Fractions in Disguise

A percentage is a fraction with denominator 100. Find a percentage of a quantity, convert between forms, and work out increases, decreases and proportions.

Key ideas

x% means x/100: 25% = 1/4 = 0.25.
To write a fraction as a percentage, multiply it by 100%.
Percentage change is always worked out on the original value.
An increase of r% multiplies by (1 + r/100); a decrease multiplies by (1 − r/100).
A rise of 10% followed by a fall of 10% does not bring you back to the start.
Four numbers are in proportion when the product of the outer two equals the product of the middle two.

Remember

Percentage of a quantity	x% of Q = x × Q ÷ 100
Percentage change	change ÷ original × 100%
After a change	new = original × (100 ± r) ÷ 100
Proportion	a : b = c : d ⇔ a × d = b × c

Watch out

Dividing the change by the new value instead of the original.
Adding percentages of different totals as if they were the same.
Comparing the wrong products when testing for a proportion.

Worked example

Are 5, 30, 7 and 43 in proportion? Give a reason.

Working: product of extremes 5 × 43 = 215; product of means 30 × 7 = 210

Answer: No