Percentages
A percent is a fraction out of 100, and every percent question is really about which quantity is the base. A rise or fall is always taken on the value just before the change. Turn each change into a multiplying factor and many long questions become one line.
Formulas and when they apply
- x% of y = x × y ÷ 100
- Percent change = (change ÷ original value) × 100
- Increase by x%: multiply by (1 + x/100). Decrease by x%: multiply by (1 − x/100)
- Two successive changes of a% and b%: net change = a + b + ab/100 (%)Applies when: increases positive, decreases negative
- If A is x% more than B, then B is x ÷ (100 + x) × 100 percent less than AApplies when: reverse comparison uses the other quantity as the base
- If a price rises by x%, consumption must fall by x ÷ (100 + x) × 100 percent to keep spending the sameApplies when: spending unchanged
- Value after n years = P × (1 + r/100)ⁿ for a rise, P × (1 − r/100)ⁿ for a fallApplies when: same rate every year
- n(A or B) = n(A) + n(B) − n(A and B)Applies when: two overlapping groups
Worked examples
Example 1. A price is increased by 20% and then decreased by 30%. What is the net change?
Solution. Take the price as 100. After the 20% rise it is 120. The 30% fall is 30% of 120 = 36, so the price becomes 84. Net change = 84 − 100 = −16, a 16% decrease.
Example 2. The price of sugar rises by 25%. By what percent must a family cut its use of sugar to keep spending the same?
Solution. Old price 100, new price 125. The same spend buys 100 ÷ 125 = 0.8 of the old quantity. The family must cut its use by 1 − 0.8 = 0.2, which is 20%.
Example 3. The population of a village is 8,000 and grows by 10% every year. What will it be after 2 years?
Solution. Each year multiplies the population by 1.10. After 2 years it is 8,000 × 1.10 × 1.10 = 8,000 × 1.21 = 9,680.
Shortcuts and their limits
- Start from 100 when the question gives only percentages. Apply the changes one by one.Does not apply when: The answer is then a percent. If the question asks for an actual quantity, scale it up at the end.
- Use a + b + ab/100 for two successive changes.Does not apply when: It handles two changes at a time. For three or more, apply it repeatedly or multiply the factors.
- Learn common fractions: 12½% = ⅛, 16⅔% = ⅙, 33⅓% = ⅓, 37½% = ⅜.Does not apply when: They help only when the percent is one of these standard values.
Common mistakes
- Adding or subtracting percentages that have different bases, such as 40% up and 25% down giving 15%.
- Using the wrong base when reversing a comparison: 'A is 25% more than B' does not make B 25% less than A.
- Undoing a percent change by applying the same percent in the opposite direction.
- Assuming two groups are independent and multiplying their percents when overlap is not given.
- Forgetting that a percent point and a percent are different things.
Independent practice made by EduVistaHQ with original questions. It is not an official examination paper, it is not affiliated with or endorsed by any examination body or company, and a practice score does not predict a real result.