Recurring Decimals GCSE Questions and Worked Answers

Let x = 0.444….

$$10x=4.444\ldots$$ $$x=0.444\ldots$$

Subtract the second equation from the first, on both sides. $10x-x=4.444\ldots-0.444\ldots$ The equal decimal tails cancel, leaving $9x=4$

$$x=\frac49$$

Tip: the answer is exact, not 0.444 rounded.

Let x = 0.363636….

$$100x=36.363636\ldots$$ $$x=0.363636\ldots$$ $$99x=36$$ $$x=\frac{36}{99}=\frac4{11}$$

Two repeating digits require a shift of two places.

The 2 occurs once; only 7 repeats. Let x = 0.2777….

$$10x=2.777\ldots$$ $$100x=27.777\ldots$$

Subtract 10x from 100x.

$$90x=25$$ $$x=\frac{25}{90}=\frac5{18}$$

Tip: x and 10x do not have matching tails here.

Let x = 0.1232323…. First shift the non-repeating digit left.

$$10x=1.232323\ldots$$

Then shift by one whole two-digit block.

$$1000x=123.232323\ldots$$ $$990x=122$$ $$x=\frac{122}{990}=\frac{61}{495}$$

Do not treat the block as 123.

Let x = 1.181818….

$$100x=118.181818\ldots$$ $$x=1.181818\ldots$$ $$99x=117$$ $$x=\frac{117}{99}=\frac{13}{11}$$

The improper fraction is sensible because the starting number exceeds 1.

5 ÷ 12 starts with 0.4 because 50 tenths ÷ 12 gives 4 tenths and a remainder of 2 tenths. Bring down a zero: 20 hundredths ÷ 12 gives 1 hundredth and a remainder of 8 hundredths. Thereafter 80 ÷ 12 gives 6 with remainder 8, so the same remainder repeats.

$$\frac5{12}=0.416666\ldots=0.41\overline6$$

Only 6 repeats.