Matematica

Percentages: Solved Exercises and Formulas Explained

How to calculate a percentage step by step: discounts, increases and reverse percentages, with five solved and verified exercises from easy to advanced.

Recommended for: Grade 6 · Grade 7 · Grade 8

A percentage is a fraction with denominator 100: 15% means 15 parts out of 100. So to find 15% of a number you just multiply it by 15/100, that is by 0.15.

The formula everything comes from

Every percentage problem links three quantities: the part, the total and the percentage. There is really only one formula:

part = total × percentage / 100

The other two cases come straight from it. If you are looking for the percentage, use percentage = (part ÷ total) × 100. If you are looking for the total, knowing the part and the percentage, use total = part × 100 / percentage.

Discounts and increases: the multiplier trick

Instead of working out the discount and then subtracting it, it is quicker to use a single number. A 20% discount means paying 80%, so you multiply by 0.80. A 15% increase means reaching 115%, so you multiply by 1.15.

The advantage shows up when there is more than one change: you simply multiply the factors one after the other, with no intermediate steps.

Reverse percentages

This is the case that trips most people up: you know the discounted price and you need the original one. The usual mistake is to add back the same percentage that was taken off, but that does not work, because the discount was calculated on the old price.

The correct method is to divide by the multiplier: if you pay £18 after a 25% discount, the original price was 18 ÷ 0.75 = £24.

How to tackle the exercises

Before calculating, always ask yourself which of the three quantities is the unknown and which total you are working on. The exercises below follow the same pattern at increasing difficulty: they start with a direct percentage calculation and end with an increase followed by a discount, where the result surprises anyone who guesses. Every solution is checked by working the calculation backwards.

Solved exercises

1. Calculate 15% of 240. base

Show solution
  1. 15% means 15 parts out of 100, that is the fraction 15/100.
  2. Multiply the fraction by the total: 240 × 15/100.
  3. Work out the numerator: 240 × 15 = 3600.
  4. Divide by 100: 3600 ÷ 100 = 36.

Answer: 36

2. In a class of 25 students, 7 scored top marks on the test. What percentage is that? base

Show solution
  1. Write the part over the total as a fraction: 7/25.
  2. To turn it into a percentage, multiply by 100: (7 ÷ 25) × 100.
  3. 7 ÷ 25 = 0.28.
  4. 0.28 × 100 = 28.

Answer: 28%

3. A hoodie costs £45 and is discounted by 20%. How much do you pay at the till? intermedio

Show solution
  1. Work out the discount: 20% of £45 = 45 × 20/100 = £9.
  2. Subtract it from the original price: 45 − 9 = £36.
  3. Check with the shortcut: after a 20% discount you pay 80% of the price.
  4. 80% of £45 = 45 × 0.8 = £36, so both methods agree.

Answer: £36

4. After a 25% discount a book costs £18. What was the price before the discount? intermedio

Show solution
  1. With a 25% discount you pay 75% of the original price, so £18 corresponds to 75%.
  2. Call the original price x: 0.75 · x = 18.
  3. Divide both sides by 0.75: x = 18 ÷ 0.75 = 24.
  4. Check: 25% of £24 is £6, and 24 − 6 = 18. Correct.

Answer: £24

5. A bike costs £200. The price first goes up by 10%, then a 10% discount is applied to the new price. What is the final price, and how much has it changed overall? avanzato

Show solution
  1. 10% increase: multiply by 1.10 → 200 × 1.10 = £220.
  2. 10% discount on the new price: multiply by 0.90 → 220 × 0.90 = £198.
  3. Compare with the starting price: 198 − 200 = −£2.
  4. Percentage change: (−2 ÷ 200) × 100 = −1%.
  5. The final price is lower because the 10% discount is calculated on £220 (that is £22), while the increase was only worth £20.

Answer: £198, that is 1% less than the starting price

FAQ

How do you turn a percentage into a decimal?

Divide by 100: 25% becomes 0.25, 7% becomes 0.07 and 120% becomes 1.2. This is the step that lets you find a percentage with a single multiplication.

Why is a 20% discount followed by a 10% discount not a 30% discount?

Because the second discount applies to the already reduced price, not to the original one. You multiply by 0.8 and then by 0.9, which is 0.72: the overall discount is 28%, not 30%.

Is there a way to work out percentages mentally?

Yes: 10% is the number divided by 10, 5% is half of 10%, and 1% is the number divided by 100. From these three building blocks you can get almost anything, for example 15% = 10% + 5% and 35% = three lots of 10% plus 5%.