3.6: Fractions and Decimals
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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.”
Example 1
Change 15/48 to a decimal.
Solution
First, reduce the fraction to lowest terms.
1548=3⋅53⋅16=516
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.
The zero remainder terminates the process. Hence, 5/16 = 0.3125.
Exercise
Change 10/16 to a decimal.
- Answer
-
0.625
Example 2
Change 3720 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.
The zero remainder terminates the process. Hence, 7/20 = 0.35. Therefore, 3720 = 3.35.
Exercise
Change 71120 to a decimal.
- Answer
-
7.55
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.
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.
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.
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
112=0.833333⋯
The ellipsis is a symbolic way of saying that the threes will repeat forever. It is the mathematical equivalent of the word “etcetera.”
Exercise
Change 5/12 to a decimal.
- Answer
-
0.41666...
There is an alternative notation to the ellipsis, namely
112=0.08¯3.
The bar over the 3 (called a “repeating bar”) indicates that the 3 will repeat indefinitely. That is,
0.08¯3=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....
Important Observation
Although 0.8¯33 will also produce 0.8333333 ..., as a rule we should use as few digits as possible under the repeating bar. Thus, 0.8¯3 is preferred over 0.8¯33.
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.
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.
Aha! Again a remainder of 23. Repetition! At this point, we are confident that
23111=0.207207....
Using a “repeating bar,” this result can be written
23111=0.¯207.
Exercise
Change 5/33 to a decimal.
- Answer
-
0.151515...
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.
Example 5
Simplify: −38−1.25.
Solution
Let’s change 1.25 to an improper fraction.
\[ \begin{aligned} 1.25 = \frac{125}{100} ~ & \textcolor{red}{ \text{ Two decimal places } \Rightarrow \text{ two zeroes.} \\ = \frac{5}{4} ~ & \textcolor{red}{ \text{ Reduce to lowest terms.}} \end{aligned}\nonumber \]
In the original problem, replace 1.25 with 5/4, make equivalent fractions with a common denominator, then subtract.
−38−1.25=−38−54 Replace 1.25 with 5/4.=−38−5⋅24⋅2 Equivalent fractions, LCD = 8.=−38−108 Simplify the numerator and denominator.=−38+(−108) Add the opposite.==138 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.
Hence, −3/8 = −0.375. Now, replace −3/8 in the original problem with −0.375, then simplify.
−38−1.25=−0.375−1.25 Replace −3/8 with −0.375.=−0.375+(−1.25) Add the opposite.=−1.625 Add.
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.
−1.625=−16251000 Three places ⇒ three zeroes.=−5⋅5⋅5⋅5⋅132⋅2⋅2⋅5⋅5⋅5 Prime factor.=−132⋅2⋅2 Cancel common factors.=−138 Simplify.
Thus, the two answers are the same.
Exercise
Simplify: −78−6.5
- Answer
-
−738 or −7.375
Example 6
Simplify: −23+0.35.
Solution
Let’s attack this expression by first changing 0.35 to a fraction.
−23+0.35=−23+35100 Change 0.35 to a fraction.=−23+720 Reduce 35/100 to lowest terms.
Find an LCD, make equivalent fractions, then add.
=−2⋅203⋅20+7⋅320⋅3 Equivalent fractions with LCD = 60.=−4060+2160 Simplify numerators and denominators.=−1960 Add.
Then, −23+0.35=−1960.
Exercise
Simplify: −49+0.25
- Answer
-
−7/36
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.
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.
Exercises
In Exercises 1-20, convert the given fraction to a terminating decimal.
1. 5916
2. 195
3. 354
4. 214
5. 116
6. 145
7. 68
8. 7175
9. 32
10. 1516
11. 119175
12. 48
13. 98
14. 52
15. 78240
16. 15096
17. 2510
18. 24
19. 924
20. 216150
In Exercises 21-44, convert the given fraction to a repeating decimal. Use the “repeating bar” notation.
21. 256180
22. 268180
23. 36412
24. 29236
25. 81110
26. 8299
27. 7615
28. 239
29. 5099
30. 5399
31. 6115
32. 3718
33. 9866
34. 305330
35. 190495
36. 102396
37. 1315
38. 6536
39. 53221
40. 4460
41. 26198
42. 686231
43. 4766
44. 41198
In Exercises 45-52, simplify the given expression by first converting the fraction into a terminating decimal.
45. 74−7.4
46. 32−2.73
47. 75+5.31
48. −74+3.3
49. 910−8.61
50. 34+3.7
51. 65−7.65
52. −310+8.1
In Exercises 53-60, simplify the given expression by first converting the decimal into a fraction.
53. 76−2.9
54. −116+1.12
55. −43−0.32
56. 116−0.375
57. −23+0.9
58. 23−0.1
59. 43−2.6
60. −56+2.3
In Exercises 61-64, simplify the given expression.
61. 56+2.375
62. 53+0.55
63. 118+8.2
64. 138+8.4
65. −710+1.2
66. −75−3.34
67. −116+0.375
68. 53−1.1
Answers
1. 3.6875
3. 8.75
5. 0.0625
7. 0.75
9. 1.5
11. 0.68
13. 1.125
15. 0.325
17. 2.5
19. 0.375
21. 1.4¯2
23. 30.¯3
25. 0.7¯36
27. 5.0¯6
29. 0.¯50
31. 4.0¯6
33. 1.¯48
35. 0.¯38
37. 0.8¯6
39. 25.¯3
41. 0.¯13
43. 0.7¯12
45. −5.65
47. 6.71
49. −7.71
51. −6.45
53. −2615
55. −12475
57. 730
59. −1915
61. 7724
63. 9.575
65. 0.5
67. −3524