Percentages Revision Notes

Percentages represent parts per hundred. To find a percentage of any amount, convert the percentage to a decimal by dividing by 100, then multiply by the amount.

For example, to find 15% of £80: first convert 15% to decimal form: 15 ÷ 100 = 0.15. Then multiply: 0.15 × 80 = £12.

This method applies to any quantity, including money, measurements, or counts. Always write the percentage as a decimal before multiplying to avoid errors.

• Convert percentage to decimal by dividing by 100.
• Multiply decimal by the amount to find the percentage value.
• Example: 25% of 60 = 0.25 × 60 = 15.

To express one amount as a percentage of another, divide the first amount by the second, then multiply the result by 100.

For example, if a student scores 45 marks out of 60, the percentage score is (45 ÷ 60) × 100 = 75%.

This technique is useful for comparing quantities and understanding proportions in percentage terms.

• Divide the part by the whole.
• Multiply the result by 100 to convert to percentage.
• Example: 30 out of 50 = (30 ÷ 50) × 100 = 60%.

Percentage change measures how much a quantity has increased or decreased relative to its original value.

Calculate the change by subtracting the original amount from the new amount.

Then divide the change by the original amount and multiply by 100 to get the percentage change: Percentage change = (Change ÷ Original amount) × 100%. If the new amount is greater, the result is a percentage increase; if smaller, a percentage decrease.

• Find the difference between new and original amounts.
• Divide difference by original amount.
• Multiply by 100 to get percentage change.
• Example: Price rises from £80 to £100: Increase = 100 - 80 = 20; Percentage increase = (20 ÷ 80) × 100 = 25%.

Simple interest is calculated only on the original principal amount over a period of time. The interest earned each period is constant.

For example, if £1000 is invested at 5% simple interest per year for 3 years, the interest each year is 5% of £1000 = £50. Total interest = £50 × 3 = £150. Total amount after 3 years = £1000 + £150 = £1150.

Compound interest is calculated on the principal plus any interest accumulated from previous periods, causing the amount to grow faster over time. This involves repeated percentage increases applied to the new total each period, not just the original principal. The formula is not provided, so calculate step-by-step: multiply the amount by (1 + interest rate as decimal) each period.

• Simple interest = Principal × Rate × Time.
• Compound interest applies interest on accumulated amount each period.
• Example simple interest: £1000 at 5% for 3 years = £150 interest.
• Example compound interest: £1000 at 5% for 2 years: Year 1: £1000 × 1.05 = £1050; Year 2: £1050 × 1.05 = £1102.50.

Reverse percentages find the original amount before a percentage increase or decrease was applied.

For example, if an item sells for £120 after a 20% increase, find the original price by dividing the selling price by 1 plus the percentage increase as a decimal: Original price = £120 ÷ 1.20 = £100.

Similarly, for a decrease, divide by 1 minus the percentage decrease as a decimal. This method is essential for problems involving profit, loss, discounts, or markups where the final amount and percentage change are known but the original amount is unknown.

• For percentage increase: Original = Final ÷ (1 + decimal increase).
• For percentage decrease: Original = Final ÷ (1 - decimal decrease).
• Example: Price after 15% discount is £85, original price = £85 ÷ 0.85 = £100.