Study notes

Percentage

Percentage means 'out of 100'. It lets us convert and compare quantities, describe change, and solve real-world money problems - profit and loss, discount, GST and commission.

Learn it step by step

Percentage means 'out of 100'

Per cent means out of 100, so $37% = \frac{37}{100}$ and $100% = 1$ . Any fraction or decimal becomes a percentage when you $\times 100%$ (so $0.72 = 72%$ ).

37%

is 37 shaded out of 100.

More than 100%, less than 1%

A percentage can be more than 100% ( $160% = 1.6$ , bigger than the whole) or less than 1% ( $0.5% = 0.005$ ). It is just a number scaled to hundredths.

160%

is longer than the whole.

A percentage of a quantity

To find a percentage of an amount, multiply: $25%$ of $40 = \frac{25}{100} \times 40 = 10$ .

25%

of the 40 is 10.

One quantity as a percentage of another

Divide, then $\times 100%$ , with both in the same unit: $\frac{12}{20} \times 100% = 60%$ . Mind the base - $x %$ of $A$ is usually not $x %$ of $B$ .

\frac{12}{20} = 60%

Percentage change

Change $= \frac{increase or decrease}{original} \times 100%$ ; the original is always the base. A $5%$ rise makes the new value $105%$ of the old.

$1600 \to $1680

is a

5%

rise.

Reverse percentage

If you know the amount after a change, scale back to $100%$ . After $+ 5%$ , $$2205 = 105%$ , so $1% = \frac{2205}{105}$ and $100% = $2100$ .

Scale

105%

back to the original

100%

Percentage point vs per cent

Going from $7%$ to $9%$ is a rise of 2 percentage points, but as a relative change it is $\frac{2}{7} \times 100% \approx 29%$ . They are different!

7% \to 9%

: 2 points, not 2%.

Percentage in real life

Profit/loss is measured against cost (or selling) price. Discount comes off the marked price - successive discounts do not add. GST ( $9%$ ) is added on top, even on the service charge. Commission is a percentage of the selling price.

Discount

= $160 - $100 = $60

Worked examples

There are 90 teachers, of whom 40 are male. Find the percentage that are female.

Female teachers $= 90 - 40 = 50$ .
As a percentage of all teachers: $\frac{50}{90} \times 100%$ .
$= 55 \frac{5}{9} %$ .

A man's monthly salary of $1600 rises by

5%

. Find the new salary.

New salary $= 105%$ of the old.
$= \frac{105}{100} \times 1600$ .
$= $1680$ .

After a

5%

rise, a salary is $2205. Find the original salary.

The new amount is $105%$ of the original: $105% = 2205$ .
$1% = \frac{2205}{105}$ , so $100% = \frac{2205}{105} \times 100$ .
Original salary $= $2100$ .

GST rose from

7%

9%

. Is that a

2%

increase?

It is a rise of 2 percentage points ( $9 - 7$ ).
As a relative increase: $\frac{2}{7} \times 100% \approx 29%$ .
So say 'up 2 percentage points', not 'up 2%'.

A bag costing $28 is sold for $35. Find the profit as a percentage of the cost price.

Profit $= 35 - 28 = 7$ .
As a percentage of cost: $\frac{7}{28} \times 100%$ .
$= 25%$ .

A $100 meal has a

10%

discount, then a

10%

service charge, then

9%

GST. Find the total.

After discount: $100 \times 0.9 = 90$ .
Add service charge: $90 \times 1.1 = 99$ .
Add GST: $99 \times 1.09 = $107.91$ .

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