Leave lesson

Ratio, proportion & rates of change · Percentages & growth

1 / 10

Percentage change and multipliers

Why every percentage increase or decrease is a single multiplication, how to build the multiplier, and why "percentage change" is always measured against the original amount.

Ratio, proportion & rates of change · Percentages & growth

Percentage change and multipliers

Why every percentage increase or decrease is a single multiplication, how to build the multiplier, and why "percentage change" is always measured against the original amount.

Why it works

"Per cent" means per hundred — a percentage is just a fraction with 100 on the bottom. 37%37\% is 37100=0.37\frac{37}{100} = 0.37. That single fact is the whole topic: once a percentage is a decimal, everything is ordinary multiplication.

A percentage change is one multiplication, not two steps. Suppose a £80\pounds 80 coat goes up by 15%15\%. The slow way is "find 15%15\%, then add it on": 0.15×80=120.15 \times 80 = 12, then 80+12=9280 + 12 = 92. But look at what that actually does — you keep the original (100%100\% of it) and add 15%15\% more, so you end up with 115%115\% of the start. That is 1.15×80=921.15 \times 80 = 92 in one go. The number 1.151.15 is the multiplier, and it is the increase.

Build the multiplier from 100%100\%:
  • Increase by p%p\% ⇒\Rightarrow multiply by 1+p1001 + \dfrac{p}{100}. A 15%15\%
rise is ×1.15\times 1.15; a 7.5%7.5\% rise is ×1.075\times 1.075.
  • Decrease by p%p\% ⇒\Rightarrow multiply by 1−p1001 - \dfrac{p}{100}. A 15%15\%
fall is ×0.85\times 0.85 (you keep 85%85\% of it); a 40%40\% discount is ×0.6\times 0.6.

The commonest slip is to multiply by p100\frac{p}{100} itself — by 0.150.15 instead of 1.151.15. That throws away the original amount and leaves you with only the change. Always start the multiplier from 11.

Percentage change is measured against the original. If a value goes from an old amount to a new one,

percentage change=new−oldold×100.\text{percentage change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100.

The denominator is the starting value, because a percentage only means something relative to a base, and the base is where you began. A price that rises from £40\pounds 40 to £50\pounds 50 has gone up 1040×100=25%\frac{10}{40}\times 100 = 25\% — not 1050×100=20%\frac{10}{50}\times 100 = 20\%. The £10\pounds 10 is being compared to the £40\pounds 40 it grew from, not the £50\pounds 50 it grew to.

Why an increase and the same-size decrease don't cancel. Put £200\pounds 200 up by 10%10\% and you get ×1.1=£220\times 1.1 = \pounds 220. Now take 10%10\% off that: ×0.9=£198\times 0.9 = \pounds 198 — not back to £200\pounds 200. The second 10%10\% is 10%10\% of the bigger £220\pounds 220, so it removes more than the first 10%10\% added. Percentages of different amounts are different sizes; the multipliers 1.11.1 and 0.90.9 multiply to 0.990.99, a 1%1\% overall fall.