Percentage meaning & conversions — notes, question types & practice · Percentage

01

What per cent means

Per cent means "out of every hundred". $40\%$ means 40 parts taken out of every 100 parts.

$40\% = \dfrac{40}{100} = \dfrac{2}{5}$
$50\%$ is half, $100\%$ is the whole thing
$200\%$ is double

Rule: Before any calculation, write the percent as a fraction with 100 below. Then cancel.

02

The fraction table

Exams reuse the same fractions again and again. Learn this table by heart; it saves a multiplication in almost every question.

Fraction	Percent	Fraction	Percent
$1/2$	50%	$1/8$	12.5%
$1/3$	33 1/3%	$1/9$	11 1/9%
$1/4$	25%	$1/11$	9 1/11%
$1/5$	20%	$1/12$	8 1/3%
$1/6$	16 2/3%	$1/15$	6 2/3%
$1/7$	14 2/7%	$1/16$	6.25%
$1/20$	5%	$1/25$	4%

Half the table comes free: $1/6$ is half of $1/3$ , so $16\dfrac{2}{3}\%$ is half of $33\dfrac{1}{3}\%$ .

Fraction to percent: multiply by 100. $\dfrac{5}{16} \times 100 = 31.25\%$ .
Percent to fraction: put over 100 and cancel. $12.5\% = \dfrac{125}{1000} = \dfrac{1}{8}$ .

03

Finding a percentage of a number

$x\%$ of $N$ means $\dfrac{x}{100} \times N$ .

$35\%$ of $200 = \dfrac{35}{100} \times 200 = 70$ .

Use the table when the percent is in it. $12.5\%$ is $\dfrac{1}{8}$ , so $12.5\%$ of $960 = 960 \div 8 = 120$ . No big multiplication needed.

Example: $16\dfrac{2}{3}\%$ of $720$ : the percent is $\dfrac{1}{6}$ , so the answer is $720 \div 6 = 120$ .

04

The 10% and 1% building blocks

Split an awkward percent into pieces of $10\%$ and $1\%$ .

$10\%$ of a number: move the decimal point one place left.
$1\%$ of a number: move it two places left.
Add pieces until you build the percent you need.

$24\%$ of $650$ : two $10\%$ pieces give $130$ , four $1\%$ pieces give $26$ , total $156$ .

Tip: $x\%$ of $y$ always equals $y\%$ of $x$ . So $4\%$ of $25$ equals $25\%$ of $4$ , which is clearly $1$ .

05

Finding the whole from a part

Here the question gives a piece of a number and its percent. The full number is asked.

Rule: whole $=$ part $\times\ 100 \div$ percent.

$35\%$ of a number is $189$ . So the number is $189 \times \dfrac{100}{35} = 540$ . Then $80\%$ of it is $\dfrac{4}{5} \times 540 = 432$ .

Sanity check: the whole (540) must be bigger than the part (189), because $35\%$ is less than the whole.

06

One quantity as a percentage of another

"A is what per cent of B?" means $\dfrac{A}{B} \times 100$ . The quantity after "of" goes below the line.

In a batch of 720 applications, 126 were rejected. Rejected applications are $\dfrac{126}{720} \times 100 = 17.5\%$ of the batch.

Watch: The number after "of" is the base. Swap the two numbers and the answer changes.

07

The remaining part of a hundred

Percentages of the same whole add up to $100\%$ .

Out of 240 students, $62.5\%$ attended a function. So $37.5\%$ were absent, and $37.5\% = \dfrac{3}{8}$ . Absent students $= \dfrac{3}{8} \times 240 = 90$ .

Note: Equal-percent statements also link two wholes. If $40\%$ of A $= 30\%$ of B, then $0.4A = 0.3B$ , so $A = 75\%$ of B.

08

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Find x% of a number

How to spot it:

A number is given and a percentage of it is asked: 'What is 24% of 650?'

\frac{x}{100} \times N

Method

Turn the percent into a fraction over 100.
Cancel it to a table fraction when possible.
Multiply by the number.
Check size: the answer should look like a fraction of the number.

Why it works:

A percent is only a fraction, so taking a percent means taking that fraction of the number.

Try this

What is 24% of 650?

Show solution

$24\%$ of $650$ : two $10\%$ pieces give $130$ .
Four $1\%$ pieces give $26$ .
$130 + 26 = 156$ .

Answer

156

Type 2very common2 practice Q

Find the number when a percent of it is given

How to spot it:

'35% of a number is 189.' The number itself, or a second percent of it, is asked.

\text{whole} = \frac{\text{part} \times 100}{x}

Method

Write whole = part x 100 / percent.
Cancel before multiplying.
Use this whole for any follow-up asked.

Why it works:

The part is a known fraction of the whole, so dividing by that fraction recovers the whole.

Try this

35% of a number is 189. What is 80% of that number?

Show solution

Whole $= 189 \times \dfrac{100}{35} = 540$ .
$80\%$ of $540 = \dfrac{4}{5} \times 540 = 432$ .

Answer

432

Type 3common2 practice Q

Change a fraction into a percentage

How to spot it:

'Express 5/16 as a percentage', or options are all in percent.

\frac{a}{b} \times 100\%

Method

Multiply the fraction by 100.
Cancel before you multiply.
Write the result as a decimal or mixed number.

Why it works:

Percent is a fraction with 100 below, so scaling the fraction to 100 gives the percent.

Try this

Express $\dfrac{5}{16}$ as a percentage.

Show solution

$\dfrac{5}{16} \times 100 = \dfrac{500}{16}$ .
$\dfrac{500}{16} = 31.25$ .

Answer

31.25%

Type 4common3 practice Q

One quantity as a percentage of another

How to spot it:

'126 out of 720 applications were rejected.' What percentage one quantity is of another is asked.

\frac{A}{B} \times 100\%

Method

Put the quantity being asked about on top.
Put the total, the quantity after 'of', below.
Multiply by 100 and simplify.

Why it works:

The percent says how many per hundred the top quantity is of the bottom one.

Try this

In a batch of 720 applications, 126 were rejected. What percentage of the applications was rejected?

Show solution

$\dfrac{126}{720} = \dfrac{7}{40}$ .
$\dfrac{7}{40} \times 100 = 17.5$ .

Answer

17.5%

Type 5common

The remaining percentage

How to spot it:

A part is described by a percent and the rest is asked: '62.5% attended. How many were absent?'

Method

Subtract the given percent from 100.
Turn the remainder into a table fraction.
Multiply by the total.

Why it works:

The two parts split the same total, so their percents add to 100.

Try this

Out of 240 students, 62.5% attended the annual function. How many students did not attend?

Show solution

Absent $= 100\% - 62.5\% = 37.5\% = \dfrac{3}{8}$ .
$\dfrac{3}{8} \times 240 = 90$ .

Answer

90 students

Type 6occasional

Equal percent statements

How to spot it:

'If 40% of A equals 30% of B...' Two different wholes are joined by one equation.

\frac{x}{100}A = \frac{y}{100}B \Rightarrow A = \frac{y}{x}B

Method

Drop the percents and write the equation: 0.4A = 0.3B.
Solve for the ratio A over B.
Turn the ratio into a percent.

Why it works:

Two equal products fix the ratio of A and B, so one percent of the other follows.

Try this

If 40% of A is equal to 30% of B, then A is what percentage of B?

Show solution

$0.4A = 0.3B$
$\dfrac{A}{B} = \dfrac{0.3}{0.4} = \dfrac{3}{4}$ .
$\dfrac{3}{4} \times 100 = 75$ .

Answer

75%

09

Formula sheet

x% of a number

\frac{x}{100} \times N

Turn the percent into a fraction first; cancel where you can.

Whole from a part

\text{whole} = \frac{\text{part} \times 100}{x}

x is the percent that the part stands for.

A as a percent of B

\frac{A}{B} \times 100\%

B is the quantity that follows the word 'of'.

Remaining part

\text{rest} = 100\% - \text{given}\%

Only when both percentages are of the same whole.

10

Shortcuts that save time

⚡ Let the fraction table do the division

Replace the percent with its fraction from the table. $12.5\%$ becomes $\dfrac{1}{8}$ , so the sum turns into a simple division.

Example

What is 12.5% of 960?

Show solution

$12.5\% = \dfrac{1}{8}$ .
$960 \div 8 = 120$ .

Answer

120

⚡ Build the percent from 10% and 1% pieces

Move the decimal point for $10\%$ and $1\%$ , then add pieces. Works without pen-and-paper multiplication.

Example

What is 24% of 650?

Show solution

$10\%$ of $650 = 65$ ; two pieces give $130$ .
$1\%$ of $650 = 6.5$ ; four pieces give $26$ .
$130 + 26 = 156$ .

Answer

156

⚡ Swap x% of y into y% of x

$x\%$ of $y$ always equals $y\%$ of $x$ . Swap when one side turns into a friendly percent.

Example

What is 4% of 25?

Show solution

Swap: $4\%$ of $25 = 25\%$ of $4$ .
$25\%$ of $4 = 1$ .

Answer

1

11

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

In 'A is what per cent of B', putting A below the line.

The quantity after 'of' is the base: $A \div B \times 100$ .

Mistake 02

Treating $16\dfrac{2}{3}\%$ as roughly $16.3\%$ .

It is exactly $\dfrac{1}{6}$ . Use the fraction from the table.

Mistake 03

Turning a fraction into a percent by dividing by 100.

Multiply by 100: $1/4$ becomes $25\%$ .

Mistake 04

Reading $0.5\%$ as $5\%$ .

$0.5\%$ is half of one per cent, that is $\dfrac{1}{200}$ of the number.

Mistake 05

Adding percents of different wholes in one line.

Percents add only when they are percents of the same whole.

Mistake 06

Finding the whole by multiplying the part by the percent.

Divide instead: whole $=$ part $\times 100 \div$ percent.

12

Quick revision

Read this the night before the exam.

Per cent means out of every hundred: $x\% = \dfrac{x}{100}$ .
$x\%$ of $N = \dfrac{x}{100} \times N$ .
Table: $1/8 = 12.5\%$ , $1/6 = 16\dfrac{2}{3}\%$ , $1/7 = 14\dfrac{2}{7}\%$ .
Whole $=$ part $\times 100 \div$ percent.
A as a per cent of B: $A \div B \times 100$ ; the number after 'of' stays below.
Percents of the same whole add to $100\%$ .
$x\%$ of $y = y\%$ of $x$ .

13

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.