# Math Lesson 3.5.4 - Converting Recurrent Fractions into Decimals

Welcome to our Math lesson on Converting Recurrent Fractions into Decimals, this is the fourth lesson of our suite of math lessons covering the topic of Converting Fractions to Decimals and Vice-versa, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

## Converting Recurrent Fractions into Decimals

This is a very special part of this tutorial because at the first sight a recurrent decimal seems impossible to convert into a fraction due to its infinity of decimal places, so the conversion a recurring decimal into a fraction is impossible through the usual methods. Therefore, a completely new approach must be used for this kind of conversion. Let's begin to explain this new method through an example where a one-digit recurring decimal is involved. For example, let's consider the number 1.3333 (or 1.3); a recurring decimal obtained when dividing 4 by 3. The method we will explain here consists on the following steps:

Step 1: We denote the original number by N, i.e.

1.3333=N

Step 2: We multiply the original number by a factor that gives a new number with the same decimal part as the original. Here, this factor is 10. Thus, we have

13.3333 = 10 × N

Step 3: We subtract the original number from the new one to eliminate the recurring part, i.e.

13.3333-1.3333 = 12

The same thing is done with the factors containing N's. Thus, we have

10 × N - N = 9 × N

In this way, we obtain the equation

9 × N = 12

Step 4: Now, we solve the above equation and simplify the result is needed. Thus,

9 × N = 12
N = 12 ÷ 9
= 12/9
= (12 ÷ 3)/(9 ÷ 3)
= 4/3

This is the number we were referring to at the beginning of this explanation.

Let's consider a few other examples with other types of recurrence.

### Example 6

Write the following recurring decimals as fractions.

a. 16.7777
b. 13.05050505
c. 5.324324324324

### Solution 6

1. We have
16.7777 = N
167.7777 = 10N
167.7777 - 16.7777 = 10N - N
151 = 9N
N = 151/9
2. This time we have to multiply the original number by 100 to obtain again the same recurrence. Thus,
13.05050505 = N
1305.05050505 = 100N
1305.05050505 - 13.05050505 = 100N - N
1292 = 99N
N = 1292/99
3. This time we have to multiply the original number by 1000 to obtain again the same recurrence. Thus,
5.324324324324 = N
5324.324324324324 = 1000N
5324.324324324324 - 5.324324324324 = 1000N - N
5319 = 999N
N = 5319/999
Both numerator and denominator are divisible by 9 (check this by using the divisibility rules explained in tutorial 1.6. Hence, we obtain
N = (5319 ÷ 9)/(999 ÷ 9) = 591/111
Again, we see that both numbers are still divisible by 3, so
N = (591 ÷ 3)/(111 ÷ 3) = 197/37

Sometimes the recurrence does not begin immediately after the decimal point but more on the right. In such cases, we first multiply the original number by a factor that leaves only the recurrence after the decimal point and then, we multiply this number by another suitable factor that gives the same recurrence, in order to make possible the elimination of recurrence through subtraction. Let's clarify this point though a couple of examples.

### Example 7

Write the following recurring decimals as fractions.

a. 5.244444
b. 37.006666
c. 8.361616161

### Solution 7

1. We have
5.24 = N
52.4 = 10N
524.4 = 100N
524.4 - 52.4 = 100N - 10N
472 = 90N
N = 472/90
= (472 ÷ 2)/(90 ÷ 2)
= 236/45
2. We have
37.006 = N
3700.6 = 100N
37006.6 = 1000N
37006.6 - 3700.6 = 1000N - 100N
33306 = 900N
N = 33306/900
= (33306 ÷ 3)/(900 ÷ 3)
= 11102/300
3. We have
8.361 = N
83.61 = 10N
8361.61 = 1000N
8361.61 - 83.61 = 1000N - 10N
8278 = 990N
N = 8278/990
= (8278 ÷ 2)/(990 ÷ 2)
= 4139/495

A special case in this process is when the recurring part contains only 9s. In this case, we obtain not a fraction but a whole number as a result. Let's consider a couple of examples to clarify this point.

### Example 8

Express the following recurring decimals as finite numbers.

a. 3.9
b. 219.9

### Solution 8

1. We have:

3.9 = N
39.9 = 10N
39.9 - 3.9 = 10N - N
36 = 9N
N = 36/9 = 4
2. We have:

219.9 = N
2199.9 = 10N
2199.9 - 219.9 = 10N - N
1980 = 9N
N = 1980/9 = 220

## More Converting Fractions to Decimals and Vice-versa Lessons and Learning Resources

Fractions Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
3.5Converting Fractions to Decimals and Vice-versa
Lesson IDMath Lesson TitleLessonVideo
Lesson
3.5.1Decimal Fractions
3.5.2The Meaning of Decimals. Converting Fractions into Decimals without a Calculator
3.5.3Converting Decimals to Fractions
3.5.4Converting Recurrent Fractions into Decimals

## Whats next?

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