How to calculate percentages: method, examples and pitfalls
Percentages are everywhere: sales, tax, exam marks, polls, interest rates. They all rest on one idea. Once you have it, every situation comes down to a multiplication or a division.
The basics: “per cent” means “out of a hundred”
15% means 15 parts out of 100. It is the fraction 15/100, or the decimal 0.15.
p% = p ÷ 100
Most errors come from forgetting to divide by 100, or doing so at the wrong moment.
Finding p% of a number
p% of N = N × p ÷ 100
Example: 15% of 80
- 80 × 15 = 1,200
- 1,200 ÷ 100 = 12
15% of 80 = 12
- 50% is half, 25% a quarter, 75% three quarters.
- 10% means dividing by 10; 20% is double that.
- 5% is half of 10%; 1% means dividing by 100.
- 15% = 10% + 5%. For 80, that is 8 + 4 = 12.
Handy trick: p% of N always equals N% of p. So 8% of 50 is the same as 50% of 8, which is 4.
Finding the rate: what share is a of b?
Divide the part by the whole, then multiply by 100.
rate = a ÷ b × 100
Example: 18 students out of 24 passed
- 18 ÷ 24 = 0.75
- 0.75 × 100 = 75
75% of the students passed
Increasing or decreasing by p%
An increase keeps 100% of the value and adds p%, so you multiply by 1 + p/100. A decrease multiplies by 1 − p/100. That number is called the multiplier.
Example: 40 increased by 25%
- Multiplier: 1 + 25 ÷ 100 = 1.25
- 40 × 1.25 = 50
The new value is 50
Finding the original price: 90 after a 10% discount
- Multiplier: 0.9
- Original = 90 ÷ 0.9 = 100
The original price was 100
Common pitfalls
- Adding percentages from successive changes: +10% then −10% multiplies by 1.1 × 0.9 = 0.99, a 1% loss overall.
- Subtracting p% from a discounted price to recover the original: divide by the multiplier instead.
- Confusing a percent change with a change in percentage points: going from 10% to 12% is a rise of 2 points, or 20%.
Worked exercises
Try to find the answer first, then open the solution.
What is 30% of 250?
Show the solution
- 250 × 30 = 7,500
- 7,500 ÷ 100 = 75
Answer: 75
What percentage is 45 of 180?
Show the solution
- 45 ÷ 180 = 0.25
- 0.25 × 100 = 25
Answer: 25%
A salary is 120 after a 20% raise. What was it before?
Show the solution
- Multiplier: 1.2
- 120 ÷ 1.2 = 100
Answer: 100