How to convert a recurring decimal into a fraction

Example 1

Write 0.77..$\mathrm{0.77..}$ as a fraction in its lowest terms.

Our first step is to form a simple equation where x=0.77..$x=\mathrm{0.77..}$. By multiplying both sides by 10$10$ we can obtain another equation with 10x=7.77..$10x=\mathrm{7.77..}$. Now we eliminate the recurring part of the decimal by subtracting x$x$ from 10x$10x$.

x10x9xx=0.77..=7.77..=7=79$$\begin{array}{rl}x& =\mathrm{0.77..}\\ 10x& =\mathrm{7.77..}\\ 9x& =7\\ x& =\frac{7}{9}\end{array}$$

So we have our answer 0.77...=79$\mathrm{0.77...}=\frac{7}{9}$.
The important part to remember is to get two equations in x$x$ where the recurring part after the decimal point is exactly the same.

Example 2

Write 0.277..$\mathrm{0.277..}$ as a fraction in its lowest terms.

Again our first step is write x=0.277..$x=\mathrm{0.277..}$. Now multiplying by 10$10$ gives us 10x=2.77..$10x=\mathrm{2.77..}$. This time we need another equation to match the recurring part of the equation. So multiplying by 10$10$ again gives 100x=27.77..$100x=\mathrm{27.77..}$. Now we have two equations with the same recurring part we subtract one from the other as before.

x10x100x90xxx=0.277..=2.77..=27.77..=25=2590=518$$\begin{array}{rl}x& =\mathrm{0.277..}\\ 10x& =\mathrm{2.77..}\\ 100x& =\mathrm{27.77..}\\ 90x& =25\\ x& =\frac{25}{90}\\ x& =\frac{5}{18}\end{array}$$

So in its lowest terms 0.277..=518$\mathrm{0.277..}=\frac{5}{18}$.

Example 3

Write 0.0855.. as a fraction in its lowest terms.

As always our first step it to write x=0.0855..$x=\mathrm{0.0855..}$. This time to get two equations with the same recurring part after the decimal point we need to multiply by 100$100$ and then 1000$1000$. This gives us:

100x1000x900xx=8.55..=85.55..=77=77900$$\begin{array}{rl}100x& =\mathrm{8.55..}\\ 1000x& =\mathrm{85.55..}\\ 900x& =77\\ x& =\frac{77}{900}\end{array}$$

This cannot be simplified any further so 0.855..=77900$\mathrm{0.855..}=\frac{77}{900}$.

Example 4

Write 0.0234234.. as a fraction in its lowest terms.

This time the '234$234$' part is the important bit. Now to get two multiples of x with 234 as the recurring part we need to multiply first by 10$10$ and then another 1000$1000$ to move the digits 3 places to the right.

x=0.0234234...10x=0.234234..10000x=234.2349990x=234xx=2349990x=13555$$\begin{array}{r}x=\mathrm{0.0234234...}\\ 10x=\mathrm{0.234234..}\\ 10000x=234.234\\ 9990x=234x\\ x=\frac{234}{9990}\\ x=\frac{13}{555}\end{array}$$

So 0.0234=13555$0.0234=\frac{13}{555}$ as a fraction in its lowest terms.