LESSON 9
Multiplying Fractions
or a Fraction of a Fraction;
Math
LESSON 9
Multiplying Fractions
or a Fraction of a Fraction; Mental Math
We’ve worked on a fraction of one and a fraction of a number; now we’ll tackle
a fraction of a fraction. This kind of problem can be the hardest to understand
or think through, but it is the easiest to do using a formula. Find 2/5 of 1/3.
Notice that we don't say, “two-fifths times one-third.” Even though it is the same
as multiplying two fractions, I want to relate it to a fraction of a number and a
fraction of one. To stress this relationship, we read it “two-fifths of one-third”. This
is also the language used in most word problems. Be sure to read the note about
word problems on lesson practice 9A in the student text.
Example 1
Find
2
1
of
(two-fifths of one-third).
5
3
1
.
3
Step 1
Start with
Step 2
Divide into 5 equal parts.
EPSILON
MULTIPLYING FRACTIONS - LESSON 9
41
Step 3Count 2 of those parts.
Pull up the pink insert to do this.
So
2
1
2
of
is
.
5
3
15
Example 2
Find
2
5
of
(two-thirds of five-sixths).
3
6
5
.
6
Step 1
Start with
Step 2
Divide into 3 equal parts.
Step 3Count 2 of those parts.
Pull up the violet insert to do this.
42
So
2
5
of
3
6
is 10 .
18
LESSON 9 - MULTIPLYING FRACTIONS
EPSILON
Another way to understand the formula for a fraction of a fraction (multiplying
fractions) is shown in figure 1, which reveals the finished answer to example 2. To
make the “factors” clearer I placed a white piece over the answer so you can see just
the two factors of the two rectangles. I hope this helps you “see” numerator times
numerator, divided by denominator times denominator.
Figure 1
5 shaded
6 # of parts
2 shaded
3 # of parts
2 5 = 10
×
3 6 = 18
The whole square is an illustration of 2/3 x 5/6, which is a multiplication
problem. In this problem we can see the two smaller problems. The numerator
problem has the factors 2 x 5. The denominator problem has the factors 3 x 6. This
exercise reveals the formula for multiplying two fractions.
2 5
2 × 5 = 10
×
=
3 6
3 × 6 = 18
Numberator times numerator (shaded rec tan gle)
Deno min ator times deno min ator (total # of parts)
EPSILON
MULTIPLYING FRACTIONS - LESSON 9
43
Mental math
Here are some more mental math problems for you to read aloud to your student.
Try a few at a time, going slowly at first.
1. Four plus five, times two, divided by three, equals ? (6)
2. Thirty-six divided by four, minus two, times five, equals ? (35)
3. Sixty-six divided by six, plus four, divided by three, equals ? (5)
4. Forty-eight divided by eight, times six, plus two, equals ? (38)
5. Six plus seven, minus four, times nine, equals ? (81)
6. Nineteen minus three, divided by four, times seven, equals ? (28)
7. Nine times five, plus three, divided by six, equals ? (8)
8. Twenty-one divided by seven, plus one, times five, equals ? (20)
9. Three times four, divided by two, times nine, equals ? (54)
44
10. Seven times six, plus two, divided by 11, equals ? (4)
LESSON 9 - MULTIPLYING FRACTIONS
EPSILON
lesson practice
Find a fraction of a fraction. These can all be built with the overlays. The first one is
done for you.
2
3
2x3
EPSILON
2 of 3 =
1. 4
6
3 of 3 =
5
4
2 of 3 =
1 of 1 =
4
6
3
2
3 of 3 =
4
2
5 of 4 =
6
5
2 of 1
3 =
1
of 1 =
3. 2
4 of 2
6 =
3
3
4
3
3
4 of 2 =
1 of
5 =
5
6 of 4
5
5
6
2
1
1
2 of
of 23 =
==
44 of
3
2
46 =
5
6
43 of
23 =
1
5
of
of
1
1 =
65 of 6
54 =
5
62
33
2
41 of
21 =
of 1
=
of
=
5. 14
26 =
of
3
2
3
4
5
6
23
63
14 of
5
of
of 212 ==
=
of
35 x
6
5=
5
6
6
34
51
51
4
12 of
=
of 221 =
=
33 of
4 =
53 x 62
2
64
32
11 of
3
of 215
=
of
==
26 x
15 =
=
65 x 536
5
42
51
14 of
222 ===
of
of
7. 3
x
33
46 ==
2
65 x 364
41
65
3
2
21 x
1 1 ==
of
of
=
x
25
16
x
6
3=
5
5
4
54
22
31 xof
42 ==
of 2
=
42
16
5
6
x
=
6
3
4
6
61
31
23
1
of 2 =
==
xx
6
5
4
553
5
2
1 of 1
22 ==
43
9. 3
xx 4 =
=
6
42
636
6
3
2
2 x
1 =
23
xx 1 ==
5
5
54
25
2
3 x
4 =
43
xx 1 ==
6
3
64
36
2
1 =
2 x
x 1
=
4
5
5
2
3
4 =
4 x
x 1
=
4
6
6
3
2 x 1 =
5
Lesson Practice
9A2
4 x 1 =
6
3
2x3
4x6
= 6
24 2x3
4x6
=
6
24
2x3 = 6
4 x 6 24
2x3
4x6
2x3
4 x 6
=
=
2.
4.
6
24
6
24
6.
8.
4
3
25
41
33
54
1
26
3
42
43
3
61
5
2
2
15
4
34
5
1
43
1
5
65
1
46
2
1
53
3
4
12
6
63
5
2
15
4
3
2
53
6
3
11
2
5
65
4
3
14
3
65
2
1
24
3
2
46
5
1
35
4
42
6
6
23
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
of
x
of
of
x
of
x
of
x
xof
of
x
xof
x
x
xof
x
xx
5
45
43 xx
3
6
46
22 xx
54
43 xx
64
2 x
5
4 x
6
=
6
3 =
34 =
61 =
32 =
42 =
1
35 =
=
2
61 =
24 =
3
55 =
4
163 =
==
2 =
46
2
6
5
23
==
1 =
6
54
3
21
1
=
=
2
62 =
4
2 =
516
=
2
35 =
6
5
21 ==
24 =
6
3
215 ===
16 =
53
25
22 ===
46 =
36
161 ==
2
513 =
422 ==
2
616 =
3
3
12
==
2
55
1
42 =
=
=
3
63
11 ==
25
14 ==
36
1 =
2
1 =
3
4x6
9A
=
6
24
2x3
4x6
=
6
24
2x3
4x6
=
6
24
2x3
4x6
=
6
24
101
4 of
2 =
1
4 of
2 =
of 1
=
of
=
of 6
=
5
3
4
3
4
6
5
3
2
6
5
4
2
1
5
1 of
of 5
=
2
1 =
4 of
2
1 =
of 2
=
of
5
6 =
5
6
5
6
3
4
6
5
3
4
1 of 1
4 9A
2
4
2 =
LESSON PRACTICe
1
5
2
1
1
5
of
=
of 3
=
of
=
of
=
6
5
6
5
6
5
6
3
4
5
6
1
2
1 of 51
1 of
1
=
4
2 =
1
4
2 =
of
of
=(Find a fraction of a fraction.) These
=
of
= with
2
6
Multiply56theof
fractions.
can
all
be
built
3
6
3
6
6
5
6
22 =
3
the overlays.
1
1
1 of 2
1
4
2 =
1 x
1 =
=
of
=
of
=
of
5
5
2
6
2
6
6
3
5
6
6
3
3 x 2
2 =
3
21 =
3
1 x
2 ==
10. 1 x
11. 6
of 2
=
of
3
5
56
5
5
2
6
3
2
6
2 x 2
1 =
3
22 =
3
3
2
1 x
3
2
x
x 5 =
=
=
4 x
6 of 5
36 =
6
3
5
2
5
5
3
4 =
2 x 2
1 =
2 x
1
3
3
2
x
x 2 =
=
x
=
4
6 =
4
5
4 x 5
6
3
5
5
6
3
2
1
3
4 =
3
4 =
2 x 2
1
3
2 x
1
x 2
=
x 6 =
x
5
12. 4
13.
4
6 =
4
5
6
3
4
5
4 x 1
1 =
2
1
2
3
4 =
2 x
3 x
4 =
x 1
=
=
x
=
6
3
5
2
5
2
6
4
5
4
6
4 x 1
1 =
4 x 1
1 =
2
3
2
x 4 =
=
x 3 =
6
3
6
5
2
4
6
5
2
4
1
2 x
4 x 1 =
=
14. 5
15.
6
3
2
6
3
4 x 1 =
6
3
Always read fraction word problems carefully and think about what is happening.
Multiplication problems involving fractions usually use the word “of,” but you should not
assume that the word “of ” somewhere in the problem automatically indicates multiplication. Read for meaning, don’t just look for key words. All of the examples on lesson
practice A, B, and C are multiplication of fractions.
16.A recipe calls for 1/3 of a cup of butter. If Matthew is making 1/4 of the
recipe, how much butter should he use?
17.One-half of the people at the picnic went home sick, but only one-fifth
of them were seriously ill. What part of the whole group was seriously ill?
18.Mom left 1/3 of a pie for Chucky to eat, but since he was not very hungry,
he ate only 1/5 of what was there. What part of a whole pie did Chucky eat?
102
EPSILON
9D
systematic review
Multiply (fraction of a fraction).
2
5
5
6
5
9
1
7
2
2
3
5
1
5
2
6
4
5
6
9
2
1
3
7
1
2
9
3
2
1
7
2
4
6
2
3
1
9
2
7
of
1 =
5
1 =
4
1 =
2
1 =
3
1 =
71 =
25 + 7
51 = 10
4
1
1 =
7
2
5
1
8 =
32
1
11 =
7
=
2 + 7
5
10
= 2 =
1 18
7
5
8
2
11
2 of 1 =
x
2.
5
5
5 x 1 =
x
2 of
1
6
4 =
5
5
5 x 1 =
+
5 x 1 =
9
2
6
4
1 + 1 =
−
5 x 1 =
3
3. 7
of
9
2
2 − 1 =
+
1
1
2 + 71 =
3
x
7 of 3 =
5
1 + 25 + 7 =
2 − 1 =
5 x 511 = 10
2
x
3 of
7 =
6
4
4
15
5
1 + 2 + 7 =
5
1
Add or subtract.
6
7
5
+
2 x
51 == 10
x
9
2
2
5
6
4
4
1
1 + 1
3
8
5
1
−
6 x 7 ==
4. 7
5.
3
1
2
9
2
=
2
5
2
1
9
11 =
1
1
−
+
3 + 8 =
3
2 = 7
7
32
=
=
1
2
7 =
1
7
28
2
1
+
+
9
11 = 10
− 5
2
4
3
=7 2 =
=
2
=
=
=
4
1
1 + 2 18
28
+ 7 =
6. 7
6
7
2
5 2 10
4
=51
=
=
2
4
18
3
8
6
7
1
2
52
=
=
9
11
3
8
2
1 =of four
2 to make
=
=
= 2 =
Use the rule
denominators
the same, then compare the fractions.
7
28
18
9
11
2
4
2
= 1 ==
=
4
2
5
7. 7 =
8.
18 =
28
6
7
3
8
4
= 2 =
=
18
1.
9.
1
9
=
=
=
=
28
4
=
28
4
=
2
11
EPSILON systematic review 9D
107
+
+
3
7
2
5
10
1 + 2 + 7 =
4
1
2
5
10
6
7
4
1
2
5
Systematic review 9D
6
7
3
8
2
5
1
2
8 numbers in the numerators or denominators
Fill in the3 missing
to
make
9
11
1 fractions.
2
2 =
equivalent
=
9
11
7
2 =
=
=
= 2 =
10.
11.
28
7
18
2
4
=
=
=
18
=
=
28
4
=
Quick Review
Here is a chance to review multiplying a two-digit number by a three-digit number.
The first one is done for you.
Estimate, then multiply to find the exact answer.
12. 6 2 3
x 4 5
1 1 1 3005
1
2482
2 8,0 3 5
600
13. 1 7 9
x 50
× 5 7
3 0, 0 0 0
(This is 5 tens x
6 hundreds.)
14. 9 0 2
× 1 1
15.Mom is making 1/3 of a recipe that calls for 2/3 of a cup of flour. How much
flour should she use?
16.If there are 365 days in one year, how many days are there in 25 years? (Don’t
worry about leap years.)
17.Nancy needs 1/2 cup of honey for bread, 1/3 cup for cookies, and 1/4 cup for
fruit salad. How much honey does she need altogether?
18.Ivan spotted 36 birds Saturday morning at Hawk Mountain. Three-fourths
of them were hawks. How many hawks did he see?
108
EPSILON
9
test
Multiply (fraction of a fraction).
1
1
of
=
4
3
1.
3.
2
1
×
=
7
4
5.
3
1
×
=
5
6
2.
3
2
×
=
4
5
4.
2
3
of
=
5
7
6.
2
1
×
=
4
3
8.
4
1
−
=
5
6
Add or subtract.
7.
3
1
+
=
11 4
9.
1
2
1
+
+
=
7
3
2
EPSILON Test 9
23
test 9
Use the rule of four to make denominators the same, then compare the fractions.
10.
2
3
5
8
11.
3
4
9
12
12.
5
6
7
9
Estimate, then multiply to find the exact answer.
13.
612
× 54
14.
15.
124
× 36
957
× 13
16.One-third of the customers at the ice cream store bought vanilla and two-fifths
of them bought chocolate. What part of the customers bought vanilla
or chocolate?
17.What is the perimeter of a triangle whose sides measure 10 feet, 12 feet,
and 12 feet?
18.Three-eighths of the guests at the picnic ate hamburgers. One-half of those
people had mustard on their hamburgers. What part of the people at the picnic
had hamburgers with mustard?
19.If there were 48 people at the picnic (#18), how many had hamburgers? How
many had hamburgers with mustard?
20.Each of the 48 people at the picnic contributed $15 for food and other
expenses. How much money was collected?
24
epsilon
15.
16.
7.
S ystema1tic
18.
19.
+
+
=
10
10 10 10
10 − 8 = 2 of his allowance
10 10
10
90
100ew 8F - LE SSON PRACTICE 9 C
Revi
5 + 3 + 5 + 3 = 16 mi
28 ÷ 7 = 4; 4x1 = 4 days
25 ÷ 4 = 6 1 chocolates
4
Remember, equivalent answers to
addition problems are correct.
20.
Lesson
Practice9B
9B
Lesson Practice
Practice
9B
Lesson
1.
1.
2.
2.
3.
3.
Lesson Practice
Lesson
Practice9A
9A
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
done
3x3 = 9
5x4
20
1x1 = 1
3x2
6
4x2 = 8
6x5
30
2x1 = 2
3x4
12
1x5 = 5
5x6
30
4x2 = 8
5x6
30
1x1 = 1
6x3
18
1x2 = 2
2x6
12
3x2 = 6
5 5
25
3 x2 = 6
6 3
18
2x 1 = 2
4 5
20
3 x 4 = 12
4 6
24
2x 1 = 2
5 2
10
4x 1 = 4
6 3
18
1 x 1 = 1 of a cu
up
3 4
12
1 x 1 = 1 of the group
2 5
10
1 x 1 = 1 of a pie
3 5
15
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
0..
10
1
11.
11.
12.
12.
13.
13.
14.
14.
15.
15.
16.
16.
17.
17.
18.
18.
Lesson Practice
Lesson
Practice9C
9C
1.
2.
3.
4.
5.
132
sol uti ons
2
6
x3
2x
3 =
6
= 30
x
6
5
6x5
30
x1
2x
1 = 2
2
2
x2
3x
2 = 6
6
3
x2
3x
2 = 6
6
3
= 32
8
x4
8x
4
32
1
1
x1
1x
1 =
1
= 20
x
4
5
4x5
20
1
2
x2
1x
2 =
2
= 18
x
6
3
6x3
18
4
1
x4
1x
4 =
= 4
x
15
3
5
3x5
15
x3
4x
3 = 12
12
4
x6
5x
6 = 30
30
5
x5
1x
5 = 5
5
1
x6
2x
6 = 12
12
2
x3
3x
3 = 9
9
3
= 16
x
4
4
4x4
16
2x 4
4 = 8
8
2
x 6 = 30
5
5 6
30
1x 1
1 = 1
1
1
x 2 = 14
7
7 2
14
3x 5
5 = 15
15
3
5x 6
6 = 30
30
5
3 x 1
1 = 3
3
3
10 x 4
4 = 40
40
10
1x 1
1 = 1
1
1
x2 =
4
8
4 2
8
7 x 1
1 = 7
7
7
x 9 = 72
8
8 9
72
3 x 1
1 = 3
3
3
of the
the class
class
x 4 = 16 of
4
4 4
16
2
3
6
2x
6 of
x3 =
= 15
of a
a bushel
bushel
3
3 5
5
15
1
1
1
1 of
1x
= 25
of the
the games
games
x 1 =
5
5
25
5 5
6.
7.
8.
9.
3x2
6x5
2x2
9x3
2x1
6x3
3x4
4x5
1x3
7x4
1x1
6x6
4x1
8x3
3x3
5x4
4x2
6x4
= 6
30
= 4
27
= 2
18
= 12
20
= 3
28
= 1
36
= 4
24
= 9
20
= 8
24
Epsilon
3 + 7 = 10
21 21
21
14
3
5.
−
= 11
21 21
21
5
4
16 or 1 6
6. P r a+c t i c e+ 97
L e sson
C - =S Y S TEMATIC
REVIEW 9E
10 10 10
10
10
28 > 6
7.
42
42
16
15
8.
>
24
24
11
18
9.
<
99
99
2
4
=
= 6 = 8
10.
7
14
21
28
1
2
3
=
=
= 4
11.
9
18
27
36
12. done
4.
Lesson Practice 9C
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
3x2 = 6
6x5
30
2x2 = 4
9x3
27
2x1 = 2
6x3
18
3x4 = 12
4x5
20
1x3 = 3
7x4
28
1x1 = 1
36
6x6
4x1 = 4
8x3
24
3x3 = 9
5x4
20
4x2 = 8
6x4
24
1x2 = 2
6x5
30
5 x 1 = 5
9 4
36
1x1 = 1
10x10
100
2x3 = 6
3x6
18
3x1 = 3
5x2
10
1x3 = 3
8x5
40
4 x 2 = 8 mi
5 4
20
2 x 1 = 2 of a cup
3 3
9
7 x 1 = 7 of a pie
8 7
56
13.
179 × 57 = 10,203
14.
(900) × (10) = (9,000)
15.
902 × 11 = 9,922
1 × 2 = 2 of a cup
3 3
9
365 × 25 = 9,125 days
16.
17.
18.
1.
2.
3.
4.
5.
6.
7.
8.
Epsilon
9.
10.
11.
12.
2 × 1 = 2
5 5
25
5 × 1 = 5
6
4
24
5 × 1 = 5
9
2
18
3 + 7 = 10
21 21
21
14 − 3 = 11
21 21
21
5 + 4 + 7 = 16 or 1 6
10 10 10
10
10
28 > 6
42
42
16 > 15
24
24
11 < 18
99
99
2 = 4 = 6 = 8
7
14
21
28
1 = 2 = 3 = 4
9
18
27
36
done
3 + 2 = 5
6
6
6
5 + 1 = 20 + 6 = 26
6
4
24
24
24
2
of 1
cups
24
36 ÷ 4 = 9
9 × 3 = 27 hawks
Systematic Review
Systematic
Review9E
9E
1.
2.
SystematicReview
Systematic
Review9D
9D
(200) × (60) = (12,000)
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
1x2 = 2
2 3
6
7 x 2 = 14
8 5
40
1x3 = 3
3 5
15
27 + 4 = 31
36
36
36
10 − 6 = 4
15 15
15
3 + 5 = 8 =1
8
8
8
2
2
=1
or 1 1
1+
4
8
8
9 = 9
27
27
16 > 15
40
40
50 > 49
70
70
(100)x(50) = (5, 000)
125x51 = 6,375
(300)x(40) = (12,sol
000u
) t i ons
254x35 = 8,890
(600)x(30) = (18, 000)
563x26 = 14, 638
107 ÷ 6 = 17 5
6
3
133
9 = 9
27
27
16
8.
> 15
40
40
50
49
9.
>
S ystema tic Revi
70 ew709E - LE SSON PRACTICE 1 0 B
10. (100)x(50) = (5, 000)
125x51 = 6,375
11. (300)x(40) = (12, 000)
254x35 = 8,890
12. (600)x(30) = (18, 000)
563x26 = 14, 638
7.
13.
14.
15.
16.
17.
18.
19.
20.
107 ÷ 6 = 17 5
6
395 ÷ 8 = 49 3
8
459 ÷ 2 = 229 1
2
1 x 4 = 4 of the chores
2 5
10
5 + 6 = 11
30
30
30
11 + 1 = 44 + 30 = 74
30
4
120 120
120
of the job
250x51 = 12,750 in
12 + 17 + 28 = 57 in
235 ÷ 5 = 47; 47x3=141
jelly beans
Systematic Review
Systematic
Review9F
9F
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
134
13.
14.
15.
16.
2x2 = 4
4 5
20
1x 2 = 2
3 6
18
1x 4 = 4
2 9
18
10 + 3 = 13
15 15
15
16 − 15 = 1
40
40
40
5 + 12 = 17
20
20
20
17 + 2 = 51 + 40 = 91 or 1 31
20
3
60
60
60
60
40 > 24
48
48
27 > 20
45
45
24 = 24
32
32
(600)x(60) = (36, 000)
558x62 = 34,596
(400)x(80) = (32, 000)
407x83 = 33,781
(300)x(10) = (3,000)
349x12 = 4,188
sol uti ons
128 ÷ 7 = 18 2
7
471÷ 3 = 157
298 ÷ 5 = 59 3
5
3 x 1 = 3 ft
1 4
4
8.
8.
9.
9.
10.
10.
11.
11.
12.
12.
13.
13.
14.
14.
15.
15.
16.
16.
17.
17.
18.
18.
19.
19.
20.
20.
27 >
> 20
45
45
45
45
24
24
24 =
24
= 32
32
32
32
((600
600))x
x((60
60)) =
= ((36
36,, 000
000))
558
x
62
=
34
,
596
558x62 = 34,596
((400
400))x
x((80
80)) =
= ((32
32,, 000
000))
407
x
83
=
33
,
781
407x83 = 33,781
300))x
x((10
10)) =
= ((3
3,,000
000))
((300
349
49x
x12
12 =
=4
4,,188
188
3
2
128 ÷
÷7
7=
= 18
18 2
128
7
7
471÷
3=
157
÷3
= 157
471
3
÷5
= 59
8÷
5=
59 3
298
29
5
5
3x 1
1 = 3
3 ft
3
1x 4
4 = 4
4 ft
1
4 − 1
1 = 12
12 − 5
5 = 7
7 mile
4
− 3 =
− 15 =
mile
5
15
15
5
3
15 15
15
1
1
1
1x
1 ;; 1
1 of
= 6
of 12
12 =
=2
2 people
people
x 1 =
3
6
3 2
2
6
6
13
13 +
+1
19
9+
+ 26
26 =
= 58
58 in
in
30
÷
2
=
15
30 ÷ 2 = 15
15
15x
x1
1=
= 15
15 days
days
Systematic
Review
10A
Lesson
Practice
10A
1.
2.
3.
4.
5.
6.
7.
8.
done
done
15 ÷ 16
24
24
12 ÷ 8
16 16
4 ÷ 2 =
8
8
1 ÷ 1 =
2 8
2 ÷ 1 =
3
6
6 ÷ 1 =
8
4
= 15 ÷ 16 = 15
1
16
÷
12
8
12
=
=
or 1 4
8
1
8
4÷2 = 2
1
8 ÷ 2 = 8 ÷ 2 = 4 times
16 16
1
12 ÷ 3 = 12 ÷ 3 = 4 pieces
18 18
1
24 ÷ 8 = 24 ÷ 8 = 3 people
32 32
1
Lesson Practice10B
Lesson
Practice 10B
1.
2.
3.
4.
5.
6.
7.
8.
12 ÷ 5
20
20
18 ÷ 4
24
24
3 ÷ 4
12 12
16 ÷ 15
24
24
5 ÷ 16
20
20
3 ÷ 1 =
4
8
9 ÷ 1
10 10
5 ÷ 1
16 16
= 12 ÷ 5 = 12 or 2 2
1
5
5
÷
18
4
18
=
=
or 4 2
1
4
4
÷
3
4
3
=
=
1
4
÷
16
15
=
= 16 or 1 1
15
1
15
÷
5
16
5
=
=
1
16
24 ÷ 4 = 24 ÷ 4 = 6 time
es
32 32
1
Epsilon
= 9 ÷ 1 = 9 volunteers
1
5
= ÷ 1 = 5 times
1
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
52 + 52 + 52 + 52 = 208 in
90
50
600
500
358 ÷ 5 = 71 3
5
541÷ 8 = 67 5
8
3
189 ÷ 6 = 31
6
67x18 = 1,206
34x26 = 884
224x22 = 4,928
2 + 1 = 6 + 5 = 11 of her money
5 3
15 15
15
15 > 12 so 5 > 2
6
18
18
3
Drumore received more snow.
15 − 12 = 3 ft difference
18
18 18
10 + 11+ 10 + 11 = 42 ft
1 of 42 = 6 ft of shelves
7
Test 9
Test
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11..
12.
13.
1 of 1 = 1
3
4
12
3x2 = 6
4 5
20
2x 1 = 2
7 4
28
2 of 3 = 6
7
5
35
3x 1 = 3
5 6
30
2x 1 = 2
4 3
12
12 + 11 = 23
44
44
44
24 − 5 = 19
30
30
30
3 + 14 = 17
21 21
21
17 + 1 = 34 + 21 = 55 or 1 13
21 2
42
42
42
42
16 > 15
24
24
36 = 36
48
48
45 > 42
54
54
(600) x (50) = (30,000)
612x54 = 33, 048
14.
16.
17.
18.
11..
12.
13.
14.
(100) x (40) = (4,000)
124x36 = 4, 464
15.
16.
17.
18.
19.
20.
(1000) x (10) = (10,000)
957x13 = 12, 441
1 + 2 = 5 + 6 = 11 of the customers
3 5
15 15
15
10 + 12 + 12 = 34 ft
3 x 1 = 3 of the people
8 2
16
3 of 48:
8
48 ÷ 8 = 6; 6x3 = 18 had burgers
3 of 48:
16
48 ÷ 16 = 3; 3x3 = 9 had mustard
48x15 = $720
Test10
Test
10
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
124x36 = 4, 464
17.
957x13 = 12, 441
1 + 2 = 5 + 6 = 11 of the customers
3 5
15 15
15
10 + 12 + 12 = 34 ft
3 x 1 = 3 of the people
18.
(1000) x (10) = (10,000)
16 > 15
24
24
36 = 36
48
48
45 > 42
U n i t T e s t I - TEST 10
54
54
(600) x (50) = (30,000)
612x54 = 33, 048
(100) x (40) = (4,000)
Epsilon
15.
10.
19.
2 ÷ 1 = 2÷1 = 2
5 5
1
15 ÷ 16 = 15 ÷ 16 = 15
24
24
1
16
4 ÷ 2 = 4÷2 = 2
8
8
1
3 ÷ 4 = 3÷ 4 = 3
12 12
1
4
27 ÷ 10 = 27 ÷ 10 = 27 or 2 7
45
45
10
1
10
12 ÷ 10 = 12 ÷ 10 = 12 or 1 2
10
15 15
1
10
3x 1 = 3
4 4
16
2x 1 = 2
3 5
15
2x 1 = 2
9 2
18
32 + 21 = 53
56 56
56
20 − 9 = 11
36
36
36
4 + 3 + 5 = 12 or 1 2
10 10 10
10
10
(500) ÷ (40) ≈ (10)
(900) ÷ (30) = (30)
(600) ÷ (60) = (10)
1 ÷ 1 = 12 ÷ 2 = 12 ÷ 2 = 6 pieces
2 12
24
24
1
7 ÷ 1 = 7 ÷ 1 = 7 pieces
199
solutions
8
8
1
3 + 1 = 4
10 10
10
4 + 3 = 20 + 30 = 50 = 1 mi
10 5
50
50
50
7 − 2 = 5 or 20 of a pizza
8
8
8
32