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 ˙ 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˙
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 ˙ 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 10, for example 10,100,or 1000 etc. such that the recurring digits overlap in the same position. |
2.
| Subtract the recurring decimal part from another, leaving a 9x, 99x etc. |
3.
| Rearrange to obtain the number as a fraction.
|
4. | Simplify the fraction (if required).
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Example 2
Write 0.4545...=0.4˙5˙ as a fraction in its simplest form.
x100x99xx=0.454545...=45.454545...=45=99451◯2◯2◯−1◯
Therefore, x=115.