Is 0.1 6 the same as 0. 1 6?

No. 0.1 6 = 0.1666 but 0. 1 6 = 0.161616

N2, N10NumberFoundation & HigherGrades 2–7

Decimals

Q: Where do the dots go on a recurring decimal?

Over the first and last digits of the repeating block. 0. 12 3 means 0.123123; a single dot means one digit repeats.

Q: How do I convert a decimal to a fraction?

For a terminating decimal, use place value: 0.375 = 3751000 = 38. For recurring decimals, use the algebraic method.

Decimals are fractions whose denominators are powers of ten. This lesson covers calculating with them by hand, converting between fractions, decimals and percentages, and the difference between decimals that stop and decimals that repeat for ever — including the Higher skill of turning a recurring decimal back into an exact fraction.

40 min Video by GCSE Maths AQA GCSE Maths

The big picture

Every fraction becomes either a terminating decimal ( $\frac{3}{8} = 0.375$ ) or a recurring one ( $\frac{2}{11} = 0.181818\ldots$ ). Which one depends only on the prime factors of the denominator (in its simplest form): if they are just 2s and 5s, the decimal terminates.

Decimal calculations reduce to whole-number calculations. Multiply $0.6 \times 0.04$ as $6 \times 4 = 24$ , then place the decimal point by counting decimal places — and let an estimate confirm it.

Multiplying Decimals — GCSE RevisionCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Add, subtract, multiply and divide decimals without a calculator
Convert between fractions, decimals and percentages
Identify terminating and recurring decimals
Convert recurring decimals to fractions (Higher)

Calculating with decimals

Adding and subtracting: line up the decimal points, fill gaps with zeros, and use column methods.

Multiplying: ignore the decimal points, multiply the whole numbers, then give the answer the same total number of decimal places as the question.

Dividing: multiply both numbers by the same power of 10 so that you divide by a whole number. $7.2 \div 0.08 = 720 \div 8 = 90$ .

Example 1Work out

0.36 \times 0.5

36 \times 5 = 180

2Decimal places in the question:

2 + 1 = 3

. So

0.180

Answer

0.18

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

Work out $1.4 \times 0.06$ .

Your answer

Show worked solution

Solution

1 $14 \times 6 = 84$ .
23 decimal places: $0.084$ .

Answer: $0.084$

Example 2Work out

4.56 \div 0.3

1Multiply both by 10:

45.6 \div 3

2Short division:

15.2

Answer

15.2

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

Work out $2.45 \div 0.07$ .

Your answer

Show hint

Multiply both numbers by 100.

Show worked solution

Solution

1 $\times 100$ : $245 \div 7$ .
2 $= 35$ .

Answer: $35$

Tip — Sense-check: dividing by a number less than 1 makes the answer bigger.

Fractions, decimals and percentages

Fraction to decimal: divide the numerator by the denominator, or find an equivalent fraction over 10, 100 or 1000: $\frac{7}{25} = \frac{28}{100} = 0.28$ .

Decimal to percentage: multiply by 100. Percentage to fraction: write over 100 and simplify: $35\% = \frac{35}{100} = \frac{7}{20}$ .

Know these by heart: $\frac{1}{2} = 0.5$ , $\frac{1}{4} = 0.25$ , $\frac{1}{5} = 0.2$ , $\frac{1}{8} = 0.125$ , $\frac{1}{3} = 0.\dot{3}$ , $\frac{2}{3} = 0.\dot{6}$ .

Example 1Write

\frac{3}{8}

0.4

and

36\%

in order of size, smallest first.

\frac{3}{8} = 0.375

and

36\% = 0.36

2Compare

0.375

0.4

0.36

Answer

36\%, \ \frac{3}{8}, \ 0.4

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

Write $\frac{9}{40}$ as a decimal.

Your answer

Show worked solution

Solution

1 $\frac{9}{40} = \frac{225}{1000}$ (multiply top and bottom by 25).
2 $= 0.225$ .

Answer: $0.225$

Terminating and recurring decimals

A terminating decimal stops ( $0.375$ ). A recurring decimal repeats for ever, shown with dots over the first and last repeating digits: $0.\dot{1}\dot{8} = 0.181818\ldots$ and $0.1\dot{6} = 0.1666\ldots$

A fraction in its simplest form terminates exactly when its denominator has no prime factors other than 2 and 5.

Example 1Without dividing, decide whether

\frac{7}{40}

and

\frac{5}{12}

terminate.

40 = 2^3 \times 5

— only 2s and 5s, so

\frac{7}{40}

terminates (

0.175

12 = 2^2 \times 3

contains a 3, so

\frac{5}{12}

recurs (

0.41\dot{6}

Answer

\frac{7}{40}

terminates;

\frac{5}{12}

recurs

Why only 2s and 5s

A decimal is a fraction over 10, 100, 1000… and those numbers are made only of 2s and 5s ( $1000 = 2^3 \times 5^3$ ). A denominator with any other prime factor, like 3, can never be scaled up to a power of 10.

Recurring decimals to fractions (Higher)

Let $x$ equal the decimal. Multiply by a power of 10 so the repeating part lines up — by 10 for one repeating digit, 100 for two, and so on. Subtract to remove the repeating tail, then solve for $x$ .

Example 1Write

0.\dot{4}\dot{5}

as a fraction in its simplest form.

x = 0.454545\ldots

, so

100x = 45.454545\ldots

2Subtract:

99x = 45

x = \frac{45}{99} = \frac{5}{11}

Answer

\frac{5}{11}

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

Write $0.\dot{2}\dot{7}$ as a fraction in its simplest form.

Your answer

Type a fraction like 5/11.

Show worked solution

Solution

1 $x = 0.2727\ldots$ , $100x = 27.2727\ldots$
2 $99x = 27$ , so $x = \frac{27}{99} = \frac{3}{11}$ .

Answer: $\frac{3}{11}$

Example 2Write

0.3\dot{8}

as a fraction in its simplest form.

x = 0.3888\ldots

10x = 3.888\ldots

100x = 38.888\ldots

2Subtract

100x - 10x

90x = 35

x = \frac{35}{90} = \frac{7}{18}

Answer

\frac{7}{18}

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

Write $0.1\dot{6}$ as a fraction in its simplest form.

Your answer

Type a fraction like 5/11.

Show hint

Use $100x - 10x$ so the repeating 6s cancel.

Show worked solution

Solution

1 $10x = 1.666\ldots$ and $100x = 16.666\ldots$
2 $90x = 15$ , so $x = \frac{15}{90} = \frac{1}{6}$ .

Answer: $\frac{1}{6}$

Think like an examiner

"Prove algebraically that $0.\dot{4}\dot{5} = \frac{5}{11}$ " requires the $x = \ldots$ , $100x = \ldots$ subtraction written out — just dividing 5 by 11 is not a proof.
On the non-calculator paper, show the whole-number calculation and how you placed the decimal point.
When ordering mixed fractions, decimals and percentages, show every value converted to the same form.

Key facts

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

Two repeating digits: multiply by 100 and subtract.

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

For a fraction in simplest form.

\frac{1}{8} = 0.125 = 12.5\%

Fraction, decimal, percentage.

Watch out for these

0.3 \times 0.2 = 0.6

Two decimal places:

0.06

6 \div 0.2 = 1.2

Dividing by a number less than 1 gives a bigger answer:

60 \div 2 = 30

0.\dot{4}\dot{5}

10x - x = 9x = 4.1

Two digits repeat, so use

100x - x = 99x = 45

\frac{1}{3} = 0.33

0.33

is a rounded value; exactly,

\frac{1}{3} = 0.\dot{3}

Stretch yourself

Show that $0.\dot{9} = 1$ .

Hint — Treat it exactly like any other recurring decimal.

Reveal solution

SolutionLet

x = 0.999\ldots

Then

10x = 9.999\ldots

Subtracting,

9x = 9

, so

x = 1

. The two decimals describe the same number — there is no gap between

0.\dot{9}

and 1.

\blacksquare

Questions students ask

Where do the dots go on a recurring decimal?

Over the first and last digits of the repeating block.

0.\dot{1}2\dot{3}

means

0.123123\ldots

; a single dot means one digit repeats.

How do I convert a decimal to a fraction?

For a terminating decimal, use place value:

0.375 = \frac{375}{1000} = \frac{3}{8}

. For recurring decimals, use the algebraic method.

0.1\dot{6}

the same as

0.\dot{1}\dot{6}

No.

0.1\dot{6} = 0.1666\ldots

but

0.\dot{1}\dot{6} = 0.161616\ldots

Key takeaways

Line up decimal points to add and subtract.
Multiply as whole numbers, then count decimal places.
Divide by making the divisor a whole number.
A simplified fraction terminates only if its denominator has prime factors 2 and 5.
Recurring to fraction: $x = \ldots$ , multiply by $10^n$ , subtract, solve.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 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 33 marks

(a)Work out $0.7 \times 0.08$ .[1]
Show mark scheme (a)
- B1 $0.056$
Answer: $0.056$
(b)Work out $3.9 \div 0.13$ .[2]
Show mark scheme (b)
- M1Converts to $390 \div 13$ or equivalent
- A1 $30$
Answer: $30$

Question 2Grade 42 marks

Write these in order of size, starting with the smallest: $\quad \frac{5}{8}, \ 0.6, \ 63\%, \ \frac{2}{3}$ [2]
Show mark scheme
- M1Converts at least three to a common form, e.g. $0.625, 0.6, 0.63, 0.666\ldots$
- A1 $0.6, \ \frac{5}{8}, \ 63\%, \ \frac{2}{3}$

Question 3Grade 73 marks

Prove algebraically that $0.5\dot{1}\dot{8} = \frac{57}{110}$ .[3]
Show mark scheme
- M1Two correct recurring decimals that differ by a whole number of repeats, e.g. $1000x = 518.1818\ldots$ and $10x = 5.1818\ldots$
- M1Subtracts: $990x = 513$
- A1 $x = \frac{513}{990} = \frac{57}{110}$ , shown fully

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

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Work out $3.8 + 2.45$ .
- $6.25$
- $5.25$
- $6.33$
- $2.83$
Show answer and explanation
Answer: $6.25$
Line up the decimal points.
Write $\frac{3}{4}$ as a decimal.
- $0.75$
- $0.34$
- $3.4$
- $0.7$
Show answer and explanation
Answer: $0.75$
$3 \div 4 = 0.75$ .
Work out $0.4 \times 0.3$ .
- $0.12$
- $1.2$
- $0.012$
- $0.7$
Show answer and explanation
Answer: $0.12$
$4 \times 3 = 12$ , two decimal places.
Work out $1.8 \div 0.06$ .
- $30$
- $0.3$
- $3$
- $300$
Show answer and explanation
Answer: $30$
$180 \div 6$ .
Which fraction gives a recurring decimal?
- $\frac{1}{6}$
- $\frac{1}{8}$
- $\frac{3}{20}$
- $\frac{7}{25}$
Show answer and explanation
Answer: $\frac{1}{6}$
$6 = 2 \times 3$ has a factor of 3.
Write $0.045$ as a percentage.
- $4.5\%$
- $45\%$
- $0.45\%$
- $0.045\%$
Show answer and explanation
Answer: $4.5\%$
Multiply by 100.
Write $0.\dot{7}$ as a fraction.
- $\frac{7}{9}$
- $\frac{7}{10}$
- $\frac{77}{100}$
- $\frac{7}{99}$
Show answer and explanation
Answer: $\frac{7}{9}$
$10x - x = 9x = 7$ .
Write $0.\dot{1}\dot{2}$ as a fraction in simplest form.
- $\frac{4}{33}$
- $\frac{12}{100}$
- $\frac{3}{25}$
- $\frac{1}{12}$
Show answer and explanation
Answer: $\frac{4}{33}$
$99x = 12$ , so $x = \frac{12}{99} = \frac{4}{33}$ .

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