Yes. Using the method, x = 0.999, 10x = 9.999, so 9x = 9 and x = 1.

1MA1 · N2, N10NumberFoundation & HigherGrades 3–7

Decimals & Recurring Decimals

Some fractions become neat decimals — $\tfrac38 = 0.375$ — while others never stop — $\tfrac{5}{11} = 0.454545\dots$ These $\textbf{recurring decimals}$ repeat forever in a fixed pattern, and a short piece of algebra converts them back into exact fractions. First, though, the everyday skills: calculating with decimals and moving between decimals and fractions.

45 min Number Fundamentals

The big picture

A decimal is just a fraction with a denominator of 10, 100, 1000 and so on. Whether a fraction terminates depends only on its denominator: in simplest form, it terminates exactly when the denominator’s only prime factors are 2 and 5.

Converting a recurring decimal to a fraction is a Higher-tier favourite on Edexcel papers, usually worth three marks, and it rewards a precise, well-laid-out method more than cleverness.

What you'll learn

Add, subtract, multiply and divide decimals without a calculator
Convert between fractions, decimals and percentages
Explain which fractions give terminating decimals
Convert recurring decimals to fractions using algebra (Higher)

Calculating with decimals

To add or subtract, line up the decimal points and fill gaps with zeros: $12.4 - 3.75 = 12.40 - 3.75 = 8.65$ .

To multiply, ignore the points, multiply as whole numbers, then give the answer as many decimal places as the question numbers have in total: $0.23 \times 1.4$ has $2 + 1 = 3$ decimal places, and $23 \times 14 = 322$ , so the answer is $0.322$ .

To divide by a decimal, multiply both numbers by a power of 10 until the divisor is whole.

Example 1Work out

0.068 \div 0.004

1Multiply both by 1000:

68 \div 4

Answer

17

Your turnSame method, new numbers — try it before moving on.

Work out $0.45 \div 0.009$ .

Your answer

Show worked solution

Solution

1Multiply both by 1000: $450 \div 9$ .
2 $= 50$ .

Answer: $50$

Fractions to decimals and back

A fraction is a division: $\tfrac{7}{8} = 7 \div 8 = 0.875$ . If the denominator is a factor of a power of 10, scale it instead: $\tfrac{7}{25} = \tfrac{28}{100} = 0.28$ .

A terminating decimal becomes a fraction over a power of 10, then simplifies: $0.625 = \tfrac{625}{1000} = \tfrac58$ .

A fraction in simplest form $\textbf{terminates}$ exactly when its denominator has no prime factors other than 2 and 5. $\tfrac{3}{40}$ terminates ( $40 = 2^{3} \times 5$ ); $\tfrac{3}{14}$ recurs ( $14 = 2 \times 7$ ).

Example 1Without dividing, decide which of

\tfrac{9}{20}

\tfrac{5}{12}

and

\tfrac{11}{16}

are terminating decimals.

20 = 2^{2} \times 5

: only 2s and 5s, so

\tfrac{9}{20}

terminates.

12 = 2^{2} \times 3

: contains 3, and

\tfrac{5}{12}

is already simplified, so it recurs.

16 = 2^{4}

: terminates.

Answer

\tfrac{9}{20}

and

\tfrac{11}{16}

terminate;

\tfrac{5}{12}

recurs.

Why this works

The test works because terminating decimals are fractions over powers of 10, and $10 = 2 \times 5$ . A factor of 3 or 7 in the denominator can never be cancelled into a power of 10.

Recurring decimal notation

A dot over a digit shows it repeats: $0.\dot{6} = 0.666\dots$

Dots over the first and last digits of a block show the whole block repeats: $0.\dot{1}2\dot{3} = 0.123123\dots$

Only part of a decimal may recur: $0.4\dot{3} = 0.4333\dots$

Tip — Write out at least two cycles of the repeat before starting a conversion. It makes it clear which power of 10 to multiply by.

Recurring decimals to fractions

Let $x$ equal the recurring decimal. Multiply by the power of 10 that shifts one full block of repeats past the decimal point — $10$ for one repeating digit, $100$ for two, $1000$ for three.

Subtract the original equation. The infinite tails cancel exactly, leaving a simple equation to solve.

If a non-repeating digit comes first, multiply twice so that both equations have the same recurring tail.

Example 1Show that

0.\dot{3}\dot{6} = \tfrac{4}{11}

1Let

x = 0.363636\dots

2Then

100x = 36.363636\dots

3Subtract:

99x = 36

, so

x = \tfrac{36}{99}

4Simplify by dividing by 9.

Answer

x = \tfrac{4}{11}

Your turnSame method, new numbers — try it before moving on.

Write $0.\dot{2}\dot{4}$ as a fraction in its simplest form.

Your answer

Type a fraction like 5/11.

Show worked solution

Solution

1 $100x - x = 24$ , so $99x = 24$ .
2 $x = \frac{24}{99} = \frac{8}{33}$ .

Answer: $\frac{8}{33}$

Example 2Write

0.4\dot{3}

as a fraction in its simplest form.

1Let

x = 0.4333\dots

10x = 4.333\dots

and

100x = 43.333\dots

3Subtract:

90x = 39

, so

x = \tfrac{39}{90}

Answer

\tfrac{13}{30}

Your turnSame method, new numbers — try it before moving on.

Write $0.1\dot{8}$ as a fraction in its simplest form.

Your answer

Type a fraction like 5/11.

Show worked solution

Solution

1 $10x = 1.888\ldots$ and $100x = 18.888\ldots$
2 $90x = 17$ , so $x = \frac{17}{90}$ .

Answer: $\frac{17}{90}$

Tip — In a "show that" question, the final line must be the fraction asked for, with the simplification shown — $\tfrac{36}{99} = \tfrac{4}{11}$ .

Think like an examiner

Write $x = \dots$ with enough repeats shown, then the multiplied equation directly beneath.
Show the subtraction step and the unsimplified fraction before simplifying.
For decimal multiplication, show the whole-number calculation and the decimal-place count.

Decimals

0.\dot{a} = \tfrac{a}{9}

One repeating digit.

0.\dot{a}\dot{b} = \tfrac{ab}{99}

Two repeating digits.

\text{terminates} \iff \text{denominator} = 2^{m}5^{n}

For a fraction in simplest form.

Watch out for these

Multiplying

0.\dot{3}\dot{6}

by 10.

Two digits repeat, so multiply by 100 to align the repeats.

Writing

0.23 \times 1.4 = 3.22

Three decimal places in the question, so

0.322

Deciding

\tfrac{6}{15}

recurs because

15 = 3 \times 5

Simplify first:

\tfrac{6}{15} = \tfrac25

, which terminates.

Leaving

\tfrac{36}{99}

unsimplified.

Divide by the HCF:

\tfrac{4}{11}

Stretch yourself

Prove that $0.2\dot{1}\dot{8} = \tfrac{12}{55}$ .

Hint — One digit (2) comes before the repeating block (18). Multiply by 10 to reach the start of the repeat, and by 1000 to move one block further.

Reveal solution

SolutionLet

x = 0.2181818\dots

Then

10x = 2.181818\dots

and

1000x = 218.181818\dots

Subtracting,

990x = 216

, so

x = \dfrac{216}{990}

. Dividing top and bottom by 18 gives

\dfrac{12}{55}

, as required. Check:

12 \div 55 = 0.21818\dots

✓.

\blacksquare

Questions students ask

Are recurring decimals rational?

Yes. Every recurring or terminating decimal can be written as a fraction of integers, which is what the algebraic method proves.

0.\dot{9}

equal to 1?

Yes. Using the method,

x = 0.999\dots

10x = 9.999\dots

, so

9x = 9

and

x = 1

How many decimal places should my multiplication answer have?

The total number of decimal places in the numbers being multiplied, before removing any trailing zeros.

Key takeaways

Line up decimal points to add or subtract; count decimal places to multiply.
Divide by a decimal by first scaling to a whole-number divisor.
A simplified fraction terminates only if its denominator’s primes are 2 and 5.
Dots mark recurring digits or blocks.
Convert recurring decimals by multiplying by a power of 10 and subtracting.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

Work out $3.2 \times 0.15$ without a calculator.[2]
Show mark scheme
- M1 $32 \times 15 = 480$
- A1 $0.48$
Answer: $0.48$

Question 2Grade 52 marks

One of $\frac{7}{40}$ and $\frac{4}{15}$ is a terminating decimal. Which one? Explain how you know without dividing.[2]
Show mark scheme
- B1 $\frac{7}{40}$
- B1 $40 = 2^3 \times 5$ has only 2 and 5 as prime factors, but $15$ has a factor of 3

Question 3Grade 73 marks

Prove algebraically that $0.2\dot{7} = \frac{5}{18}$ .[3]
Show mark scheme
- M1 $10x = 2.777\ldots$ and $100x = 27.777\ldots$
- M1 $90x = 25$
- A1 $\frac{25}{90} = \frac{5}{18}$ shown

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

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