25
30
7. !! " !!
9
2
6
8
6. !! " !!
5
12
24
8. !! " !!
5
5
5
3. !! " !!
6
5. !! " !!
9
16
20
3
! " !!
2. !
18 $ 3
16
9. !! " !!
4
Write the next three fractions in each pattern.
2
4
6
3 6
!, !!, . . .
10. !5!, !
10 15
© Scott Foresman, Gr. 5
9
!, . . .
11. !4!, !8!, !
12
(227)
Use with Chapter 7, Lesson 7.
Name _____________________________________________________________________________________________________
Equivalent Fractions
H 7-7
Find each equivalent fraction.
1.
4
!!
6
!
"!
18
! " !!
2. !
5
15
12
! "! !
3. !
20
4.
2
!!
5
!
"!
40
! " !!
5. !
7
21
7.
7
!!
21
18
15
6. !8! " !!
4
5
!
9. !9! " !
27
2
4
!
12. !6! " !
54
2
12
! " !!
11. !
15
3
12
14. !9! " !!
10. !6! " !!
2
13. !8! " !!
20
35
1
! " !!
8. !
49
" !3!
9
1
6
3
!
15. !4! " !
24
Write the next three fractions in each pattern.
7 14 21
3
6
9
32
!, !!, . . .
16. !8!, !
16 24
!, !!, . . .
17. !7!, !
14 21
2 3
!, !!, . . .
19. !1!, !
9 18 27
!, !!, !!, . . .
20. !
100 200 300
7
14
16
8
!, !!, !!, . . .
18. !
160 80 40
21
9
18 27
!, !!, !!, . . .
21. !
20 40 60
Test Prep Circle the correct letter for each answer.
21
!?
22. Which of the following is equivalent to !
56
56
8
!
A !
21
9
B !3!
3
2
D !8!
36
!
J !
112
C !5!
7
! and !! are equivalent to
23. !
54
42
3
1
F !7!
© Scott Foresman, Gr. 5
G !6!
(228)
H !6!
16
Use with Chapter 7, Lesson 7.
Name _____________________________________________________________________________________________________
Fractions in Simplest Form
R 7-8
A fraction is in simplest form if the greatest common factor (GCF) of the
numerator and the denominator is 1.
Write
2
1 4
2
3
!
3!
5, !
7!
0, and !5! are equivalent
8!
!
in simplest form.
24
but only
3
!5!
is in simplest form.
Step 1
Find the GCF of the numerator and denominator.
Factors of 8: 1, 2, 4, 8
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The GCF of 8 and 24 is 8.
Step 2
Divide the numerator and
the denominator by the GCF.
1
!!
3
is the simplest form or
8
!!
24
8 " 8
!!
24 " 8
# !13!
because the GCF of 1 and 3 is 1.
1. Write the factors of 12.
2. Write the factors of 16.
3. Write the greatest common factor of 12 and 16.
4. Divide by the greatest common factor:
12 "
!!
16 "
#
12
!?
5. What is the simplest form of the fraction !
16
Write each fraction in simplest form.
!6!
24
4!
7. !
28
5!
8. !
25
1!
8
9. !
36
7!
10. !
21
6!
11. !
30
9!
12. !
72
8!
13. !
48
6.
© Scott Foresman, Gr. 5
(230)
Use with Chapter 7, Lesson 8.
Name _____________________________________________________________________________________________________
Fractions in Simplest Form
H 7-8
Write each fraction in simplest form. Write yes if the fraction is already in simplest form.
3
9
3. !!
9
10
4. !!
16
18
7. !!
4
12
8. !!
6
30
12. !!
96
100
5
10
2. !!
11
15
6. !!
!!
1. !
5. !!
26
36
10. !!
13. !!
32
40
16
17. !!
100
9. !!
6
10
18
1
7
11. !!
14. !!
3
100
15. !!
16. !!
5
18. !!
39
8
19. !!
70
3
20. !!
13
13
!
21. !
21
3
12
14
20
60
100
15
11
!
22. !
52
!
23. !
44
!
24. !
85
Circle the fractions that are in simplest form.
25. a. !!
12
15
b. !!
4
10
c. !!
5
6
d. !!
9
15
e. !!
26. a. !!
5
12
b. !!
9
18
c. !!
3
6
d. !!
7
14
e. !!
27. a. !!
6
16
b. !!
11
12
c. !!
12
15
d. !!
10
14
e. !!
14
28
b. !!
6
21
c. !!
5
40
d. !!
13
39
e. !!
28. a. !!
10
20
8
12
15
25
7
18
9
!
29. Math Reasoning Sal noticed that both the numerator and denominator of !
27
3
are odd numbers, so he divided by 3 to get !9!. Explain why the fraction is not
yet in simplest form. What can Sal do now to find the simplest form?
Test Prep Circle the correct letter for each answer.
30. Which of the following is in simplest form?
11
A !
3!
3
3
B !
1!
1
9!
C !
111
13
D !
3!
9
9!
H !
111
9
J !
1!
4
31. Which of the following is in simplest form?
13
F !
18!
2
© Scott Foresman, Gr. 5
18
G !
2!
4
(231)
Use with Chapter 7, Lesson 8.
Name _____________________________________________________________________________________________________
Relating Fractions to One Half
R 7-9
A number line is helpful when comparing fractions to 1"2".
Find
Is
4
"
1"
0
4
"
1"
0
on the number line below.
less than 1"2", equal to 1"2", or greater than 1"2"?
1
""
2
0
0
""
10
1
""
10
2
""
10
is to the left of
so
4"
"
10
is less than
1
"2" on
1
"2".
is equal to
5
""
10
6
""
10
7
""
10
8
""
10
9
""
10
10
""
10
the number line,
Which fraction is equal to
5
"
1"
0
4
""
10
!"
4
"
1"
0
3
""
10
1
"1
2"
on the number line above?
1
"2"
1
Remember: A fraction is equal to "2" if its numerator is half the denominator.
A fraction is less than
the denominator.
1
"2"
A fraction is more than
the denominator.
Compare. Write >, <, or ! for each
"
1. "
11
!
"1
2"
1
2. "2"
5
5. "9"
!
"1
2"
1"
5
6. "
27
!
!
"1
2"
1"
2
10. "
24
!
8
8
"
9. "
15
© Scott Foresman, Gr. 5
(233)
!
if its numerator is less than half
"1
2"
if its numerator is greater than half
!.
3
3. "7"
!
1
"2"
1
4. "2"
!
6"
"
10
"1
2"
1
7. "2"
!
"3
5"
3
8. "6"
!
"1
2"
"1
2"
9
11. "
1"
8
"2
9"
!
"1
2"
19
12. "
3"
8
!
"1
2"
Use with Chapter 7, Lesson 9.
Name _____________________________________________________________________________________________________
Relating Fractions to One Half
Compare. Write >, <, or ! for each
!
!
!
!
1
"2"
1
2. "3"
5
"
1"
1
5
6. "6"
5
1. "
1"
0
1
5. "2"
3
9. "6"
1
13. "2"
!
!
!
!
1
"2"
7
10. "8"
"3
5"
3
14. "
1"
6
!.
1
"2"
1
3. "2"
1
"2"
1
7. "2"
1
"2"
5
11. "7"
1
"2"
H 7-9
1
15. "2"
!
!
!
!
!
!
!
!
6
"
1"
2
8
4. "
1"
4
"4
5"
1
8. "2"
1
"2"
5
12. "8"
2
"7"
1
16. "2"
"1
2"
4
"7"
1
"2"
17
"
2"
0
1
2
17. To cover expenses for a school play, at least "" of the tickets must be sold. The box
office reports that "34" of the tickets have been sold. Is that enough to cover the
expenses? Explain.
18. Of 15 cast members, 11 came to the cast party. An article in the school
newspaper reported that less than half the cast attended the cast party. Was
that correct? Explain.
19.
Students sold 180 items of gift wrap, and 100 of them were gift bags.
Did gift bags account for more than or less than half of all the items that
were sold?
Test Prep Circle the correct letter for each answer.
1
20. Which of the following is greater than "2"?
A
2
"5"
B
3
"
1"
2
C
5
"9"
D
4
"
1"
0
H
6
3"
0
J
9
1"
6
1
21. Which of the following is less than "2"#?
F
3
"9"
© Scott Foresman, Gr. 5
G
(234)
1
0
"
"
12
Use with Chapter 7, Lesson 9.
Name _____________________________________________________________________________________________________
Problem-Solving Strategy
R 7-10
Make a List
Aaron, Barry, Celia, and Debbie are walking to school. Only 2 students at a time
can walk next to one another. Each of the 4 students can walk with each of the
others. How many different pairs are possible?
Understand
You need to find how many different pairs these
4 students can form.
Plan
You can make a list of all the possible ways to pair
the students.
Solve
Use the first initial of each student.
Start with Aaron.
AB
AC
Next list the combinations that begin
with Barry. Don’t list the combination
of Barry and Aaron since BA is the
same as AB and you have already
counted this combination.
BC
BD
List the combinations with Celia that
are not already listed.
CD
AD
All the possible combinations with Debbie are listed,
so the list is complete.
Count the combinations. Six different pairs are possible.
Look Back
Did you list all possible pairs?
Make a list to solve each problem.
1. Aaron is getting dressed. How many outfits can he make with a yellow shirt, a blue
shirt, blue jeans, and black pants? Use Y for yellow shirt, B for blue shirt, J for blue
jeans, and P for black pants.
2. Aaron, Barry, Celia, and Debbie must walk single file at one point of the path.
Aaron is always the leader, but Barry, Celia, and Debbie can follow in any order.
List the possible ways the 4 students can walk single file. Use A for Aaron, B for Barry,
C for Celia, and D for Debbie.
© Scott Foresman, Gr. 5
(236)
Use with Chapter 7, Lesson 10.
Name _____________________________________________________________________________________________________
Problem-Solving Strategy
H 7-10
Make a List
Make a list to solve each problem.
1. A rock band has 3 members: a guitarist, a drummer, and a bass guitarist. They are
taking publicity photos. In how many different ways can these 3 musicians stand
together in a row? Describe the order.
2. The musicians are choosing a costume to wear on stage. How many different outfits
can be made with a white shirt, a striped shirt, a blue shirt, black pants, and gray
pants? Describe the outfits.
3. Ria is making a poster to advertise a concert. She can use any two of these colors:
red, blue, yellow, or green. List all the different possible color combinations she
could use.
4. Three singers are each looking for a piano player to accompany them. There are
4 piano players available. How many different pairs of singer and piano player
are possible?
5. A string quartet has members that play these instruments: cello, bass, violin, and
viola. In how many different ways can these four musicians arrange themselves in
a row onstage? Use c for cello, b for bass, v for violin, and a for viola.
6. Three bands are in a competition. One band will finish in first place, one in second
place, and one in third place. Describe all the possible ways the bands can finish.
7. Yasu is making programs for the next band concert. He is using tan, gray, ivory, and
lemon-colored paper. The cover design will be printed in either black, maroon, or
purple ink. List all the different combinations he can make.
© Scott Foresman, Gr. 5
(237)
Use with Chapter 7, Lesson 10.
Name _____________________________________________________________________________________________________
Fractions Greater Than One
An improper fraction is a fraction
in which the numerator is greater
than or equal to the denominator.
4
!!
4
numerator
!!
denominator
Write
11
!!
5
11
!!
5
as a mixed number.
2!2! is a mixed number.
Write 3
to change an improper fraction
to a mixed number. Write the
remainder as a fraction in
simplest form.
2
5!1
"1
"
'10
!!
1
A mixed number is a number written
as a whole number and a fraction.
5
12
5
!! $ !!
5
5
12
!!
5
means 11 % 5. Use division
11
!!
5
R 7-11
5
!!
5
3
!!
8
as an improper fraction.
&
2
!!
5
$ 22!5!
&
First, multiply the denominator by the
whole number and add the numerator.
Then, write the sum over the
denominator.
24 ! 3 " 27
1
!!
5
3!3!
3!3!
8
3!3!
8
8
27
!!
8
8 # 3 " 24
Write each mixed number as an improper fraction.
Write each improper fraction as a mixed number.
3
18
1. 1!4!
2. !9!
15
!
3. !
10
!.
!1
!1
7
7
!
4. 7!
10
5. 2 !8!
8
6. !5!
Compare. Write ", #, or $ for each
12
!
7. !
12
15
!
10. !
16
5
!1
!1
© Scott Foresman, Gr. 5
8. !2!
14
11. !9!
(239)
3
9. !7!
9!
12. !
8
!1
!1
Use with Chapter 7, Lesson 11.
Name _____________________________________________________________________________________________________
Fractions Greater Than One
H 7-11
Write each mixed number as an improper fraction. Write each improper
fraction as a mixed number.
!
1. !
13
84
2. !4!
25
5. 7!8!
3
6. 3!7!
9. 1!4!
3
10. !8!
19
11. !9!
5
14. !9!
71
15. !3!
5
13. 2!7!
8
12
8. !5!
12. !4!
25
16. !6!
18. 1
!
10
!!
11
!
19. !
13
5
!1
22. 1
!
5
!!
4
23. 1
2
!1
26. !5!
!1
27. !1!
5
17
!.
!2
25. !3!
9
10
7
21. !6!
!
4. !
10
7. !5!
Compare. Write ", #, or $ for each
17. !5!
17
1
3. 5!5!
12
6
!1
!
8
!1
6
!1
7
!1
20. !8!
3
!!
2
24. !6!
!1
28. !1!
Test Prep Circle the correct letter for each answer.
5
!
25. Which improper fraction is equal to 7!
16 ?
112
80
!
A !
16
!
B !
16
35
117
!
C !
16
!
D !
16
114
26. Which mixed number is equal to !8!?
1
4
F 14!4!
© Scott Foresman, Gr. 5
G 11!8!
(240)
1
H 142!8!
1
!
J 14!
25
Use with Chapter 7, Lesson 11.
Name _____________________________________________________________________________________________________
Comparing and Ordering
Fractions and Mixed Numbers
R 7-12
To compare fractions and mixed numbers, write them with their least common
denominator (LCD). Then compare the numerators.
3
7
11
Order 1$4$, 1$8$, and 1$16$ from least to greatest.
Find the LCD of the fractions. The
LCD is the least common multiple.
Step 1
3
$$: multiples of 4: 4,
4
7
$$: multiples of 8: 8,
8
11
$$: multiples of 16:
16
8, 16 …
16…
16, 32:
The LCD is 16.
Write an equivalent fraction for each
fraction using the LCD as the denominator.
Step 2
Because 11 " 12,
Because 12 " 14,
11
1$1$
6
"
3
1$4$
"
7
1$8$.
11
$$
16
12
$$
16
"
"
Compare. Write !, ", or # for each
3
!
#
7
$$
8
#
3%4
$$
4%4
#
12
$$
16
7%2
14
$ $ # $$
8%2
16
11
11
$$ # $$
16
16
Compare the numerators of the new fractions.
Then think back to the original fractions.
Step 3
1. $7$
3
$$
4
6
$$
14
1
2. $7$
!
1
$$
8
12
$$,
16
14
$$,
16
!.
so
so
11
3
$ $ " $$ .
16
4
3
7
$$ " $$.
4
8
7
3. $8$
!
5
$$
6
7
$
4. $
12
!
13
$$
24
Order numbers from the least to greatest.
2 2 5
5 1 1
5. $5$, $7$, $8$
© Scott Foresman, Gr. 5
6. $9$, $2$, $3$
(242)
3 3
7
$
7. $8$, $4$, $
16
4 2 11
$
8. $5$, $3$, $
15
Use with Chapter 7, Lesson 12.
Name _____________________________________________________________________________________________________
Comparing and Ordering
Fractions and Mixed Numbers
Compare. Write >, <, or ! for each
4
9
1. "
2
3
4. "
!
!
!
7
"
18
2. "
17
"
32
5. "
2
13
7. "
1
2
!.
!
!
!
19
"
22
11
24
1
"
4
4
5
8. "
H 7-12
4
7
3. "
9
"
16
1
3
6. "
7
"
12
3
5
9. "
!
!
!
5
"
9
4
"
15
3
"
14
Order the numbers from least to greatest.
1 1 4 3
3 4 3 4
10. ", ", ", "
2
1
1
3
7
1
1
2
3
4
5
6
1
3
11. 2", 2"5", 1", 2"
8
8
4
1
5
12. 1", 1"5", 1", 1"
6
10
6
13. 4", 4", 4", 4"
14. In the opening scene, 17 of the 25 cast members are on stage. Is this more or less
than half the cast?
15. The first act is 35 minutes long. The whole show is 90 minutes long. What fraction of
the whole show is the first act? Write the fraction in simplest form.
16. The PTA sold juice during the intermission. Sixteen out of 24 orange juice cartons
were sold and 10 out of 12 cranberry juice cartons were sold. How does the fraction
of cranberry juice cartons sold compare with the fraction of orange juice cartons sold?
Test Prep Circle the correct letter for each answer.
17. Order the numbers from least to greatest.
3
1
7
9
1
3
9
7
7
3
1
9
", 5"2", 5"", 5"
5"
1"
0
10
10
3
7
1
9
3
7
1
9
"", 5"", 5"", 5 1
""
"", 51
""
"", 5"", 51
""
"", 51
""
", 5"" C 5"", 52
", 5"", 52
A 52
0 B 52
0, 5"
0 D 5"
0
10
10
10
10
10
10
10
10
18. Order the numbers from least to greatest.
1
2
1
F 7, 6"3", 6"5", 6"2"
© Scott Foresman, Gr. 5
(243)
1
1
2
G 6"2", 6"3", 6"5", 7
7, 62"5", 61"3", 61"2"
2
1
1
H 6"5", 6"3", 6"2", 7
1
2
1
J 6"3", 6"5", 6"2", 7
Use with Chapter 7, Lesson 12.
Name _____________________________________________________________________________________________________
Problem-Solving Application
R 7-13
Representing Remainders
Madeline has 15 lunch boxes to fill. She has 33 sandwiches. How many
sandwiches can she put in each lunch box?
Understand
You need to find the number of sandwiches Madeline will
pack in each lunch box.
Plan
You can divide to find how many sandwiches
are in each lunch box.
Solve
33 " 15 # 2 R3 or 33 " 15 # 2.2
1
3
! or 2!!
or
33 " 15 # 2!
5
15
There are 3 possible solutions. You must ask yourself
which makes more sense:
•
•
•
To pack 2 sandwiches per box with a remainder of 3?
To pack 2.2 sandwiches per box?
To pack 2!15! sandwiches per box?
In this case, it makes more sense to pack 2 sandwiches
per box with 3 sandwiches left over. So the solution is 2.
Did you divide correctly? Does your answer make sense?
Look Back
Madeline has 15 lunch boxes to transport. She has cartons that hold 10 lunch boxes
each. How many cartons does she need to transport all the lunch boxes?
1. How many lunch boxes does she have to transport?
2. How many lunch boxes does each carton hold?
3. Which of the following solutions best represents the number of cartons Madeline needs?
1
2
1
2
a. She needs 1!! cartons because 15 " 10 # 1!.
b. She needs 2 cartons to pack all 15 lunch boxes, even though she will have
5 extra spaces in 1 carton.
c.
She needs 1 carton. She can pack 10 lunch boxes and have 5 lunch boxes left over.
© Scott Foresman, Gr. 5
(245)
Use with Chapter 7, Lesson 13.
Name _____________________________________________________________________________________________________
Problem-Solving Application
H 7-13
Representing Remainders
For each problem, decide how to express the remainder. Then, in the Code Box, find
the letter that goes with the number from that answer. Write the letter on the blank next
to the exercise. Read down to learn what the mystery item is that goes through a door
all the time but never goes in or out.
1. Neil made 50 bread rolls. He packed 3 bread rolls per package. How many
full packages of bread rolls did he pack?
2. Paulette has 35 apples to ship. She has cartons that hold 1 dozen apples
each. How many cartons does she need to ship all the apples?
3. The cost of 16 muffins is $24. What is the cost of 1 muffin?
4. A carton of 1 dozen eggs weighs 24 ounces. The empty carton weighs
3 ounces. How much does 1 egg weigh?
5. Tammy made 75 peanut butter and jelly sandwiches. She used 60
ounces of peanut butter. How many ounces of peanut butter were on
each sandwich?
6. Al packed 600 peaches into boxes. Each box held 48 peaches.
How many boxes did he use?
7. One dozen tuna pies cost $15. What is the cost of each tuna pie?
Code Box
0.5 0.6
A
R
0.7 0.8
N
O
0.9
1
2
3
4
12
13
14
15
16
17
I
B
T
E
J
M
L
P
S
K
D
$1.00 $1.25 $1.50 $1.75 $2.00
C
E
© Scott Foresman, Gr. 5
Y
(246)
F
N
1.25
1.5
1.75
2.25
2.5
X
Q
H
U
G
Use with Chapter 7, Lesson 13.