Percentage Change
To change an amount by a percentage, the quick way is to multiply by a single
multiplier. The whole amount is 100\%, so
increasing by 20\% gives 120\% of the
original, and decreasing by 20\% leaves 80\%.
Written as decimals those are 1.20 and 0.80.
- 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)
Use + for an increase and - for a
decrease. A 20\% rise on \pounds 50 is
50 \times 1.20 = \pounds 60; a 20\% fall
is 50 \times 0.80 = \pounds 40.
Going the other way, suppose you already know the old and new amounts and want the
percentage change. Compare the change to the
original amount:
\frac{\text{change}}{\text{original}} \times 100\%
If a price rises from \pounds 40 to \pounds 50,
the change is \pounds 10, so the increase is
\tfrac{10}{40} \times 100\% = 25\%. The same idea answers "one amount
as a percentage of another" — that is just part over whole.
-
Increase by p\%: multiply by
\left(1 + \tfrac{p}{100}\right).
-
Decrease by p\%: multiply by
\left(1 - \tfrac{p}{100}\right).
-
Percentage change:
\dfrac{\text{change}}{\text{original}} \times 100.
-
One amount as a % of another:
\dfrac{\text{part}}{\text{whole}} \times 100.
See it as a bar
The whole amount is a bar of 100. An increase adds a block on top;
a decrease cuts a block off. Step through both.