4.5. MULTIPLYING AND DIVIDING MIXED FRACTIONS
4.5
291
Multiplying and Dividing Mixed Fractions
We begin with definitions of proper and improper fractions.
Proper and Improper Fractions. A proper fraction is a fraction whose
numerator is smaller than its denominator. An improper fraction is a fraction
whose numerator is larger than its denominator.
For example,
23
2
, − ,
3
39
and
119
127
are all examples of proper fractions. On the other hand,
4
317
, −
,
3
123
and
−
233
101
are all examples of improper fractions.
A mixed fraction 1 is part whole number, part fraction.
Mixed Fractions. The number
3
4
is called a mixed fraction. It is defined to mean
5
5
3
3
=5+ .
4
4
In the mixed fraction 5 34 , the 5 is the whole number part and the 3/4 is the
fractional part.
Changing Mixed Fractions to Improper Fractions
We have all the tools required to change a mixed fraction into an improper
fraction. We begin with an example.
You Try It!
EXAMPLE 1. Change the mixed fraction 4 87 into an improper fraction.
1A
mixed fractions is sometimes called a mixed number.
Change 5 34 to an improper
fraction.
292
CHAPTER 4. FRACTIONS
Solution. We employ the definition of a mixed fraction, make an equivalent
fraction for the whole number part, then add.
4
Answer: 23/4
7
7
=4+
8
8
4·8 7
=
+
8
8
4·8+7
=
8
39
=
8
By definition.
Equivalent fraction with LCD = 8.
Add numerators over common denominator.
Simplify the numerator.
Thus, 4 87 is equal to 39/8.
There is a quick technique you can use to change a mixed fraction into an
improper fraction.
Quick Way to Change a Mixed Fraction to an Improper Fraction. To
change a mixed fraction to an improper fraction, multiply the whole number
part by the denominator, add the numerator, then place the result over the
denominator.
Thus, to quickly change 4 87 to an improper fraction, multiply the whole number
4 by the denominator 8, add the numerator 7, then place the result over the
denominator. In symbols, this would look like this:
4
4·8+7
7
=
.
8
8
This is precisely what the third step in Example 1 looks like; we’re just eliminating a lot of the work.
You Try It!
Change 7 83 to an improper
fraction.
EXAMPLE 2. Change 4 23 to an improper fraction.
Solution. Take 4 23 , multiply the whole number part by the denominator, add
the numerator, then put the result over the denominator.
4
2
4·3+2
=
3
3
Thus, the result is
4
14
2
=
.
3
3
Answer: 59/8
4.5. MULTIPLYING AND DIVIDING MIXED FRACTIONS
293
It is very easy to do the intermediate step in Example 2 mentally, allowing
you to skip the intermediate step and go directly from the mixed fraction to
the improper fraction without writing down a single bit of work.
You Try It!
EXAMPLE 3. Without writing down any work, use mental arithmetic to
change −2 53 to an improper fraction.
5
Change −3 12
to an improper
fraction.
Solution. To change −2 53 to an improper fraction, ignore the minus sign,
proceed as before, then prefix the minus sign to the resulting improper fraction.
So, multiply 5 times 2 and add 3. Put the result 13 over the denominator 5,
then prefix the resulting improper fraction with a minus sign. That is,
−2
3
13
=− .
5
5
Answer: −41/12
Changing Improper Fractions to Mixed Fractions
The first step in changing the improper fraction 27/5 to a mixed fraction is to
write the improper fraction as a sum.
27
25 2
=
+
5
5
5
(4.1)
Simplifying equation 4.1, we get
27
2
=5+
5
5
2
=5 .
5
Comment. You can’t just choose any sum. The sum used in equation 4.1
is constructed so that the first fraction will equal a whole number and the
second fraction is proper. Any other sum will fail to produce the correct mixed
fraction. For example, the sum
27
23 4
=
+
5
5
5
is useless, because 23/5 is not a whole number. Likewise, the sum
27
20 7
=
+
5
5
5
is no good. Even though 20/5 = 4 is a whole number, the second fraction 7/5
is still improper.
294
CHAPTER 4. FRACTIONS
You Try It!
Change 25/7 to a mixed
fraction.
EXAMPLE 4. Change 25/9 to a mixed fraction.
Solution. Break 25/9 into the appropriate sum.
25
18 7
=
+
9
9
9
7
=2+
9
7
=2
9
Answer: 3 74
Comment. A pattern is emerging.
• In the case of 27/5, note that 27 divided by 5 is equal to 5 with a remainder of 2. Compare this with the mixed fraction result: 27/5 = 5 52 .
• In the case of Example 4, note that 25 divided by 9 is 2 with a remainder
of 7. Compare this with the mixed fraction result: 25/9 = 2 79 .
These observations motivate the following technique.
Quick Way to Change an Improper Fraction to a Mixed Fraction.
To change an improper fraction to a mixed fraction, divide the numerator by
the denominator. The quotient will be the whole number part of the mixed
fraction. If you place the remainder over the denominator, this will be the
fractional part of the mixed fraction.
You Try It!
Change 38/9 to a mixed
fraction.
EXAMPLE 5. Change 37/8 to a mixed fraction.
Solution. 37 divided by 8 is 4, with a remainder of 5. That is:
4
8)37
32
5
The quotient becomes the whole number part and we put the remainder over
the divisor. Thus,
37
5
=4 .
8
8
4.5. MULTIPLYING AND DIVIDING MIXED FRACTIONS
295
Note: You can check your result with the “Quick Way to Change a Mixed
Fraction to an Improper Fraction.” 8 times 4 plus 5 is 37. Put this over 8 to
get 37/8.
Answer: 4 92
You Try It!
EXAMPLE 6. Change −43/5 to a mixed fraction.
Solution. Ignore the minus sign and proceed in the same manner as in
Example 5. 43 divided by 5 is 8, with a remainder of 3.
Change −27/8 to a mixed
fraction.
8
5)43
40
3
The quotient is the whole number part, then we put the remainder over the
divisor. Finally, prefix the minus sign.
−
3
43
= −8 .
5
5
Answer: −3 83
Multiplying and Dividing Mixed Fractions
You have all the tools needed to multiply and divide mixed fractions. First,
change the mixed fractions to improper fractions, then multiply or divide as
you did in previous sections.
You Try It!
EXAMPLE 7. Simplify:
1
−2 12
· 2 45 .
Solution. Change to improper fractions, factor, cancel, and simplify.
4
25 14
1
−2 · 2 = − ·
12 5
12 5
25 · 14
=−
12 · 5
(5 · 5) · (2 · 7)
(2 · 2 · 3) · (5)
5·5·
2·7
=−
2
·
2
·
3·5
35
=−
6
=−
Change to improper fractions.
Multiply numerators; multiply denominators.
Unlike signs; product is negative.
Prime factor.
Cancel common factors.
Multiply numerators and denominators.
Simplify:
3 2
−3 · 2
4 5
296
CHAPTER 4. FRACTIONS
This is a perfectly good answer, but if you want a mixed fraction answer, 35
divided by 6 is 5, with a remainder of 5. Hence,
−2
1
4
5
· 2 = −5 .
12 5
6
Answer: −9
You Try It!
Simplify:
−4 54 ÷ 5 35 .
EXAMPLE 8. Simplify:
4 2
−2 · 3
9 3
Solution. Change to improper fractions, invert and multiply, factor, cancel,
and simplify.
3
24 28
4
−4 ÷ 5 = − ÷
5
5
5
5
24 5
=− ·
5 28
2·2·2·3
=−
5
2·
2·2·3
=−
5
6
=−
7
Change to improper fractions.
Invert and multiply.
5
2·2·7
5
·
2
·
2
·7
·
Prime factor.
Cancel common factors.
Multiply numerators and denominators.
Answer: −2/3
4.5. MULTIPLYING AND DIVIDING MIXED FRACTIONS
❧ ❧ ❧
Exercises
297
❧ ❧ ❧
In Exercises 1-12, convert the mixed fraction to an improper fraction.
1
3
8
1
11
1
1
19
1
−1
5
3
−1
7
3
1
17
1
9
5
1
11
1
−1
2
5
−1
8
1
1
3
5
−1
7
1. 2
7. 1
2.
8.
3.
4.
5.
6.
9.
10.
11.
12.
In Exercises 13-24, convert the improper fraction to a mixed fraction.
13.
14.
15.
16.
17.
18.
13
7
17
−
9
13
−
5
10
−
3
16
−
5
16
13
19.
20.
21.
22.
23.
24.
9
8
16
5
6
−
5
17
−
10
3
−
2
7
−
4
In Exercises 25-48, multiply the numbers and express your answer as a mixed fraction.
1 1
25. 1 · 2
7 2
27. 4 · 1
1 1
26. 1 · 1
8 6
28. 1
1
6
7
·4
10
298
1
3
−1
3
12
4
1
1
30. −3
3
2
3
29.
1
1
31. 7 · 1
2 13
1
5
32. 2 · 1
4 11
2
2
33. 1
−4
13
3
1
2
34. 1
−2
14
5
3
3
35. 1
−3
7
4
3
4
36. 1
−3
5
4
2
37. 9 · −1
15
5
38. 4 · −2
6
CHAPTER 4. FRACTIONS
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
1
(−6)
−2
8
1
(−9) −3
6
2
1
−4
−2
2
5
3
3
−1
−3
7
4
1
−2
·4
6
1
(−6) · 1
9
1
4
−1
2
15
2
1
5
−1
1
5
9
7
1
−2
−1
2
11
7
7
−1
−1
11
12
In Exercises 49-72, divide the mixed fractions and express your answer as a mixed fraction.
2
2
49. 8 ÷ 2
÷4
56. −4
9
3
2
2
1
50. 4 ÷ 4
57. −5
÷ −2
3
3
6
1
1
2
1
51. −3
÷ 1
58. −2
÷ −2
2
16
2
9
1
1
2
1
52. −1
59. −6
÷ 1
÷ 4
5
15
2
4
1
1
7
1
60. −1
÷ 1
53. 6 ÷ 1
6
8
2
12
9
1
3
61. (−6) ÷ −1
54. 5 ÷ 1
11
2
10
5
2
55. (−4) ÷ 1
62. −6
÷ (−6)
9
3
4.5. MULTIPLYING AND DIVIDING MIXED FRACTIONS
299
2
2
68. 1 ÷ 1
3
9
1
2
69. −7
÷ −2
2
5
5
1
70. −5
÷ −2
3
6
2
1
71. 3
÷ −1
3
9
1
3
÷ −1
72. 8
2
4
2
4
÷ (−4)
3
2
64. 6
÷ (−6)
3
3
1
65. 1
÷ −1
4
12
1
4
66. 2
÷ −1
7
5
63.
2
1
67. 5 ÷ 1
3
9
73. Small Lots. How many quarter-acre lots
can be made from 6 12 acres of land?
75. Jewelry. To make some jewelry, a bar of
1
silver 4 21 inches long was cut into pieces 12
inch long. How many pieces were made?
74. Big Field. A field was formed from 17 21
half-acre lots. How many acres was the
resulting field ?
76. Muffins. This recipe will make 6 muffins:
1 cup milk, 1 32 cups flour, 2 eggs, 1/2 teaspoon salt, 1 21 teaspoons baking powder.
Write the recipe for six dozen muffins.
❧ ❧ ❧
Answers
❧ ❧ ❧
1.
7
3
15. −2
3
5
3.
20
19
17. −3
1
5
5. −
7.
10
7
19. 1
1
8
10
9
21. −1
1
5
3
2
23. −1
1
2
9. −
4
11.
3
25. 2
6
7
6
13. 1
7
27. 4
2
3
300
29. −4
31. 8
1
16
1
13
CHAPTER 4. FRACTIONS
53. 4
2
19
55. −2
33. −5
5
13
57. 2
35. −5
5
14
59. −1
37. −10
1
5
61. 4
4
7
8
13
9
17
5
7
39. 12
3
4
63. −1
41. 10
4
5
1
6
65. −1
8
13
43. −8
45. −3
47. 4
49. 3
2
3
1
6
1
11
3
5
51. −3
67. 5
1
10
69. 3
1
8
71. −3
3
10
73. 26 quarter-acre lots
5
17
75. 54 pieces