Decimals

Spec reference: Number - Fractions, decimals and percentages
Key idea: Calculate with decimals, including addition, subtraction, multiplication and division.

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Adding and subtracting decimals

Line up the decimal points and add/subtract as normal.

Example

Calculate $12.73+5.8$

$$\begin{array}{r}12.73\\ +5.80\\ 18.53\end{array}$$

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Multiplying decimals

Ignore the decimal point, multiply as integers, then put the decimal point back in.

Count total decimal places in both numbers - the answer has the same total.

Example

Calculate $0.4\times 0.7$

$4\times 7=28$

Total decimal places: 1 + 1 = 2

$$0.4\times 0.7=\boxed{{\displaystyle 0.28}}$$

Example

Calculate $3.6\times 1.2$

$36\times 12=432$

Total decimal places: 1 + 1 = 2

$$3.6\times 1.2=\boxed{{\displaystyle 4.32}}$$

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Dividing decimals

Dividing by a decimal: multiply both numbers by a power of 10 to make the divisor a whole number.

Example

Calculate $5.4÷0.6$

Multiply both by 10: $54÷6=\boxed{{\displaystyle 9}}$

Example

Calculate $3.6÷0.04$

Multiply both by 100: $360÷4=\boxed{{\displaystyle 90}}$

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Recurring decimals

A recurring decimal has digits that repeat forever.

Fraction	Decimal
${\displaystyle \frac{1}{3}}$	$0.\bar{3}=0.333\dots$
${\displaystyle \frac{1}{6}}$	$0.1\bar{6}=0.1666\dots$
${\displaystyle \frac{1}{7}}$	$0.\overline{142857}$
${\displaystyle \frac{1}{9}}$	$0.\bar{1}=0.111\dots$
${\displaystyle \frac{2}{3}}$	$0.\bar{6}=0.666\dots$

TIP

A dot over a digit means it recurs: $0.\dot{3}=0.333\dots$
A dot over the first and last digit of a group means that group recurs: $0.\dot{1}\dot{4}=0.141414\dots$

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Converting recurring decimals to fractions

Example

Convert $0.\bar{7}$ to a fraction.

Let $x=0.777\dots$

$10x=7.777\dots$

$10x-x=7$

$9x=7$

$$x=\boxed{{\displaystyle \frac{7}{9}}}$$

Example

Convert $0.\overline{27}$ to a fraction.

Let $x=0.272727\dots$

$100x=27.272727\dots$

$100x-x=27$

$99x=27$

$$x=\frac{27}{99}=\boxed{{\displaystyle \frac{3}{11}}}$$

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Exam tips

Watch out for

• Always line up decimal points when adding/subtracting
• Count decimal places carefully when multiplying
• $0.\bar{9}=1$ exactly - this is a known result

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Test Yourself

Question 1 of 5

Calculate $0.3 \times 0.7$