Mr Kidman's Classroom
Revise the key ideas
Percentage change
- A percentage means parts per hundred. To express A as a percentage of B, use × 100%, with B ≠ 0 and matching units.
- For percentage change, use × 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 = × 100% = 15%.
- An increase of p% multiplies by 1 + ; a decrease multiplies by 1 − . A 12% reduction uses multiplier 0.88.
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 percentages and repeated changes
- 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 (the starting amount of money) 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. With starting amount P, final amount = P(1 + )n for n equal periods at rate r% per period.
After each period, the interest is added to the balance before the next period’s interest is calculated. A period is the time interval used for the rate, such as one year.
- 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 (a fall in value) 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.
Higher — iterative financial models
- If money is added or withdrawn each period, a simple power formula may no longer apply. Write a recurrence (a rule for finding the next balance from the current one) 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.
Here Bₙ is the balance after n years and Bₙ₊₁ is the next year’s balance. For year 2, use the year-1 balance: 1.02 × 610 + 100 = £722.20.
- 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 (the target amount) is reached. Do not round a time up or down without interpreting the context.
