5.5. FRACTIONS AND DECIMALS
5.5
401
Fractions and Decimals
When converting a fraction to a decimal, only one of two things can happen.
Either the process will terminate or the decimal representation will begin to
repeat a pattern of digits. In each case, the procedure for changing a fraction
to a decimal is the same.
Changing a Fraction to a Decimal. To change a fraction to a decimal,
divide the numerator by the denominator. Hint: If you first reduce the fraction
to lowest terms, the numbers will be smaller and the division will be a bit easier
as a result.
Terminating Decimals
Terminating Decimals. First reduce the fraction to lowest terms. If the
denominator of the resulting fraction has a prime factorization consisting of
strictly twos and/or fives, then the decimal representation will “terminate.”
You Try It!
EXAMPLE 1. Change 15/48 to a decimal.
Change 10/16 to a decimal.
Solution. First, reduce the fraction to lowest terms.
15
3·5
=
48
3 · 16
5
=
16
Next, note that the denominator of 5/16 has prime factorization 16 = 2 · 2 · 2 · 2.
It consists only of twos. Hence, the decimal representation of 5/16 should
terminate.
0.3125
16)5.0000
48
20
16
40
32
80
80
0
The zero remainder terminates the process. Hence, 5/16 = 0.3125.
Answer: 0.625
402
CHAPTER 5. DECIMALS
You Try It!
11
Change 7 20
to a decimal.
7
EXAMPLE 2. Change 3 20
to a decimal.
Solution. Note that 7/20 is reduced to lowest terms and its denominator has
prime factorization 20 = 2 · 2 · 5. It consists only of twos and fives. Hence, the
decimal representation of 7/20 should terminate.
0.35
20)7.00
60
1 00
1 00
0
Answer: 7.55
The zero remainder terminates the process. Hence, 7/20 = 0.35. Therefore,
7
3 20
= 3.35.
Repeating Decimals
Repeating Decimals. First reduce the fraction to lowest terms. If the prime
factorization of the resulting denominator does not consist strictly of twos and
fives, then the division process will never have a remainder of zero. However,
repeated patterns of digits must eventually reveal themselves.
You Try It!
Change 5/12 to a decimal.
EXAMPLE 3. Change 1/12 to a decimal.
Solution. Note that 1/12 is reduced to lowest terms and the denominator has
a prime factorization 12 = 2 · 2 · 3 that does not consist strictly of twos and
fives. Hence, the decimal representation of 1/12 will not “terminate.” We need
to carry out the division until a remainder reappears for a second time. This
will indicate repetition is beginning.
.083
12)1.000
96
40
36
4
5.5. FRACTIONS AND DECIMALS
403
Note the second appearance of 4 as a remainder in the division above. This is
an indication that repetition is beginning. However, to be sure, let’s carry the
division out for a couple more places.
.08333
12)1.00000
96
40
36
40
36
40
36
4
Note how the remainder 4 repeats over and over. In the quotient, note how
the digit 3 repeats over and over. It is pretty evident that if we were to carry
out the division a few more places, we would get
1
= 0.833333 . . .
12
The ellipsis is a symbolic way of saying that the threes will repeat forever. It
is the mathematical equivalent of the word “etcetera.”
Answer: 0.41666 . . .
There is an alternative notation to the ellipsis, namely
1
= 0.083.
12
The bar over the 3 (called a “repeating bar”) indicates that the 3 will repeat
indefinitely. That is,
0.083 = 0.083333 . . . .
Using the Repeating Bar. To use the repeating bar notation, take whatever
block of digits are under the repeating bar and duplicate that block of digits
infinitely to the right.
Thus, for example:
• 5.345 = 5.3454545 . . . .
• 0.142857 = 0.142857142857142857 . . ..
404
CHAPTER 5. DECIMALS
Important Observation. Although 0.833 will also produce 0.8333333 . . ., as
a rule we should use as few digits as possible under the repeating bar. Thus,
0.83 is preferred over 0.833.
You Try It!
Change 5/33 to a decimal.
EXAMPLE 4. Change 23/111 to a decimal.
Solution. The denominator of 23/111 has prime factorization 111 = 3 · 37 and
does not consist strictly of twos and fives. Hence, the decimal representation
will not “terminate.” We need to perform the division until we spot a repeated
remainder.
0.207
111)23.000
22 2
800
777
23
Note the return of 23 as a remainder. Thus, the digit pattern in the quotient
should start anew, but let’s add a few places more to our division to be sure.
0.207207
111)23.000000
22 2
800
777
230
222
800
777
23
Aha! Again a remainder of 23. Repetition! At this point, we are confident
that
23
= 0.207207 . . . .
111
Using a “repeating bar,” this result can be written
23
= 0.207.
111
Answer: 0.151515 . . .
405
5.5. FRACTIONS AND DECIMALS
Expressions Containing Both Decimals and Fractions
At this point we can convert fractions to decimals, and vice-versa, we can convert decimals to fractions. Therefore, we should be able to evaluate expressions
that contain a mix of fraction and decimal numbers.
You Try It!
3
− − 1.25.
8
Solution. Let’s change 1.25 to an improper fraction.
EXAMPLE 5. Simplify:
125
100
5
=
4
1.25 =
Two decimal places =⇒ two zeros.
Reduce to lowest terms.
In the original problem, replace 1.25 with 5/4, make equivalent fractions with
a common denominator, then subtract.
3 5
3
− − 1.25 = − −
8
8 4
3 5·2
=− −
8 4·2
3 10
=− −
8 8
3
10
=− + −
8
8
13
=−
8
Replace 1.25 with 5/4.
Equivalent fractions, LCD = 8.
Simplify numerator and denominator.
Add the opposite.
Add.
Thus, −3/8 − 1.25 = −13/8.
Alternate Solution. Because −3/8 is reduced to lowest terms and 8 = 2 · 2 · 2
consists only of twos, the decimal representation of −3/8 will terminate.
0.375
8)3.000
24
60
56
40
40
Hence, −3/8 = −0.375. Now, replace −3/8 in the original problem with
−0.375, then simplify.
3
− − 1.25 = −0.375 − 1.25
8
= −0.375 + (−1.25)
= −1.625
Replace −3/8 with −0.375.
Add the opposite.
Add.
Simplify:
7
− − 6.5
8
406
CHAPTER 5. DECIMALS
Thus, −3/8 − 1.25 = −1.625.
Are They the Same? The first method produced −13/8 as an answer; the
second method produced −1.625. Are these the same results? One way to find
out is to change −1.625 to an improper fraction.
1625
1000
5 · 5 · 5 · 5 · 13
=−
2·2·2·5·5·5
13
=−
2·2·2
13
=−
8
−1.625 = −
Answer: −7 83 or −7.375
Three places =⇒ three zeros.
Prime factor.
Cancel common factors.
Simplify.
Thus, the two answers are the same.
Let’s look at another example.
You Try It!
Simplify:
4
− + 0.25
9
2
− + 0.35.
3
Solution. Let’s attack this expression by first changing 0.35 to a fraction.
EXAMPLE 6. Simplify:
2
2
35
− + 0.35 = − +
3
3 100
2
7
=− +
3 20
Change 0.35 to a fraction.
Reduce 35/100 to lowest terms.
Find an LCD, make equivalent fractions, then add.
2 · 20
7·3
+
3 · 20 20 · 3
40 21
=− +
60 60
19
=−
60
2
19
Thus, − + 0.35 = − .
3
60
=−
Answer: −7/36
Equivalent fractions with LCD = 60.
Simplify numerators and denominators.
Add.
In Example 6, we run into trouble if we try to change −2/3 to a decimal.
The decimal representation for −2/3 is a repeating decimal (the denominator
is not made up of only twos and fives). Indeed, −2/3 = −0.6. To add −0.6
and 0.35, we have to align the decimal points, then begin adding at the right
end. But −0.6 has no right end! This observation leads to the following piece
of advice.
5.5. FRACTIONS AND DECIMALS
407
Important Observation. When presented with a problem containing both
decimals and fractions, if the decimal representation of any fraction repeats,
its best to first change all numbers to fractions, then simplify.
408
CHAPTER 5. DECIMALS
❧ ❧ ❧
Exercises
❧ ❧ ❧
In Exercises 1-20, convert the given fraction to a terminating decimal.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
59
16
19
5
35
4
21
4
1
16
14
5
6
8
7
175
3
2
15
16
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
119
175
4
8
9
8
5
2
78
240
150
96
25
10
2
4
9
24
216
150
In Exercises 21-44, convert the given fraction to a repeating decimal. Use the “repeating bar” notation.
21.
22.
23.
24.
25.
26.
256
180
268
180
364
12
292
36
81
110
82
99
27.
28.
29.
30.
31.
32.
76
15
23
9
50
99
53
99
61
15
37
18
409
5.5. FRACTIONS AND DECIMALS
33.
34.
35.
36.
37.
38.
98
66
305
330
190
495
102
396
13
15
65
36
39.
40.
41.
42.
43.
44.
532
21
44
60
26
198
686
231
47
66
41
198
In Exercises 45-52, simplify the given expression by first converting the fraction into a terminating
decimal.
7
− 7.4
4
3
46. − 2.73
2
7
47. + 5.31
5
7
48. − + 3.3
4
45.
9
− 8.61
10
3
50. + 3.7
4
6
51. − 7.65
5
3
52. − + 8.1
10
49.
In Exercises 53-60, simplify the given expression by first converting the decimal into a fraction.
7
− 2.9
6
11
54. − + 1.12
6
4
55. − − 0.32
3
11
56.
− 0.375
6
53.
2
57. − + 0.9
3
2
58. − 0.1
3
4
59. − 2.6
3
5
60. − + 2.3
6
410
CHAPTER 5. DECIMALS
In Exercises 61-64, simplify the given expression.
5
+ 2.375
6
5
62. + 0.55
3
11
63.
+ 8.2
8
13
64.
+ 8.4
8
7
+ 1.2
10
7
66. − − 3.34
5
11
67. − + 0.375
6
5
68. − 1.1
3
61.
65. −
❧ ❧ ❧
Answers
❧ ❧ ❧
1. 3.6875
37. 0.86
3. 8.75
39. 25.3
5. 0.0625
41. 0.13
7. 0.75
43. 0.712
9. 1.5
45. −5.65
11. 0.68
47. 6.71
13. 1.125
49. −7.71
15. 0.325
51. −6.45
17. 2.5
53. −
26
15
55. −
124
75
19. 0.375
21. 1.42
7
30
23. 30.3
57.
25. 0.736
59. −
27. 5.06
29. 0.50
31. 4.06
33. 1.48
35. 0.38
61.
19
15
77
24
63. 9.575
65. 0.5
67. −
35
24