# 3.6: Fractions and Decimals (2023)

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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.

\begin{aligned} \frac{15}{48} = \frac{3 \cdot 5}{3 \cdot 16} \\ = \frac{5}{16} \end{aligned}\nonumber

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.

0.625

Example 2

Change $$3 \frac{7}{20}$$ 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. The zero remainder terminates the process. Hence, 7/20 = 0.35. Therefore, $$3 \frac{7}{20}$$ = 3.35.

Exercise

Change $$7 \frac{11}{20}$$ to a decimal.

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.

(Video) How to convert 3.6 to Fraction || 3.6 as a Fraction ( 3.6 Decimal to Fraction) 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

$\frac{1}{12} = 0.833333 \cdots\nonumber$

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.

0.41666...

There is an alternative notation to the ellipsis, namely

$\frac{1}{12} = 0.08 \overline{3}.\nonumber$

The bar over the 3 (called a “repeating bar”) indicates that the 3 will repeat indefinitely. That is,

$0.08 \overline{3} = 0.083333 ....\nonumber$

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:

• $$5. \overline{345} = 5.3454545 ....$$
• $$0. \overline{142857} = 0.142857142857142857 ....$$

Important Observation

Although $$0.8 \overline{33}$$ will also produce 0.8333333 ..., as a rule we should use as few digits as possible under the repeating bar. Thus, $$0.8 \overline{3}$$ is preferred over $$0.8 \overline{33}$$.

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

$\frac{23}{111} = 0.207207 ....\nonumber$

Using a “repeating bar,” this result can be written

$\frac{23}{111} = 0. \overline{207}.\nonumber$

Exercise

Change 5/33 to a decimal.

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: $$- \frac{3}{8} - 1.25$$.

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.

\begin{aligned} - \frac{3}{8} - 1.25 = - \frac{3}{8} - \frac{5}{4} ~ & \textcolor{red}{ \text{ Replace 1.25 with 5/4.}} \\ = - \frac{3}{8} - \frac{5 \cdot \textcolor{red}{2}}{4 \cdot \textcolor{red}{2}} ~ & \textcolor{red}{ \text{ Equivalent fractions, LCD = 8.}} \\ = - \frac{3}{8} - \frac{10}{8} ~ & \textcolor{red}{ \text{ Simplify the numerator and denominator.}} \\ = - \frac{3}{8} + \left( - \frac{10}{8} \right) ~ & \textcolor{red}{ \text{ Add the opposite.}} \\ = = \frac{13}{8} ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber

Thus, −3/8 − 1.25 = −13/8.

(Video) 3.6 Convert Fraction to Decimal Ex41

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.

\begin{aligned} - \frac{3}{8} - 1.25 = -0.375 - 1.25 ~ & \textcolor{red}{ \text{ Replace } -3/8 \text{ with } -0.375.} \\ =-0.375 + (-1.25) ~ & \textcolor{red}{ \text{ Add the opposite.}} \\ = -1.625 ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber

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.

\begin{aligned} -1.625 = - \frac{1625}{1000} ~ & \textcolor{red}{ \text{ Three places } \Rightarrow \text{ three zeroes.}} \\ = - \frac{5 \cdot 5 \cdot 5 \cdot 5 \cdot 13}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} ~ & \textcolor{red}{ \text{ Prime factor.}} \\ = - \frac{13}{2 \cdot 2 \cdot 2} ~ & \textcolor{red}{ \text{ Cancel common factors.}} \\ = - \frac{13}{8} ~ & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber

Thus, the two answers are the same.

Exercise

Simplify: $$− \frac{7}{8} − 6.5$$

$$−7 \frac{3}{8}$$ or −7.375

Example 6

Simplify: $$-\frac{2}{3} + 0.35$$.

Solution

Let’s attack this expression by first changing 0.35 to a fraction.

\begin{aligned} - \frac{2}{3} + 0.35 = - \frac{2}{3} + \frac{35}{100} ~ & \textcolor{red}{ \text{ Change 0.35 to a fraction.}} \\ = - \frac{2}{3} + \frac{7}{20} ~ & \textcolor{red}{ \text{ Reduce 35/100 to lowest terms.}} \end{aligned}\nonumber

Find an LCD, make equivalent fractions, then add.

\begin{aligned} = - \frac{2 \cdot 20}{3 \cdot 20} + \frac{7 \cdot 3}{20 \cdot 3} ~ & \textcolor{red}{ \text{ Equivalent fractions with LCD = 60.}} \\ = - \frac{40}{60} + \frac{21}{60} ~ & \textcolor{red}{ \text{ Simplify numerators and denominators.}} \\ = - \frac{19}{60} ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber

Then, $$-\frac{2}{3} +0.35 = - \frac{19}{60}$$.

Exercise

Simplify: $$-\frac{4}{9} + 0.25$$

−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 = $$−0. \overline{6}$$. To add $$−0. \overline{6}$$ and 0.35, we have to align the decimal points, then begin adding at the right end. But $$−0. \overline{6}$$ 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. $$\frac{59}{16}$$

2. $$\frac{19}{5}$$

3. $$\frac{35}{4}$$

4. $$\frac{21}{4}$$

5. $$\frac{1}{16}$$

6. $$\frac{14}{5}$$

7. $$\frac{6}{8}$$

8. $$\frac{7}{175}$$

9. $$\frac{3}{2}$$

10. $$\frac{15}{16}$$

11. $$\frac{119}{175}$$

12. $$\frac{4}{8}$$

13. $$\frac{9}{8}$$

(Video) Math Antics - Converting Base-10 Fractions

14. $$\frac{5}{2}$$

15. $$\frac{78}{240}$$

16. $$\frac{150}{96}$$

17. $$\frac{25}{10}$$

18. $$\frac{2}{4}$$

19. $$\frac{9}{24}$$

20. $$\frac{216}{150}$$

In Exercises 21-44, convert the given fraction to a repeating decimal. Use the “repeating bar” notation.

21. $$\frac{256}{180}$$

22. $$\frac{268}{180}$$

23. $$\frac{364}{12}$$

24. $$\frac{292}{36}$$

25. $$\frac{81}{110}$$

26. $$\frac{82}{99}$$

27. $$\frac{76}{15}$$

28. $$\frac{23}{9}$$

29. $$\frac{50}{99}$$

30. $$\frac{53}{99}$$

31. $$\frac{61}{15}$$

32. $$\frac{37}{18}$$

33. $$\frac{98}{66}$$

34. $$\frac{305}{330}$$

35. $$\frac{190}{495}$$

36. $$\frac{102}{396}$$

37. $$\frac{13}{15}$$

38. $$\frac{65}{36}$$

39. $$\frac{532}{21}$$

40. $$\frac{44}{60}$$

41. $$\frac{26}{198}$$

42. $$\frac{686}{231}$$

43. $$\frac{47}{66}$$

44. $$\frac{41}{198}$$

In Exercises 45-52, simplify the given expression by first converting the fraction into a terminating decimal.

45. $$\frac{7}{4} − 7.4$$

46. $$\frac{3}{2} − 2.73$$

47. $$\frac{7}{5} + 5.31$$

48. $$− \frac{7}{4} + 3.3$$

49. $$\frac{9}{10} − 8.61$$

50. $$\frac{3}{4} + 3.7$$

51. $$\frac{6}{5} − 7.65$$

52. $$− \frac{3}{10} + 8.1$$

In Exercises 53-60, simplify the given expression by first converting the decimal into a fraction.

53. $$\frac{7}{6} − 2.9$$

54. $$− \frac{11}{6} + 1.12$$

55. $$− \frac{4}{3} − 0.32$$

56. $$\frac{11}{6} − 0.375$$

(Video) MATH085 Section 3.6 Percents, Fractions, and Decimals

57. $$− \frac{2}{3} + 0.9$$

58. $$\frac{2}{3} − 0.1$$

59. $$\frac{4}{3} − 2.6$$

60. $$− \frac{5}{6} + 2.3$$

In Exercises 61-64, simplify the given expression.

61. $$\frac{5}{6} + 2.375$$

62. $$\frac{5}{3} + 0.55$$

63. $$\frac{11}{8} + 8.2$$

64. $$\frac{13}{8} + 8.4$$

65. $$− \frac{7}{10} + 1.2$$

66. $$− \frac{7}{5} − 3.34$$

67. $$− \frac{11}{6} + 0.375$$

68. $$\frac{5}{3} − 1.1$$

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. $$1.4 \overline{2}$$

23. $$30. \overline{3}$$

25. $$0.7 \overline{36}$$

27. $$5.0 \overline{6}$$

29. $$0. \overline{50}$$

31. $$4.0 \overline{6}$$

33. $$1. \overline{48}$$

35. $$0. \overline{38}$$

37. $$0.8 \overline{6}$$

39. $$25. \overline{3}$$

41. $$0. \overline{13}$$

43. $$0.7 \overline{12}$$

45. −5.65

47. 6.71

49. −7.71

51. −6.45

53. $$− \frac{26}{15}$$

55. $$− \frac{124}{75}$$

57. $$\frac{7}{30}$$

59. $$− \frac{19}{15}$$

61. $$\frac{77}{24}$$

63. 9.575

(Video) How to Divide a Decimal by a Decimal | Math with Mr. J

65. 0.5

67. $$− \frac{35}{24}$$

## FAQs

### What is 3.6 as a decimal as a fraction? ›

Solution: 3.6 as a fraction is 18/5.

Is 3.6 a terminating decimal? ›

First of all, 3.6 is a terminating decimal and all terminating decimals are rational numbers. The number 3.6 can also be represented as a mixed number 3 6/10 or as the fraction 36/10.

What is 3 6 as decimal? ›

Solution: 3/6 as a decimal is 0.5

And finally, you get 0.5 as your answer when you convert 3/6 to a decimal.

What is 3.5 as a fraction? ›

3.5 in fraction form is 7/2 or 35/10. The number of digits after the decimal point is equal to the number of zeros following the digit 1 in the denominator of the fractional form.

What is 3/4 fraction as a decimal? ›

The decimal form is 0.75.

Is 0.35 as a fraction? ›

Solution: 0.35 as a fraction is 7/20.

What is 3 6 10 as decimal? ›

Solution: 3 6/10 as a decimal is 3.6.

What is 3 and 6 tenths as a decimal? ›

Three ones and six-tenths = 3 + 6/10 = 3 + 0.6 = 3.6.

How do you turn a fraction into a decimal? ›

The fraction bar separating the “part” and the “whole” represents division. This means that all fractions can be converted into decimals by dividing the numerator by the denominator. For example, the fraction 45 represents 4 out of 5, or 4 divided by 5. This fraction can be converted into a decimal by dividing 4 by 5.

What is 3 6 equal to as a fraction? ›

Since, both the values are equal, therefore, 1/2 and 3/6 are equivalent fractions.

### How do you write 3 6 as a fraction? ›

3/6 = 12 = 0.5

Spelled result in words is one half.

How do you write 3.6 as a mixed number? ›

Answer: 3.6 as a fraction would be written as 18/5.

Let's convert 3.6 into a fraction. Explanation: 3.6 being a decimal number can easily be converted into a fraction. This is done by dividing and multiplying both the numerator and denominator by 10.

What is 3.56 as a fraction? ›

Solution: 3.56 as a fraction is 89/25.

What is 3 5 as a decimal? ›

Solution: 3/5 as a decimal is 0.6

And finally, you get 0.6 as your answer when you convert 3/5 to a decimal.

What is 3 6 as a percentage? ›

So 3/6 is equal to 50%.

What is 3 and 3 4 as a decimal? ›

Solution: 3 3/4 as a decimal is 3.75.

What is 3/4 as a fraction? ›

or in a line of text as one number with a slash or solidus then the other number, like this: 3/4. The fraction 3/4 or three quarters means 3 parts out of 4. The upper number, 3, is called the numerator and the lower number, 4, is the denominator.

What is 36 as a fraction? ›

Answer: 36 as a fraction would be written as 36/1 or 360/10.

What is 0.35 in fraction simplest form? ›

Answer: 0.35 as a fraction is 7/20

Let us see how to convert 0.35 to a fraction. Explanation: Use the steps given below to convert a decimal number into a fraction.

What is 35% as a fraction and decimal? ›

This decimal number has two digits after the decimal, so put 35 over a 1 and two zeros, that is, 35/100 and then finally simplify the fraction. 35/100 is simplified to 7/20, giving us the final answer.

### What is 3 and 2 6 as a decimal? ›

Solution: 3 2/6 as a decimal is 3.33.

What is 3 and 5/6 as a decimal? ›

Solution: 3 5/6 as a decimal is 3.83

We got 3.83 as the answer when you convert 3 5/6 (or 23/6) to a decimal.

Is 0.33333 as a fraction? ›

So we can see that our original decimal of 0.333333... is equal to the fraction 1/3.

What is 3 and 3/10 as a decimal? ›

And finally, you get 3.3 as your answer when you convert 3 3/10 (or 33/10) to a decimal.

What is 3/10 as a fraction and decimal? ›

Solution: 3/10 as a decimal is 0.3.

How do you write 0.375 as a fraction? ›

Solution: 0.375 as a fraction is 3/8.

How do you change a fraction to a decimal without a calculator? ›

This can be an easy way to change a common fraction to a decimal without using your calculator or doing long division.
...
Multiply both the numerator and denominator of the fraction by that number.
1. 3/5 x 2/2 = 6/10.
2. 3/4 x 25/25 = 75/100.
3. 5/16 x 625/625 = 3,125/10,000.

How do you write 1 3 as a decimal? ›

Therefore, we can write as 13= 0.33333....

What is 3/6 as a whole number? ›

👍 That's right! You have 2 wholes. That means 6/3 is equivalent to 2 wholes! Here are some examples of fractions equivalent to one whole.

Is 3/6 is a proper fraction? ›

A proper fraction is a fraction which has its numerator value less than the denominator. For example, ⅔, 6/7, 8/9, etc. are proper fractions.

### How to simplify a fraction? ›

Steps:
1. Find the highest common factor (HCF) of numerator and denominator of the fraction.
2. Divide both the numerator and the denominator by HCF to get the simplified fraction.
3. Write the whole and the simplified fraction together.

What is 6 and 3/4 as a fraction? ›

Answer and Explanation: The mixed number 6 3/4 can be changed into the improper fraction 27/4.

What is the answer of 6 of 3? ›

Hence, The value of 6% of 3 is 0.18.

What is 3.3 as a fraction? ›

3.3 as a fraction is 3 3/10. The first 3 is a whole number because it is to the left of the decimal point, and it represents 3 ''ones''. However, the second 3 to the right of the decimal point represents a decimal amount. Because this 3 is in the tenths place, we say it as ''3 tenths.

What is 3.2 as a fraction? ›

Solution: 3.2 as a fraction is 16/5.

What is 3.65 as a fraction? ›

Solution: 3.65 as a fraction is 73/20.

What is 3.66 as a fraction? ›

Solution: 3.66 as a fraction is 183/50.

What is 3.68 as a fraction? ›

Solution: 3.68 as a fraction is 92/25.

What is 0.036 as a fraction? ›

Solution: 0.036 as a fraction is 9/250.

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