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Writing recurring decimals as fractions

Writing recurring decimals as fractions

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Explainer Video

Tutor: Daniel

Summary

Writing recurring decimals as fractions

​​In a nutshell

Not all fractions can be expressed as a terminating decimals (decimals with only a finite number of digits). In some cases, their decimal form may be infinitely long and so you cannot write down the entire number. 



Recurring decimals

Definition

A recurring decimal is a number containing an infinitely repeating digit or series of digits. The repeating digit(s) are indicated with a  ˙\dot{}˙  above the numbers.


Example 1

The following are examples of recurring decimals:


0.1111...=0.1˙, 0.345345...=0.3˙45˙, 0.298454545...=0.2984˙5˙0.1111... = 0.\dot{1},\text{ } 0.345345...=0.\dot{3}4\dot{5},\text{ } 0.298454545... = 0.298\dot{4}\dot{5}0.1111...=0.1˙, 0.345345...=0.3˙45˙, 0.298454545...=0.2984˙5˙



Converting between recurring decimals and fractions

Fraction to recurring decimal

The process of converting a fraction to a recurring decimal is the same as before, except ensure that the  ˙\dot{}˙  goes above the correct numbers to indicate which are repeating.


Recurring decimal to fraction - higher only

​​Procedure

1.
Multiply the recurring decimal by a multiple of 101010, for example 10,100,or 100010 , 100, or \space100010,100,or 1000 etc. such that the recurring digits overlap in the same position.​
2.
Subtract the recurring decimal part from another, leaving a 9x, 99x9x, \text{ } 99x9x, 99x etc.​
3.
Rearrange to obtain the number as a fraction.
4.
Simplify the fraction (if required).


Example 2

Write 0.4545...=0.4˙5˙0.4545... = 0.\dot{4}\dot{5}0.4545...=0.4˙5˙ as a fraction in its simplest form.


x=0.454545...1◯100x=45.454545...2◯99x=452◯−1◯x=4599\begin{aligned}x &= 0.454545... &\qquad \textcircled{1} \\100x &= 45.454545... &\qquad \textcircled{2} \\ 99x &= 45 &\qquad \textcircled{2}- \textcircled{1}\\x &= \dfrac {45}{99}\end {aligned}x100x99xx​=0.454545...=45.454545...=45=9945​​1◯2◯2◯−1◯​


Therefore, x=511‾\underline{ x= \dfrac {5}{11}}x=115​​.



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Learn with Basics

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Length:
Methods for multiplication and division

Unit 1

Methods for multiplication and division

Multiplying and dividing decimals

Unit 2

Multiplying and dividing decimals

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Writing recurring decimals as fractions

Unit 3

Writing recurring decimals as fractions

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FAQs - Frequently Asked Questions

Can all decimals be written as fractions?

No, the irrational numbers are decimals and these cannot be written as fractions.

Are recurring decimals rational numbers?

Yes, as when you put them into fractional form, both the numerator and denominator become non-fractional whole numbers.

How do you identify a recurring decimal?

A recurring decimal exists when decimal numbers repeat forever. The repeating numbers are indicated with a dot above them.

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