Rukasha Maths

GCSE maths Beta

Rates, units and best buys · Foundation notes

Foundation · Ratio · AQA references N13, R1, R11

A rate compares quantities with different units, such as kilometres per hour or pounds per kilogram. The word per means for each one, so dividing often finds a unit rate. Comparisons are fair only when they use matching units. A journey time of 1 hour 30 minutes is 1.5 hours, not 1.30 hours. Speed connects distance and time; density connects mass and volume. Write units beside your calculations to help choose the correct conversion. In a best-buy problem, compare cost per item or per unit mass rather than just comparing the packet prices.

Learning objectives

  • Calculate speed and density using consistent units.
  • Convert area units and compare unit costs.

Key ideas

A compound unit combines the units of both quantities.

speed=distancetime,density=massvolume\text{speed}=\frac{\text{distance}}{\text{time}},\qquad\text{density}=\frac{\text{mass}}{\text{volume}}

Convert minutes to hours before finding speed in kilometres per hour.

hours=minutes60\text{hours}=\frac{\text{minutes}}{60}

Area units square the length conversion.

1 m=100 cm,1 m2=10 000 cm21\text{ m}=100\text{ cm},\qquad1\text{ m}^2=10\,000\text{ cm}^2

For best buys, compare costs for one item or one common unit.

unit cost=total costquantity\text{unit cost}=\frac{\text{total cost}}{\text{quantity}}

Worked examples

Example 1

A cyclist covers thirty-six kilometres in one hour and thirty minutes. Find the average speed.

t=1+3060=1.5 ht=1+\frac{30}{60}=1.5\text{ h}
v=361.5=24 km/hv=\frac{36}{1.5}=24\text{ km/h}

Answer

24 km/h24\text{ km/h}

Example 2

Convert the area from square metres to square centimetres.

2.5 m2=2.5×10 000 cm2=25 000 cm22.5\text{ m}^2=2.5\times10\,000\text{ cm}^2=25\,000\text{ cm}^2

Answer

25 000 cm225\,000\text{ cm}^2

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Convert units and calculate a rate or unit price.

AO2 · Reasoning

Compare offers using a common quantity and explain your decision.

AO3 · Problem solving

Combine a rate calculation with a time, quantity or cost constraint.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Square the conversion factor for area. Thirty minutes is half an hour, not three tenths of an hour.

Try it yourself

Attempt each question before opening its answer.

Check 1: Convert 2.5 m² to cm².

Answer: 25 000 cm²

A square metre contains 100 × 100 = 10 000 square centimetres. Multiply 2.5 by 10 000.

Check 2: A block has mass 540 g and volume 200 cm³. Find its density.

Answer: 2.7 g/cm³

Density = 540 ÷ 200 = 2.7 grams per cubic centimetre.

Explain it in your own words

  • Explain why 1 hour 45 minutes is 1.75 hours.
  • Compare the unit prices of two packs you invent.
  • Draw a 1 m by 1 m square to explain the area conversion factor.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All foundation notes

Specification source

AQA Ratio specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.