Percentages

The word percent means “out of one hundred”.

\[ \% \text{ means } \div 100 \]

Percentages are just fractions with a denominator of 100.

Examples

\[ 25\% = \frac{25}{100} \]

\[ 40\% = \frac{40}{100} \]

\[ 12\% = \frac{12}{100} \]

Simplified fractions:

\[ 25\% = \frac{25}{100} = \frac{1}{4} \] \[ 40\% = \frac{40}{100} = \frac{2}{5} \] \[ 12\% = \frac{12}{100} = \frac{3}{25} \]

And as decimals:

\[ 25\% = 0.25 \] \[ 40\% = 0.40 \] \[ 12\% = 0.12 \]

Calculating a Score as a Percentage

This is the same as turning a fraction into a percentage.

Example

Fred attained 23 marks out of 40 for his test.

What was his score as a percentage?

\[ \frac{23}{40} \] \[ = \frac{23}{40} \times 100 \] \[ = 57.5\% \]

Sometimes it is easier to convert the fraction so the denominator becomes 100:

Examples

\[ \frac{18}{50} \] \[ = \frac{36}{100} \] \[ = 36\% \]

\[ \frac{12}{75} \] \[ = \frac{16}{100} \] \[ = 16\% \]

Calculating Percentages – Using Fractions

Convert the percentage into a fraction, then follow the rules for calculating fractions .

Example

Find 20% of £45

\[ 20\% = \frac{20}{100} \] \[ \frac{20}{100} \times 45 \] \[ = \frac{1}{5} \times 45 \] \[ = 9 \]

Calculating Percentages – Using Decimals

Convert the percentage to a decimal, then multiply.

Example

Find 20% of £45

\[ 20\% = 0.20 \] \[ 0.20 \times 45 = 9 \]

Calculating Percentages – Using Units

To find 1% of a value – divide by 100
To find 10% of a value – divide by 10
To find 100% of a value – divide by 1

The required percentage can then be built up.

Example

Find 20% of £45

\[ 10\% = \frac{45}{10} = 4.5 \] \[ 20\% = 4.5 + 4.5 = 9 \]

Useful tips:

To find 0.1%, divide 1% by 10
To find 1%, divide by 100
To find 2%, double 1%
To find 3%, add 2% and 1%
To find 4%, double 2%, or subtract 1% from 5%
To find 5%, halve 10% or multiply 1% by 5
To find 6%, add 5% and 1%
To find 7%, add 5% and 2%
To find 8%, add 5% and 3%, or subtract 2% from 10%
To find 9%, add 5% and 4%, or subtract 1% from 10%

To find 20%, double 10% or divide by 5 (since \(20\% = \frac{1}{5}\))
To find 25%, add 20% and 5% or divide by 4 (since \(25\% = \frac{1}{4}\))
To find 30%, multiply 10% by 3
To find \(33\frac{1}{3}\%\), divide by 3 (since \(33\frac{1}{3}\% = \frac{1}{3}\))
To find 40%, multiply 10% by 4 or use \(40\% = \frac{2}{5}\)
To find 50%, divide by 2 or multiply 10% by 5
To find 60%, multiply 10% by 6 or add 50% and 10%
To find \(66\frac{2}{3}\%\), divide by 3 then multiply by 2
To find 70%, multiply 10% by 7 or add 50% and 20%
To find 75%, add 50% and 25% or use \(75\% = \frac{3}{4}\)
To find 80%, multiply 10% by 8 or subtract 20%
To find 90%, multiply 10% by 9 or subtract 10%

Examples

\[ \text{Find } 15\% \text{ of } 260 \] \[ 10\% = 26 \] \[ 5\% = 13 \] \[ 15\% = 26 + 13 = 39 \]

\[ \text{Find } 48\% \text{ of } 350 \] \[ 50\% = 175 \] \[ 2\% = \frac{175}{50} = 7 \] \[ 48\% = 175 - 7 = 168 \]

Value Added Tax

VAT is a tax added to most goods and services in the UK.

VAT is currently 20%, but used to be 17.5%.

To calculate VAT at 20%, divide the value by 10 and double.

Example

A lawnmower costs £450 ex VAT.

What is the price inclusive of VAT?

\[ 10\% \text{ of } £450 = 450 \div 10 = £45 \] \[ 20\% = 2 \times 10\% \] \[ \text{so } 20\% \text{ of } £450 = 2 \times £45 \] \[ \text{VAT } = £90 \] \[ \text{Price to pay } = £450 + £90 \] \[ = £540 \]

The price inclusive of VAT is £540

VAT at 17.5% is very easy to calculate:

17.5% = 10% + 5% + 2.5%

Divide by 10 to get 10%
Half that answer to get 5%
Half that answer again to get 2.5%
Add all three up

Example

Calculate the VAT due at 17.5% on an article costing £360.

\[ 10\% = \frac{360}{10} = 36 \] \[ 5\% = \frac{36}{2} = 18 \] \[ 2.5\% = \frac{18}{2} = 9 \] \[ \text{VAT} = 36 + 18 + 9 = 63 \]

VAT due is £63

Reverse Percentage VAT

Calculating Percentages – Using the Calculator

Save yourself effort – divide the number in front of the percentage sign by 100.

\[ 100\% = 1 \] \[ 10\% = 0.1 \] \[ 1\% = 0.01 \] \[ 0.1\% = 0.001 \]

Examples

\[ 90\% \text{ of } £34{,}567 \] \[ = 0.9 \times 34{,}567 \] \[ = £31{,}110.30 \]

\[ 235.6\% \text{ of } £34{,}567 \] \[ = 2.356 \times 34{,}567 \] \[ = £81{,}439.85 \]