Percentages appear in prices, taxes, salaries, grades, interest rates and statistics, yet many people still reach for a calculator for even simple questions. The core idea is short: a percentage is a fraction of 100. Once you know a handful of formulas, you can work out discounts, increases, tips and taxes quickly and avoid the mistakes that cause costly errors.

The Basic Formulas

  • Percent of a number: part = whole × percentage ÷ 100. For example, 15% of 240 is 240 × 15 ÷ 100 = 36.
  • Finding the percentage: percentage = part ÷ whole × 100. If you scored 42 out of 60, that is 42 ÷ 60 × 100 = 70%.
  • Finding the whole: whole = part ÷ percentage × 100. If 36 is 15% of a number, the number is 36 ÷ 15 × 100 = 240.

Percentage Increase and Decrease

Percentage change = (new value − old value) ÷ old value × 100. A positive result is an increase, and a negative one is a decrease.

  • A salary that rises from 50,000 to 55,000 has increased by 5,000 ÷ 50,000 × 100 = 10%.
  • A price that moves from 80 to 100 has risen by 20 ÷ 80 × 100 = 25%.
  • A price that moves from 100 to 80 has fallen by 20 ÷ 100 × 100 = 20%, not 25%.

That last pair shows why percentage changes are not symmetrical: the starting value is always the base, so the same 20-unit move is a different percentage depending on the direction.

Percent Versus Percentage Points

If an interest rate goes from 4% to 5%, it has risen by 1 percentage point, but by 25% in relative terms (1 ÷ 4 × 100). Reports often mix these up, so check which one is meant. When a central bank raises rates by 0.25, that is a quarter of a percentage point, not 0.25%.

How Discounts Work

The sale price after a discount of d% is: price × (1 − d ÷ 100). A coat priced at 80 with 20% off costs 80 × 0.80 = 64, so you save 16.

Successive discounts do not add up. If an item is reduced by 20% and then a further 10% is taken off the new price, the result is 100 × 0.80 × 0.90 = 72, a total discount of 28%, not 30%. This applies to coupon codes stacked on top of a sale. Our discount calculator handles stacked discounts automatically.

Reverse Percentages: Finding the Original Price

If a sale price already includes a discount, divide by the remaining fraction. An item costs 80 after a 20% discount, so the original price was 80 ÷ 0.80 = 100. It is not 80 plus 20% of 80, which would give 96. The same logic works for tax: a price of 120 that includes 20% VAT has a net price of 120 ÷ 1.20 = 100, so the VAT is 20.

Percentages That Cancel Out Differently

A rise of 50% followed by a fall of 50% does not return you to the start. If 100 grows by 50% to 150 and then falls by 50%, you reach 75, a net loss of 25%. The same effect matters in investing, where a 50% loss requires a 100% gain to recover. Percentages applied one after another multiply: the combined factor is 1.50 × 0.50 = 0.75.

Mental Maths Shortcuts

  • To find 10%, move the decimal point one place left: 10% of 64 is 6.4.
  • 5% is half of 10%, and 1% is moving the decimal two places.
  • 15% is 10% plus 5%: for 64 that is 6.4 + 3.2 = 9.6.
  • 20% is double 10%, and 25% is a quarter.
  • Percentages can be swapped: 8% of 50 equals 50% of 8, which is much easier to work out as 4.

Everyday Uses

  • Sales tax: 8.25% of 40 is 3.30, so the total is 43.30.
  • Tips: our tip guide shows how to calculate and split gratuities.
  • Grades: our weighted grade guide explains how percentages combine in a course.
  • Interest: see how compounding applies a percentage repeatedly in our guide to compound versus simple interest.

Common Mistakes to Avoid

  • Using the wrong base. Always divide by the original value when calculating a change.
  • Adding successive percentages instead of multiplying them.
  • Confusing percentage points with percent.
  • Forgetting that a percentage of a percentage multiplies: 20% of 30% is 6%.

Habits that prevent errors

Name the base before you divide, multiply stacked discounts instead of adding them, and say percentage points when you compare rates. When an answer looks wrong, estimate it with 10% first.