Units and Conversions

Converting between units is a quietly essential skill in OCR GCSE Mathematics (J560): it underpins compound measures, ratio problems, area and volume questions, and money calculations. This lesson covers metric length, mass and capacity conversions, the trickier area and volume unit conversions (where the factor is squared or cubed), compound-unit conversions such as km/h to m/s, and simple currency conversion. These appear on every paper and on both tiers.

This lesson builds AO1 fluency with conversion factors, AO2 reasoning when you decide whether to multiply or divide, and AO3 problem-solving when a conversion is one step inside a larger calculation.

Key Vocabulary

Term	Meaning
Metric units	The decimal system of units (mm, cm, m, km; g, kg; ml, l)
Conversion factor	The number you multiply or divide by to change units
Area conversion	Uses the square of the length factor
Volume conversion	Uses the cube of the length factor
Compound-unit conversion	Converting a rate, e.g. km/h to m/s
Exchange rate	The rate for converting one currency to another

Metric Length, Mass and Capacity

The metric system is built on powers of $10$10, which is what makes converting between metric units so much easier than the old imperial system. Every step up or down a unit is a multiplication or division by $10$10, $100$100 or $1{,}000$1,000. The hardest part is simply deciding whether to multiply or divide, and there is a reliable way to settle it: ask whether the new unit is smaller or larger than the old one. A smaller unit means you need more of them to make the same amount, so you multiply; a larger unit means you need fewer, so you divide. Learn these conversion factors:

Quantity	Conversions
Length	$1$1 cm $= 10$=10 mm; $1$1 m $= 100$=100 cm; $1$1 km $= 1{,}000$=1,000 m
Mass	$1$1 g $= 1{,}000$=1,000 mg; $1$1 kg $= 1{,}000$=1,000 g; $1$1 tonne $= 1{,}000$=1,000 kg
Capacity	$1$1 litre $= 1{,}000$=1,000 ml; $1$1 cl $= 10$=10 ml; $1$1 litre $= 100$=100 cl

Rule: going to a smaller unit, multiply (more of them); going to a larger unit, divide (fewer of them).

Worked Example 1

Convert $3.6$3.6 km to metres.

A kilometre is larger than a metre, and $1$1 km $= 1{,}000$=1,000 m, so multiply:

$3.6 \times 1{,}000 = 3{,}600 \text{ m}.$3.6×1,000=3,600 m.

Answer: $3{,}600$3,600 m. The answer has more of the new unit (metres) than the original number of kilometres, which is the check that multiplying was correct — a metre is smaller than a kilometre.

Worked Example 2

Convert $4{,}500$4,500 g to kilograms.

A gram is smaller than a kilogram, so divide by $1{,}000$1,000:

$4{,}500 \div 1{,}000 = 4.5 \text{ kg}.$4,500÷1,000=4.5 kg.

Answer: $4.5$4.5 kg.

Worked Example 3

Convert $250$250 cl to litres.

Since $1$1 litre $= 100$=100 cl, divide by $100$100:

$250 \div 100 = 2.5 \text{ litres}.$250÷100=2.5 litres.

Answer: $2.5$2.5 litres. Here a centilitre is smaller than a litre, so you expect fewer litres than centilitres — and $2.5$2.5 is indeed smaller than $250$250, confirming that dividing was correct.

Common error: multiplying when you should divide. Always ask whether the answer should have more or fewer of the new unit.

Worked Example 3b

Convert $0.75$0.75 m to millimetres.

This is a two-step conversion, or one step with a combined factor. Since $1$1 m $= 100$=100 cm and $1$1 cm $= 10$=10 mm, then $1$1 m $= 1{,}000$=1,000 mm. Going to a smaller unit, multiply:

$0.75 \times 1{,}000 = 750 \text{ mm}.$0.75×1,000=750 mm.

Answer: $750$750 mm.

Area Conversions

Area conversions catch out more students than any other part of this topic, so it is worth understanding why they work rather than just memorising factors. For area, the conversion factor is squared, because area has two dimensions. Think of a square that is $1$1 m by $1$1 m: that is $1$1 m². But $1$1 m is $100$100 cm, so the same square is $100$100 cm by $100$100 cm, which is $100 \times 100 = 10{,}000$100×100=10,000 cm². So $1$1 m² $= 100^2 = 10{,}000$=1002=10,000 cm² — the length factor of $100$100 has to be applied twice, once for each dimension. The same reasoning gives every area conversion below.

Area conversion	Factor
$1$1 cm² $=$= ? mm²	$10^2 = 100$102=100
$1$1 m² $=$= ? cm²	$100^2 = 10{,}000$1002=10,000
$1$1 km² $=$= ? m²	$1{,}000^2 = 1{,}000{,}000$1,0002=1,000,000

Worked Example 4

Convert $5$5 m² to cm².

Multiply by $100^2 = 10{,}000$1002=10,000:

$5 \times 10{,}000 = 50{,}000 \text{ cm}^2.$5×10,000=50,000 cm2.

Answer: $50{,}000$50,000 cm². It can be surprising how large the numbers become — a $5$5 m² room is $50{,}000$50,000 cm², because each square metre contains ten thousand square centimetres.

Common error: multiplying by $100$100 instead of $10{,}000$10,000. Area uses the square of the length factor, so the conversion number is always much bigger than for a length.

Worked Example 4b

Convert $30{,}000$30,000 cm² to m².

Going from the smaller unit (cm²) to the larger unit (m²), so divide by $100^2 = 10{,}000$1002=10,000:

$30{,}000 \div 10{,}000 = 3 \text{ m}^2.$30,000÷10,000=3 m2.

Answer: $3$3 m².

Common error: dividing by $100$100 to get $300$300 m². The factor for area is the square, $10{,}000$10,000, not the length factor $100$100.

Volume Conversions

Volume conversions follow the same logic as area, but the factor is cubed, because volume has three dimensions. A cube that is $1$1 cm on each side is $1$1 cm³; since $1$1 cm $= 10$=10 mm, the same cube is $10$10 mm by $10$10 mm by $10$10 mm $= 10^3 = 1{,}000$=103=1,000 mm³. So $1$1 cm³ $= 1{,}000$=1,000 mm³, and similarly $1$1 m³ $= 100^3 = 1{,}000{,}000$=1003=1,000,000 cm³. The length factor is applied three times, once for each dimension.

A useful link to remember is that $1$1 litre $= 1{,}000$=1,000 cm³, and $1$1 ml $= 1$=1 cm³. This is how volume in cubic centimetres connects to capacity in litres and millilitres.

Worked Example 5

Convert $2.5$2.5 m³ to cm³.

Multiply by $100^3 = 1{,}000{,}000$1003=1,000,000:

$2.5 \times 1{,}000{,}000 = 2{,}500{,}000 \text{ cm}^3.$2.5×1,000,000=2,500,000 cm3.

Answer: $2{,}500{,}000$2,500,000 cm³.

Worked Example 6

A container holds $3{,}500$3,500 cm³ of water. Work out this volume in litres.

Since $1$1 litre $= 1{,}000$=1,000 cm³, divide by $1{,}000$1,000:

$3{,}500 \div 1{,}000 = 3.5 \text{ litres}.$3,500÷1,000=3.5 litres.

Answer: $3.5$3.5 litres.

Compound-Unit Conversions

A compound unit combines two units, such as kilometres per hour (km/h) or grams per cubic centimetre (g/cm³). Converting a compound unit means converting both the top and bottom units, which is the step students most often forget. To change km/h to m/s, you must convert kilometres to metres (multiply by $1{,}000$1,000) and hours to seconds (divide by $3{,}600$3,600). Doing only one of the two gives a badly wrong answer. The most common compound-unit conversion at GCSE is between km/h and m/s, so it is worth knowing the shortcut: divide by $3.6$3.6 to go from km/h to m/s, and multiply by $3.6$3.6 to go the other way.

Worked Example 7

Convert $72$72 km/h to m/s.

$72$72 km/h $= 72 \times 1{,}000 = 72{,}000$=72×1,000=72,000 m per hour. There are $3{,}600$3,600 seconds in an hour, so:

$\dfrac{72{,}000}{3{,}600} = 20 \text{ m/s}.$3,60072,000​=20 m/s.

Answer: $20$20 m/s. (Shortcut: divide km/h by $3.6$3.6, and indeed $72 \div 3.6 = 20$72÷3.6=20.)

Worked Example 8

Convert $15$15 m/s to km/h.

Multiply by $3.6$3.6 (the reverse of the shortcut above):

$15 \times 3.6 = 54 \text{ km/h}.$15×3.6=54 km/h.

Answer: $54$54 km/h.

Common error: converting only one of the two units. A rate has units top and bottom, so both must change.

Currency Conversion

Currency conversion is simply direct proportion using an exchange rate as the constant. The rate tells you how much of one currency you get for one unit of another, so converting is a single multiplication or division. The key decision is which way round: to convert from the stated base currency to the other, multiply by the rate; to convert back, divide by the rate.

Worked Example 9

The exchange rate is £1 $= 1.18$=1.18 euros. Convert £250 to euros.

Multiply by the rate:

$250 \times 1.18 = 295 \text{ euros}.$250×1.18=295 euros.

Answer: $295$295 euros.

Worked Example 10

Using the same rate £1 $= 1.18$=1.18 euros, convert $531$531 euros to pounds.

Divide by the rate:

$531 \div 1.18 = 450.$531÷1.18=450.

So the amount is £450.

Answer: £450.

Worked Example 10b

The exchange rate is £1 $= 1.40$=1.40 Australian dollars. A camera costs $315$315 Australian dollars. Work out the cost in pounds.

To convert from dollars back to pounds, divide by the rate:

$315 \div 1.40 = 225.$315÷1.40=225.