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.

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*

N = 12 ÷ 9

=

=

=

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.

Write the following recurring decimals as fractions.

a. 16.7777

b. 13.05050505

c. 5.324324324324

b. 13.05050505

c. 5.324324324324

- We have 16.7777 = N

167.7777 = 10N

167.7777 - 16.7777 = 10N - N

151 = 9N

N =*151**/**9* - 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* - This time we have to multiply the original number by 1000 to obtain again the same recurrence. Thus,5.324324324324 = NBoth numerator and denominator are divisible by 9 (check this by using the divisibility rules explained in tutorial 1.6. Hence, we obtain

5324.324324324324 = 1000N

5324.324324324324 - 5.324324324324 = 1000N - N

5319 = 999N

N =*5319**/**999*N =Again, we see that both numbers are still divisible by 3, so=*(5319 ÷ 9)**/**(999 ÷ 9)**591**/**111*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.

Write the following recurring decimals as fractions.

a. 5.244444

b. 37.006666

c. 8.361616161

b. 37.006666

c. 8.361616161

- We have 5.2
*4*= N

52.*4*= 10N

524.*4*= 100N

524.*4*- 52.*4*= 100N - 10N

472 = 90N

N =*472**/**90*

=*(472 ÷ 2)**/**(90 ÷ 2)*

=*236**/**45* - We have 37.00
*6*= N

3700.*6*= 100N

37006.*6*= 1000N

37006.*6*- 3700.*6*= 1000N - 100N

33306 = 900N

N =*33306**/**900*

=*(33306 ÷ 3)**/**(900 ÷ 3)*

=*11102**/**300* - We have 8.3
*61*= 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.

Express the following recurring decimals as finite numbers.

a. 3.*9*

b. 219.*9*

b. 219.

- We have: 3.
*9*= N

39.*9*= 10N

39.*9*- 3.*9*= 10N - N

36 = 9N

N == 4*36**/**9* - We have: 219.
*9*= N

2199.*9*= 10N

2199.*9*- 219.*9*= 10N - N

1980 = 9N

N == 220*1980**/**9*

Enjoy the "Converting Recurrent Fractions into Decimals" math lesson? People who liked the "Converting Fractions to Decimals and Vice-versa lesson found the following resources useful:

- Recurrent Fraction To Decimal Feedback. Helps other - Leave a rating for this recurrent fraction to decimal (see below)
- Fractions Math tutorial: Converting Fractions to Decimals and Vice-versa. Read the Converting Fractions to Decimals and Vice-versa math tutorial and build your math knowledge of Fractions
- Fractions Video tutorial: Converting Fractions to Decimals and Vice-versa. Watch or listen to the Converting Fractions to Decimals and Vice-versa video tutorial, a useful way to help you revise when travelling to and from school/college
- Fractions Revision Notes: Converting Fractions to Decimals and Vice-versa. Print the notes so you can revise the key points covered in the math tutorial for Converting Fractions to Decimals and Vice-versa
- Fractions Practice Questions: Converting Fractions to Decimals and Vice-versa. Test and improve your knowledge of Converting Fractions to Decimals and Vice-versa with example questins and answers
- Check your calculations for Fractions questions with our excellent Fractions calculators which contain full equations and calculations clearly displayed line by line. See the Fractions Calculators by iCalculator™ below.
- Continuing learning fractions - read our next math tutorial: The Meaning of Fractions. Equivalent Fractions. Simplifying Fractions

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