Percentages and unit conversions

Percentages as multipliers

$p\%$ of a quantity is $\frac{p}{100}$ times it. A change by a percentage is a multiplication:

Change	Multiplier
increase by 22%	$1.22$
increase by 5%	$1.05$
decrease by 30%	$0.70$
decrease by 2.5%	$0.975$

This one idea handles three common tasks:

• New value: multiply. A €80 item with 22% tax costs $80 \times 1.22 = 97.60$ euros.
• Several changes in a row: multiply the multipliers. Up 20% then down 25% is $1.2 \times 0.75 = 0.9$, a 10% fall overall. Percentages do not add.
• Original value: divide. If a price after a 22% tax is €61, the price before tax is $\frac{61}{1.22} = 50$ euros.

Without a calculator, convert awkward multipliers into fractions: $1.25 = \frac{5}{4}$, $0.75 = \frac{3}{4}$, $1.2 = \frac{6}{5}$, $0.8 = \frac{4}{5}$, $1.125 = \frac{9}{8}$. Dividing by $1.25$ is multiplying by $\frac{4}{5}$.

Percent change and its base

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

The base is always the starting value. A rise from 50 to 60 is $+20\%$; the fall from 60 back to 50 is about $-16.7\%$. "$A$ is 25% more than $B$" means $A = 1.25B$, and then $B = 0.8A$, which is 20% less than $A$.

Percentage points

When the quantity itself is a percentage, such as an interest rate, a tax rate or a share of votes, two different changes can be asked for. A rate that moves from 8% to 10% has risen by 2 percentage points (a subtraction) and by 25% (a percent change, $\frac{2}{8}$). Read which one the question wants; both numbers will be among the options.

Unit conversion: multiply by one

A conversion factor is a fraction equal to 1, such as $\frac{1000 \text{ m}}{1 \text{ km}}$ or $\frac{1 \text{ h}}{60 \text{ min}}$. Multiply by factors arranged so that the unknown units cancel and the wanted ones remain:

$$72 \ \frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 20 \ \frac{\text{m}}{\text{s}}.$$

So km/h to m/s is division by 3.6, and m/s to km/h is multiplication by 3.6. Before you trust a result, ask whether it should be bigger or smaller: a quantity in a smaller unit needs a bigger number.

Squares and cubes of units

An area unit is a length unit squared, so its factor is squared too; a volume unit is cubed:

Relation	Length	Area	Volume
m and cm	$1 \text{ m} = 100 \text{ cm}$	$1 \text{ m}^2 = 10\,000 \text{ cm}^2$	$1 \text{ m}^3 = 1\,000\,000 \text{ cm}^3$
km and m	$1 \text{ km} = 1000 \text{ m}$	$1 \text{ km}^2 = 1\,000\,000 \text{ m}^2$

Two named units are worth knowing: $1$ litre $= 1 \text{ dm}^3 = 1000 \text{ cm}^3$, so $1 \text{ m}^3 = 1000$ litres; and $1$ hectare $= 10\,000 \text{ m}^2$, the area of a square 100 m on each side.

The safest method for areas and volumes is to convert each length first, then multiply. A 2.5 m by 1.2 m floor is 250 cm by 120 cm, which is $30\,000 \text{ cm}^2$, the same as $3 \text{ m}^2 \times 10\,000$.

Time

Time is not decimal. $1.25$ hours is 1 hour and $0.25 \times 60 = 15$ minutes, not 1 hour 25 minutes. Convert the decimal part by multiplying by 60, and convert minutes to hours by dividing by 60: 45 minutes is $0.75$ h.

A habit that saves marks

Write the units next to every number in your working. A result in "km times hours" when you wanted km/h shows at once that something was multiplied instead of divided.