Percentage Change

Every sale sign, pay rise and price hike is a percentage change: 20% off a jacket, rent up 5%, a 15% tip on a bill. Getting these right tells you exactly what you will pay or take home — so it is worth being quick and sure with them.

A percentage change means an amount grows or shrinks by some percent. The most important idea is this: the amount you start with is always the whole — it is 100\%. Every percentage you add or take away is measured against that starting amount.

There are two short steps. First, find the change — that is just the percentage of the original. Then, for an increase you add the change on top, and for a decrease you subtract it:

\text{change} = \frac{p}{100}\times\text{original}, \qquad \text{new} = \text{original} \pm \text{change}

Use + for a rise and - for a fall. For example, to increase \pounds 50 by 20\%: the change is \tfrac{20}{100}\times 50 = \pounds 10, so the new price is 50 + 10 = \pounds 60. To decrease it by 20\% instead, take the same \pounds 10 away: 50 - 10 = \pounds 40.

The quick way: one multiplier

Once you are comfortable, there is a faster trick. Because the original is 100\%, increasing by 20\% leaves you with 120\% of it, and decreasing by 20\% leaves 80\%. Written as decimals those are 1.20 and 0.80 — your single multiplier.

To increase by 20\%: multiply by 1.20.
To decrease by 20\%: multiply by 0.80.

\text{new} = \text{old} \times \left(1 \pm \tfrac{p}{100}\right)

So 50 \times 1.20 = \pounds 60 and 50 \times 0.80 = \pounds 40 — exactly the answers we got the long way.

Three worked examples

A jacket, 15% off. It costs \pounds 80. The discount is \tfrac{15}{100}\times 80 = \pounds 12, so you pay 80 - 12 = \pounds 68. (Quick way: 80 \times 0.85 = \pounds 68.)
A plant grows 30%. It was 20\text{ cm} tall. The growth is \tfrac{30}{100}\times 20 = 6\text{ cm}, so now it is 20 + 6 = 26\text{ cm}. (Quick way: 20 \times 1.30 = 26.)
Going backwards. A class grew from 200 to 250 children. The change is 50, and we compare it to the original: \tfrac{50}{200}\times 100\% = 25\%. So that is a 25\% increase.

That last example shows how to find a percentage change when you already know the old and new amounts — compare the change to the original:

\frac{\text{change}}{\text{original}} \times 100\%

Increase by p\%: multiply by \left(1 + \tfrac{p}{100}\right).
Decrease by p\%: multiply by \left(1 - \tfrac{p}{100}\right).
Find a percentage change: \dfrac{\text{change}}{\text{original}} \times 100.
One amount as a % of another: \dfrac{\text{part}}{\text{whole}} \times 100.

Two traps that catch almost everyone:

The percentage is always taken of the original amount, not the new one. A 10\% rise then a 10\% fall use different wholes, so they do not cancel.
Because of that, a 20\% rise followed by a 20\% fall does not bring you back to the start. Take \pounds 100: up 20\% gives \pounds 120; now 20\% of 120 is \pounds 24, so down to \pounds 96 — not \pounds 100! 100 \times 1.2 \times 0.8 = 96.

pizza

A pizza normally costs \pounds 10, and today the sign says 20% off. The discount is \tfrac{20}{100}\times 10 = \pounds 2, so you pay 10 - 2 = \pounds 8. A discount is just a percentage decrease — you take a slice off the price.

balloon

You blow a balloon up so it is 50\% bigger than before. If it held 40 puffs of air, the extra is \tfrac{50}{100}\times 40 = 20 puffs, so it now holds 40 + 20 = 60. Growing by 50\% means multiplying by 1.5 — half as much again on top.

See it: original vs new

The top bar is the original amount — the whole, 100\%. The bottom bar is the new amount: an increase adds a coloured block on the end, and a decrease leaves a faded block missing. Press Refresh for a new amount and a new change.

Step through both

Here the same whole bar of 100 is changed two ways: first an increase adds a block on top, then a decrease cuts a block off.