M23 · Percentages and financial maths | Edexcel GCSE Maths

Revise the key ideas

A percentage means parts per hundred. To express A as a percentage of B, use AB × 100%, with B ≠ 0 and matching units.
For percentage change, use changeoriginal amount × 100%. The denominator is the original amount, not the new amount.
Worked example: A price rises from £40 to £46. Increase = £6 and percentage increase = 640 × 100% = 15%.
An increase of p% multiplies by 1 + p100; a decrease multiplies by 1 − p100. A 12% reduction uses multiplier 0.88.
Forward and reverse percentage multipliers
Enlarge diagram: Forward and reverse percentage multipliers
Worked example: £250 reduced by 12% becomes 250 × 0.88 = £220. Find the new amount directly with the multiplier.

Reverse a percentage change by dividing by its multiplier. A sale price of £72 after a 20% reduction came from 72 ÷ 0.8 = £90.
A 20% rise followed by a 20% fall is not no change: 1.2 × 0.8 = 0.96, giving an overall 4% decrease.
Simple interest is calculated on the original principal every year. At 4% simple interest, £500 earns £20 per year, so after 3 years the total is £560.
Compound interest applies the percentage to the changing balance. Final amount = P(1 + r100)ⁿ for n equal periods at rate r% per period.
Worked example: £500 at 4% compound interest for 3 years becomes 500 × 1.04³ = £562.432, or £562.43 to the nearest penny. Round the final result, unless the account's rules specify rounding each year.
Depreciation or decay uses a multiplier below 1. A £12 000 car losing 15% of its value each year is worth 12 000 × 0.85² = £8670 after 2 years.
Rates and periods must match: an annual rate is not applied once per month without converting the model. State any assumption that the rate stays constant.

If money is added or withdrawn each period, a simple power formula may no longer apply. Write a recurrence that reflects the order of interest and payments.
Worked example: A balance earns 2% interest, then receives a £100 deposit each year. Bₙ₊₁ = 1.02Bₙ + 100. Starting at £500 gives £610 after year 1 and £722.20 after year 2.
Depositing before interest instead gives Bₙ₊₁ = 1.02(Bₙ + 100), a different result. Use the wording to decide the order.
Find an unknown whole number of periods by repeated calculation or trial powers and check the first period at which the threshold is reached. Do not round a time up or down without interpreting the context.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.