Percentages and successive changes

Percentages in your head

A percent is a fraction out of 100, so $p\%$ of $x$ is $\frac{p}{100} \times x$. Without a calculator, build awkward percentages from easy ones:

• $10\%$: move the decimal point one place left. $10\%$ of 260 is 26.
• $5\%$: half of $10\%$. $1\%$: move the point two places.
• $15\%$ of 260 $= 26 + 13 = 39$; $19\%$ of 260 $= 20\% - 1\% = 52 - 2.6 = 49.4$.

Some percentages are simple fractions, and the fraction is faster: $25\% = \frac{1}{4}$, $20\% = \frac{1}{5}$, $12.5\% = \frac{1}{8}$, $33\frac{1}{3}\% = \frac{1}{3}$, $75\% = \frac{3}{4}$.

One more trick: $x\%$ of $y$ equals $y\%$ of $x$, because both are $\frac{xy}{100}$. So $18\%$ of 50 is $50\%$ of 18, which is 9.

Every change is a multiplier

An increase of $p\%$ multiplies by $1 + \frac{p}{100}$; a decrease of $p\%$ multiplies by $1 - \frac{p}{100}$. Up 6% is $\times 1.06$; down 35% is $\times 0.65$; up 100% is $\times 2$.

Multipliers make three things easy:

1. New value in one step. €420 cut by 15% is $420 \times 0.85 = 357$.
2. Successive changes multiply. Up 20% then down 25% is $1.2 \times 0.75 = 0.9$, a net fall of 10%. The order does not matter, since multiplication is commutative.
3. Reversing a change divides. If a price after a 25% rise is 1,150, the price before was $\frac{1150}{1.25} = 920$.

For two successive changes of $a\%$ and $b\%$ (negative for a fall), the net change is

$$a + b + \frac{ab}{100} \text{ percent}.$$

For $+20\%$ then $-25\%$: $20 - 25 + \frac{20 \times (-25)}{100} = -5 - 5 = -10\%$, matching the multiplier route. The extra term $\frac{ab}{100}$ is why successive percents never simply add.

The base of a percent change

$$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$$

The base is always the starting value. A rise from 80 to 100 is $\frac{20}{80} = 25\%$; the fall from 100 back to 80 is $\frac{20}{100} = 20\%$. In the same way, "A is 25% more than B" means $A = 1.25B$, so $B = \frac{A}{1.25} = 0.8A$, which is 20% less than $A$, not 25%.

The same idea separates markup from margin. An item that costs 80 to make and sells for 100 has a markup of $\frac{20}{80} = 25\%$ on cost and a margin of $\frac{20}{100} = 20\%$ on price. Read which base the question names.

Percentage points

When the quantity is itself a rate, there are two different changes. An interest rate going from 4% to 5% rises by 1 percentage point (a subtraction) and by 25 percent (a percent change of the rate, $\frac{1}{4}$). Questions often offer both numbers; decide which one was asked for before you look at the options.

Percent of a percent, and "what percent of"

A percent of a percent is a product of fractions: $40\%$ of $25\%$ of a budget is $0.4 \times 0.25 = 0.1$, which is $10\%$ of the budget. Questions of the form "what percent of A is B?" are a single division, $\frac{B}{A} \times 100\%$, with A, the thing after "of", underneath. So 18 is $\frac{18}{72} = \frac{1}{4} = 25\%$ of 72, while 72 is $\frac{72}{18} = 400\%$ of 18. Simplify the fraction before converting: a fraction you recognise, such as $\frac{1}{4}$ or $\frac{3}{8}$, turns into a percent at once, which spares you long division.

Estimate before you choose

A quick sense check catches most slips: a discount should leave a smaller number, two rises should give more than either alone, and an original price recovered from a sale price must be larger than the sale price. If an answer breaks such a rule, a multiplier went the wrong way.