KS3MAT10 · Percentages, rates and finance | KS3 Maths

Revise the key ideas

Increase and decrease — A 15% increase uses multiplier 1.15; a 15% decrease uses 0.85. £80 increased by 15% is £92. A change percentage is measured against the original amount.
Finding a change percentage — A price rises from £40 to £46. The change is £6 and percentage increase is 6/40 × 100 = 15%. Dividing by the new price answers a different comparison.
Reverse percentages — After a 20% reduction, a coat costs £56, so 0.8 × original = 56. Divide by 0.8 to find £70. Adding 20% to £56 gives £67.20 because it uses a different base.
Repeated changes — Increasing £100 by 10% then decreasing the result by 10% gives £99: 100 × 1.1 × 0.9. The percentages have different reference amounts, so equal percentage changes do not cancel.

Simple interest — Simple interest is calculated on the original amount borrowed or saved, called the principal. In I = Prt, I is interest, P is principal, r is the rate as a decimal per time period, and t is the number of those periods. For £500 at 4% per year for 3 years: I = 500 × 0.04 × 3 = £60; the total is £500 + £60 = £560. It is not recalculated on each growing balance.
Budgeting — Income is money coming in; outgoings are money spent. Subtract total outgoings from income: a positive result is a surplus (money left over), and a negative result is a deficit (a shortfall). With £45 income and costs £12.50, £8.75 and £15, the surplus is £8.75. Include units, one-off charges and the time period.
Best buys — Compare the same quantity: 750 g for £2.40 costs £3.20/kg, while 1.2 kg for £3.60 costs £3/kg. The second is cheaper per kilogram, but affordability and waste can affect the decision.

Speed — Average speed = distance/time. A 150 km journey in 2.5 hours averages 60 km/h. The average does not imply the traveller maintained that speed at every moment; include stops in total journey time when asked.
Convert speed — 1 m/s = 3.6 km/h because 1 hour has 3600 seconds and 1 km has 1000 metres. Thus 72 km/h = 20 m/s. Convert the full compound unit, not just the distance part.
Rearrange rate formulae — From v = d/t, distance d = vt and time t = d/v for v ≠ 0. At 8 m/s for 45 s, distance is 360 m. Check units to decide which operation produces the requested quantity.
Density — Density = mass/volume. A 240 g block occupying 80 cm³ has density 3 g/cm³. Mass = density × volume. A larger object need not have greater density; it may simply contain more of the same material.
Other compound units — £/hour is a rate of pay; litres/minute is flow rate. At £12.50/hour for 6.4 hours, pay is £80. State what is being counted per what and check whether the rate is constant.

Multi-step planning — A 180 km trip at 60 km/h takes 3 h driving; adding a 25-minute stop gives 3 h 25 min. At 12 km per litre it uses 15 litres. These models assume constant average rates and omit other costs.
Communicate a conclusion — A saving of £12 on £60 is 20%, while £12 on £80 is 15%. Explain both the absolute change (the number of pounds saved) and the relative change (the saving as a percentage of the original price) when comparing offers, and state any missing information such as delivery charges.

50 questions · Random sets of 10. These quick checks support revision; practise longer explanations and justified judgements too.