Recurring Decimals

Turn a recurring decimal into a fraction with algebra: call it x, multiply by 10, 100 or 1000 so the repeating tail lines up, then subtract.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 32 real past-paper questions below

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What a Recurring Decimal Is

Some fractions never stop as decimals. $5 \div 11$ gives 0.454545... and the 45 repeats for ever. This is a recurring decimal.

On the exam paper, dots sit over the digits that repeat. One dot: that one digit repeats. Two dots: everything from the first dot to the second dot repeats, including any digits between them.

Dots over	Written out	What repeats
the 4 and the 5 in 0.45	$0.454545 \dots$	the 45
only the 7 in 0.17	$0.17777 \dots$	the 7, after the 1
the 1 and the 8 in 0.318	$0.3181818 \dots$	the 18, after the 3
the 1 and the 2 in 0.142	$0.142142 \dots$	the 142: the 4 between the dots repeats too

Memory trick

Know these by heart: $\frac{1}{3} = 0.333 \dots$ , $\frac{2}{3} = 0.666 \dots$ , $\frac{1}{9} = 0.111 \dots$ .

Key idea

Puzzle: is $0.999 \dots$ a tiny bit less than 1? We know $\frac{1}{3} = 0.333 \dots$ . Multiply both sides by 3: $1 = 0.999 \dots$ . So $0.999 \dots$ is exactly 1.

Prove It with Algebra

The 2-mark question: "Use algebra to show that $0.454545 \dots = \frac{5}{11}$ ." The 45 repeats. Line up the repeating tails, then subtract so they vanish.

Rule

Count the digits in the repeating block. One digit: multiply by 10. Two digits: by 100. Three digits: by 1000.

Worked example

1
$x = 0. 454545 \dots$ — Call the number x. The 45 repeats, so the repeating block has 2 digits.
2
$100 x = 45. 454545 \dots$ — Multiply both sides by 100, because the block has 2 digits. The point moves past one whole block, so the tail is still 4545….
3
$100 x - x = 45. 4545 \dots - 0. 4545 \dots$ — Take the first line from the second. The tails are identical, so they cancel.
4
$99 x = 45$ — 100x take away x is 99x. On the right only the whole number 45 is left.
5
$x = \frac{45}{99}$ — Divide both sides by 99 to get x on its own.
6
$x = \frac{5}{11}$ — Simplify: 9 goes into 45 and 99. This is the fraction we had to show, so the proof is complete.

Common mistake

Multiplying by 10. Then $10 x = 4.5454 \dots$ and the tail is 5454, not 4545, so nothing cancels. Match the length of the repeating block.

When the Repeat Starts Late

The most common 2-mark question: the repeat starts late, like $0.3181818 \dots$ . The 18 repeats, but the 3 does not. Make two numbers whose tails match, then subtract.

Worked example

1
$x = 0.3 181818 \dots$ — Call the number x. The 18 repeats, but the 3 in front does not.
2
$10 x = 3. 181818 \dots$ — Multiply by 10 to move the point past the 3 only. Now the tail after the point is just the repeat.
3
$1000 x = 318. 1818 \dots$ — Multiply by 1000 to move the point past the 3 and one whole 18. The tail is the same repeat again.
4
$1000 x - 10 x = 318. 1818 \dots - 3. 1818 \dots$ — Take the 10x line from the 1000x line. The tails match, so they cancel.
5
$990 x = 315$ — 1000x − 10x is 990x, and 318 − 3 is 315.
6
$x = \frac{315}{990}$ — Divide both sides by 990 to get x on its own.
7
$x = \frac{7}{22}$ — Simplify: divide top and bottom by 45. Exams want the simplest form.

Common mistake

Using $x$ and $100 x$ . $x = 0.31818 \dots$ has 31818… after the point, but $100 x = 31.818 \dots$ has 818…. They do not match, so nothing cancels. Both lines must have the same digits after the point.

Key idea

Check on a calculator: $7 \div 22 = 0.3181818 \dots$ . But most of the marks are for the algebra. A right answer with no working scores 1 mark at most, so write every line.

Shortcut: The 9s and 0s Rule

For a 1-mark “write as a fraction” question, or to check your algebra, there is a shortcut. It is the same subtraction, done in your head.

Formula

fraction = \frac{N - P}{9 \dots 9 0 \dots 0}

Symbol	What it means
$N$	the digits after the point, up to the end of the first repeating block, read as one whole number
$P$	the digits after the point that do not repeat (0 if the repeat starts straight away)
$9 \dots 90 \dots 0$	one 9 for each repeating digit, then one 0 for each digit that does not repeat

Worked example

1
$x = 0.2 3636 \dots$ — The 36 repeats; the 2 in front does not.
2
$N = 236 = 236$ — Read the digits up to the end of the first 36.
3
$P = 2$ — Read the digits before the repeat starts: just the 2.
4
$N - P = 236 - 2 = 234$ — This is the top. It is the same sum as 1000x − 10x, so the tails have already cancelled.
5
$990$ — This is the bottom: two 9s for the two repeating digits, then one 0 for the 2.
6
$x = \frac{234}{990}$ — Put the top over the bottom.
7
$x = \frac{13}{55}$ — Simplify: divide top and bottom by 18.

Key idea

When the repeat starts straight away there are no 0s: $0.070707 \dots = \frac{7}{99}$ .

Common mistake

If the question says “use algebra” or “show all your working”, the shortcut alone scores 1 mark at most. Write the x, 10x, 1000x lines, and use the shortcut only to check.

Recurring Decimal Plus a Fraction

Some questions add a recurring decimal to a fraction, like $0.4777 \dots + \frac{2}{9}$ . A never-ending decimal cannot be added exactly, so change it to a fraction first.

Worked example

1
$x = 0.4 777 \dots$ — Call the decimal x. Only the 7 repeats; the 4 does not.
2
$10 x = 4. 777 \dots$ — Multiply by 10 to move the point past the 4. The tail is now just the repeat.
3
$100 x = 47. 777 \dots$ — Multiply by 100 to move the point past the 4 and one 7. Same tail again.
4
$100 x - 10 x = 47 - 4$ — Subtract the two lines. The tails cancel, leaving 47 − 4.
5
$90 x = 43$ — 100x − 10x is 90x, and 47 − 4 is 43.
6
$x = \frac{43}{90}$ — Divide by 90. This is 0.4777… as a fraction.
7
$\frac{2}{9} = \frac{20}{90}$ — Make the bottoms match: 9 × 10 = 90, so multiply top and bottom by 10.
8
$\frac{43}{90} + \frac{20}{90} = \frac{63}{90}$ — The bottoms are the same, so add the tops: 43 + 20 = 63.
9
$\frac{63}{90} = \frac{7}{10}$ — Simplify: divide top and bottom by 9. Check: 0.4777… + 0.2222… = 0.7.

Key idea

Most of the marks are for changing the decimal into a fraction. Show those lines, even if the adding goes wrong.

Key idea

If the question asks for a decimal answer instead, write the repeat out to more places than you need, then work: $0.636363 \dots - 0.2 = 0.436363 \dots$ , which is 0.43636 to 5 decimal places.

In Terms of a Letter (Higher)

The 3-mark question: $k$ is a digit from 1 to 9. Write $0.2 k k k \dots$ as a fraction, where the digit k repeats. The method is exactly the same.

Worked example

1
$x = 0.2 k k k \dots$ — Call the number x. The digit k repeats; the 2 in front does not.
2
$10 x = 2 + 0. k k k \dots$ — Multiply by 10 to move the point past the 2. The tail after the point is now just kkk….
3
$100 x = 20 + k + 0. k k k \dots$ — Multiply by 100 to move the point past the 2 and one k. In front of the point, the 2 is worth 20 and the k is worth k.
4
$100 x - 10 x = (20 + k) - 2$ — Take the 10x line from the 100x line. The tails are the same, so they cancel.
5
$90 x = 18 + k$ — 100x − 10x is 90x, and 20 − 2 is 18. The k stays as it is.
6
$x = \frac{18 + k}{90}$ — Divide both sides by 90. This is the fraction in terms of k.

Key idea

Test it with $k = 5$ : $\frac{23}{90} = 0.2555 \dots$ . It works.

Which Method? Read the Question

Look at what you are given and what you must give back.

You see	You do
Dots from the first digit	x, then 10x, 100x or 1000x, subtract
Dots start later	two multiples of x with matching tails
A whole number in front, like 2.3636…	same method, the whole part comes along; give a mixed number if the question shows one
A recurring decimal plus a fraction	change the decimal to a fraction first, then add with a common bottom
1 mark, no working asked	the 9s and 0s shortcut is enough
A fraction to write as a decimal	Converting Fractions to Decimals
A decimal that stops, like 0.45	Converting Decimals to Fractions

Key idea

"Use algebra to show" means the working is the answer. Always write the lines for x, the multiple, and the subtraction.

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