What is a percentage?

A percentage expresses a proportion out of 100. The “%” symbol means “per cent” or “divided by 100”.

50% = 50/100 = 0.5 = half.

The three basic formulas

Find the percentage

Percentage = (Part / Total) × 100

Example: 20 students out of 25 pass an exam.
(20 / 25) × 100 = 80% pass rate.

Find the part (how much it amounts to)

Part = (Percentage × Total) / 100

Example: 30% off an item priced at 80€.
(30 × 80) / 100 = 24€ saved. Final price: 56€.

Find the total (the original price)

Total = (Part × 100) / Percentage

Example: You pay 45€ after a 25% discount. What was the price before?
Watch out: 45€ is 75% of the price (100% - 25%).
(45 × 100) / 75 = 60€.

Calculating a variation

Variation = ((Final value - Initial value) / Initial value) × 100

Salary going from 2000€ to 2300€:
((2300 - 2000) / 2000) × 100 = +15%

Price going from 800€ to 600€:
((600 - 800) / 800) × 100 = -25%

The successive variations trap

A 20% increase followed by a 20% decrease does not bring you back to the starting point.

100€ + 20% = 120€
120€ - 20% = 96€

Why? Because the 20% drop applies to 120€, not 100€.

The same principle applies to discounts: two 50% discounts don’t add up to 100% off. They add up to 75%.

Mental math

A few shortcuts:

  • 10% = divide by 10
  • 1% = divide by 100
  • 50% = divide by 2
  • 25% = divide by 4
  • 20% = divide by 5

For the others:

  • 15% = 10% + half of 10%
  • 30% = 10% × 3
  • 75% = 50% + 25%

Common mistakes

Percentage points vs percentage. If a rate goes from 5% to 7%, it has risen by 2 percentage points. But in relative terms, that’s a 40% rise (you go from 5 to 7, that is +2/5).

Forgetting to convert. To run the calculation, you need to put the percentage into decimal form: 25% = 0.25. Otherwise the results are wrong.