ArithmeticArithmetic FoundationsMiddle school

Percentages

Percent of a number, percent change and reverse percentages.

Percent means per hundred, so $35\%$ is the fraction $\frac{35}{100}$ or the decimal $0.35$ . Almost every percentage question is one of three forms: find a percent of a number, find what percent one number is of another, or find the original number before a percentage change.

Percent change compares a change to the starting value, not to the ending value. Successive changes multiply rather than add: a $20\%$ rise followed by a $20\%$ fall leaves you at $1.2 \times 0.8 = 0.96$ of the start, a $4\%$ loss overall.

Key terms

Percent change: The change divided by the original value, times 100.
$\%\Delta = \frac{\text{new} - \text{old}}{\text{old}} \times 100$

Key ideas

To increase by $p\%$ , multiply by $1 + \frac{p}{100}$ ; to decrease, multiply by $1 - \frac{p}{100}$ .
Reverse percentage: divide the final value by the multiplier to recover the original.
Percentage points measure the difference between two percentages; percent change measures relative change.
Write the multiplier first, then compute, to avoid adding percentages that should multiply.

Worked example

A jacket costs $\$68$ after a $15\%$ discount. What was the original price?

A $15\%$ discount leaves $85\%$ , a multiplier of $0.85$ .
Original $\times 0.85 = 68$ .
Original $= 68 \div 0.85 = 80$ .

Answer: $\$80$

Common mistakes

Finding $15\%$ of the sale price and adding it back, which gives $\$78.20$ instead of $\$80$ .
Dividing by the new value when computing percent change.
Adding successive percentage changes.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.What is $12\%$ of $250$ ?
Show answer
$30$
2.A price rises from $40$ to $46$ . What is the percent increase?
Show answer
$15\%$
3.After a $25\%$ increase a value is $150$ . What was it before?
Show answer
$120$

Keep practising

Calculators

Percentage Calculator