Per cent means "per hundred". A percentage is simply a fraction whose denominator is $100$.
| Percentage | Fraction | Decimal |
|---|---|---|
| $10\%$ | $\dfrac{1}{10}$ | $0.1$ |
| $25\%$ | $\dfrac{1}{4}$ | $0.25$ |
| $50\%$ | $\dfrac{1}{2}$ | $0.5$ |
| $75\%$ | $\dfrac{3}{4}$ | $0.75$ |
| $100\%$ | $1$ | $1$ |
| $150\%$ | $\dfrac{3}{2}$ | $1.5$ |
| $0.5\%$ | $\dfrac{1}{200}$ | $0.005$ |
Find $23\%$ of £$640$.
Answer: £$147.20$
Without a calculator, build the answer from easy percentages:
| Percentage | Non-calculator method |
|---|---|
| $50\%$ | halve |
| $25\%$ | halve, then halve again |
| $10\%$ | divide by $10$ |
| $5\%$ | half of $10\%$ |
| $1\%$ | divide by $100$ |
Find $37\%$ of $840$ g.
Answer: $310.8$ g
Nadia scored $57$ out of $75$. What percentage is this?
Answer: $76\%$
Express $360$ g as a percentage of $1.5$ kg.
Answer: $24\%$
The single most important technique in this whole subtopic is the percentage multiplier. Learn it and almost every percentage question becomes one calculation.
Decrease by $p\%$: $\times \left(1 - \dfrac{p}{100}\right)$
| Change | Multiplier | Change | Multiplier |
|---|---|---|---|
| $+5\%$ | $1.05$ | $-5\%$ | $0.95$ |
| $+12\%$ | $1.12$ | $-12\%$ | $0.88$ |
| $+20\%$ | $1.2$ | $-20\%$ | $0.8$ |
| $+7.5\%$ | $1.075$ | $-7.5\%$ | $0.925$ |
| $+100\%$ | $2$ | $-100\%$ | $0$ |
A season ticket costing £$680$ rises by $8\%$. Find the new price.
Answer: £$734.40$
A laptop costing £$540$ is reduced by $35\%$. Find the sale price.
Answer: £$351$
- Work out the change: new $-$ original (a negative answer means a decrease).
- Divide by the original.
- Multiply by $100$.
- State clearly whether it is an increase or a decrease.
A trader buys a painting for £$450$ and sells it for £$603$. Find the percentage profit.
Answer: $34\%$ profit
A car bought for £$18\,500$ is sold three years later for £$11\,470$. Find the percentage loss.
Answer: a $38\%$ loss
These are the questions students lose most marks on. The giveaway wording is "before the increase", "original price", "pre-VAT" or "what was it worth last year".
A television costs £$564$ including $20\%$ VAT. Find the price before VAT.
Check: $470 \times 1.2 = 564$ ✓
The wrong method would give $564 \times 0.8 = £451.20$ — over £$18$ out.
In a sale, all prices are cut by $30\%$. A coat now costs £$87.50$. What was its original price?
Check: $125 \times 0.7 = 87.50$ ✓
With simple interest the interest is calculated on the original amount every year, so you earn the same amount each year.
$P$ = principal, $r$ = rate per year (%), $n$ = number of years
£$3200$ is invested at $4.5\%$ simple interest per year for $6$ years. Find the interest earned and the final total.
Percentages let you compare fairly when the totals are different.
In Maths, Tom scored $34$ out of $40$. In Physics he scored $51$ out of $60$. In which subject did he do better?
Equally well — both are $85\%$, even though the raw marks differ.
Definition
$p\% = \dfrac{p}{100}$ — a fraction out of one hundred.
Of an amount
$p\%$ of $A = \dfrac{p}{100} \times A$.
As a percentage
$\dfrac{A}{B} \times 100$, with matching units.
Increase / decrease
$\times (1 + \tfrac{p}{100})$ or $\times(1 - \tfrac{p}{100})$.
Percentage change
$\dfrac{\text{change}}{\text{original}} \times 100$. Divide by the original.
Reverse
Original $=$ new $\div$ multiplier.
Simple interest
$I = \dfrac{Prn}{100}$ — the same amount every year.
Combining
Multiply multipliers; $1.1 \times 0.9 = 0.99$, a $1\%$ fall.
Find (a) $18\%$ of £$250$ (b) $6.5\%$ of $840$ kg.
▶ Show solution
(a) $0.18 \times 250 = £45$
(b) $0.065 \times 840 = 54.6$ kg
Express $27$ minutes as a percentage of $2$ hours.
▶ Show solution
$2$ hours $= 120$ minutes.
$\dfrac{27}{120} \times 100 = 22.5$
Answer: $22.5\%$
A phone contract costs £$28$ per month. The price rises by $7.5\%$. Find the new monthly cost.
▶ Show solution
Multiplier $= 1.075$
$28 \times 1.075 = 30.1$
Answer: £$30.10$ per month
A bicycle priced at £$425$ is reduced by $16\%$ in a sale. Find the sale price.
▶ Show solution
Multiplier $= 1 - 0.16 = 0.84$
$425 \times 0.84 = £357$
The population of a village grew from $1250$ to $1425$. Find the percentage increase.
▶ Show solution
Change $= 1425 - 1250 = 175$
$\dfrac{175}{1250} \times 100 = 14$
Answer: a $14\%$ increase
A meal costs £$46.20$ including a $10\%$ service charge. Find the cost of the meal before the service charge was added.
▶ Show solution
This is a reverse percentage. Multiplier $= 1.1$.
$46.20 \div 1.1 = £42$
Check: $42 \times 1.1 = 46.20$ ✓
After a $12\%$ pay rise, Aisha earns £$29\,120$ per year. What did she earn before the rise?
▶ Show solution
Multiplier $= 1.12$.
$29\,120 \div 1.12 = £26\,000$
£$4500$ is invested for $5$ years at $3.2\%$ simple interest per year. Find the total value at the end.
▶ Show solution
Interest per year $= 4500 \times 0.032 = £144$
Over $5$ years $= 144 \times 5 = £720$
Total $= 4500 + 720 = £5220$
A shop increases a price by $25\%$, then later reduces the new price by $25\%$. Show that the final price is not the same as the original, and find the overall percentage change.
▶ Show solution
Overall multiplier $= 1.25 \times 0.75 = 0.9375$
Since $0.9375 \neq 1$, the price has changed.
$0.9375 = 93.75\%$, so the overall change is a decrease of $100 - 93.75 = \mathbf{6.25\%}$.
Why? The $25\%$ rise is calculated on the small original, but the $25\%$ fall is calculated on the larger increased price, so more is taken off than was put on.
A jeweller buys a ring for £$820$. She adds $65\%$ to get the shop price, then in a sale reduces the shop price by $20\%$.
(a) Find the sale price. (b) Find her percentage profit on the original £$820$. (c) What single multiplier takes £$820$ to the sale price?
▶ Show solution
(a) Shop price $= 820 \times 1.65 = £1353$
Sale price $= 1353 \times 0.8 = £1082.40$
(b) Profit $= 1082.40 - 820 = £262.40$
$\dfrac{262.40}{820} \times 100 = 32$, so a $\mathbf{32\%}$ profit.
(c) $1.65 \times 0.8 = \mathbf{1.32}$ — which confirms the $32\%$ increase ✓