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Maths · Ratio and Proportion

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Units, Conversions and Scale

Converting between metric units, using scales on maps and drawings, and changing units of area and volume.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Convert between metric units of length, mass, capacity and time without a calculator.
  2. 2Convert between units of area and volume (Higher tier).
  3. 3Use a scale such as 1 cm to 2 m, or a map ratio such as 1 : 50 000, to find real lengths and drawing lengths.
  4. 4Solve problems that need a conversion before the main calculation.

Why units matter

Many marks in this topic are lost before any real maths starts, by mixing centimetres with metres or grams with kilograms. On a non-calculator paper the conversions are always multiplications or divisions by 10, 100 or 1000, so they are quick once you know which way to go. The golden rule is to decide whether the number should get bigger or smaller before you move the decimal point.

Facts you must know

These are never given on the formula sheet, so they have to be learned.

  • Length

    10 mm \(=\) 1 cm, 100 cm \(=\) 1 m, 1000 m \(=\) 1 km.

  • Mass

    1000 g \(=\) 1 kg, 1000 kg \(=\) 1 tonne.

  • Capacity

    1000 ml \(=\) 1 litre, 100 cl \(=\) 1 litre, and 1 cm\(^3\) \(=\) 1 ml.

  • Time

    60 seconds in a minute, 60 minutes in an hour, 24 hours in a day. Time is not decimal, so 1.5 hours is 90 minutes, not 150.

  • Imperial units

    Edexcel, AQA and OCR expect you to know rough equivalents only when they are given in the question, such as 1 inch \(\approx\) 2.5 cm, 1 kg \(\approx\) 2.2 pounds, 1 gallon \(\approx\) 4.5 litres. Learn the common ones in case they are not.

Converting units

(a) Change 3.6 km to metres. (b) Change 2450 g to kilograms. (c) Change 0.75 litres to millilitres.

Show the solutionHide the solution
  1. 1 (a) Big to small, so multiply \(3.6 \times 1000 = 3600\) m.
  2. 2 (b) Small to big, so divide \(2450 \div 1000 = 2.45\) kg.
  3. 3 (c) Big to small, so multiply \(0.75 \times 1000 = 750\) ml.
  4. 4 Check the size A kilometre is much longer than a metre, so there should be many more metres than kilometres.

Answer(a) 3600 m (b) 2.45 kg (c) 750 ml

Problems with mixed units

A question will often give one quantity in one unit and another in a different unit. Put them in the same unit before doing anything else.

  • Make the units match first

    Do the conversion as its own line of working so the method mark is easy to see.

  • Choose the easier unit

    Work in whichever unit leads to whole numbers. Changing 2.5 m to 250 cm is usually easier than changing 80 cm to 0.8 m.

  • Convert the answer back

    If the question asks for the answer in kilograms, finish in kilograms.

  • Check with common sense

    A bag of sugar does not weigh 2500 kg.

Using two different units

A recipe needs 250 g of flour for one cake. Mia has a 2 kg bag of flour. How many cakes can she make?

Show the solutionHide the solution
  1. 1 Make the units match \(2\text{ kg} = 2 \times 1000 = 2000\) g.
  2. 2 Divide the flour by the amount for one cake \(2000 \div 250\).
  3. 3 Calculate \(2000 \div 250 = 8\).
  4. 4 Write the answer 8 cakes.

Answer8 cakes

Scale drawings and maps

A scale tells you how a length on a drawing or map is linked to the real length.

  • Written as a statement

    "1 cm represents 2 m" means every centimetre on the drawing is 2 metres in real life.

  • Written as a ratio

    A scale of \(1 : 50\,000\) means 1 unit on the map is 50 000 of the same unit in real life. The units on both sides are the same, so 1 cm on the map is 50 000 cm, which is 500 m or 0.5 km.

  • Drawing to real life

    Multiply the drawing length by the scale factor.

  • Real life to drawing

    Divide the real length by the scale factor.

Maps with a ratio scale

A map has a scale of \(1 : 25\,000\). Two towns are 8 cm apart on the map. Work out the real distance in kilometres.

Show the solutionHide the solution
  1. 1 Multiply by the scale \(8 \times 25\,000 = 200\,000\) cm.
  2. 2 Centimetres to metres \(200\,000 \div 100 = 2000\) m.
  3. 3 Metres to kilometres \(2000 \div 1000 = 2\) km.
  4. 4 Write the answer 2 km.

Answer2 km

Units of area and volume (Higher tier)

When the unit is squared or cubed, the conversion factor is squared or cubed too. This is a Higher tier skill that catches out many students.

  • Why it is different

    1 m \(=\) 100 cm, but 1 m\(^2\) is a square of side 100 cm, so it is \(100 \times 100 = 10\,000\) cm\(^2\).

  • Volume

    1 m\(^3\) is a cube of side 100 cm, so it is \(100 \times 100 \times 100 = 1\,000\,000\) cm\(^3\).

  • Square millimetres

    1 cm\(^2\) \(= 10 \times 10 = 100\) mm\(^2\).

  • Quick check

    Draw a unit square on the diagram above and label its sides. It is much safer than trying to remember the number.

Changing an area

(Higher tier) Change 3.2 m\(^2\) to cm\(^2\).

Show the solutionHide the solution
  1. 1 Find the conversion for area \(1\text{ m}^2 = 100 \times 100 = 10\,000\text{ cm}^2\).
  2. 2 Multiply \(3.2 \times 10\,000 = 32\,000\).
  3. 3 Write the answer 32 000 cm\(^2\).

Answer32 000 cm\(^2\)

Test yourself

  1. 1

    How many metres are in a kilometre?

    Show answerHide answer

    1000.

  2. 2

    How many grams are in a kilogram?

    Show answerHide answer

    1000.

  3. 3

    How many millilitres are in a litre?

    Show answerHide answer

    1000.

  4. 4

    A map scale is \(1 : 50\,000\). What does 1 cm on the map represent in real life?

    Show answerHide answer

    50 000 cm, which is 500 m.

  5. 5

    How many square centimetres are in a square metre?

    Show answerHide answer

    10 000.

Exam technique: units and scale

These questions look easy, which is why a careless slip costs marks.

  • Say which way you are going

    Write "\(\times 100\)" next to a conversion so you can check it.

  • Write the units on each line

    A number with no unit is easy to misuse.

  • Do not convert the area by the length

    4 m\(^2\) is not 400 cm\(^2\).

  • Read the scale carefully

    "1 cm represents 2 m" and "1 : 2" are very different scales.

Summary and exam focus

  • Big unit to small unit means multiply, and small to big means divide.
  • Know 10, 100 and 1000 for length, mass and capacity, and that 1 cm\(^3\) is 1 ml.
  • Put every quantity in the same unit before calculating.
  • Multiply by the scale for real lengths and divide by it to draw.
  • Area and volume conversions use the square and cube of the length conversion (Higher tier).

Exam focus

A map has a scale of 1 cm to 5 km. Two towns are 7.5 cm apart on the map. Work out the real distance between them. (2 marks) (2 marks)

Multiply \(7.5 \times 5\) to get 37.5 km. Decimals are likely here, so write \(7 \times 5 = 35\) and \(0.5 \times 5 = 2.5\) separately and add them. Always put the unit in your answer.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Metric units
Units based on powers of ten, such as millimetres, centimetres, metres and kilometres.
Conversion factor
The number you multiply or divide by to change from one unit to another.
Capacity
The amount a container can hold, measured in litres or millilitres.
Tonne
A unit of mass equal to 1000 kilograms.
Scale
The ratio between a length on a drawing or map and the real length.
Scale factor
The number you multiply by to turn a drawing length into a real length.
Map
A scale drawing of an area, where lengths on the paper represent real distances.
Area
The amount of surface a shape covers, measured in square units.
Volume
The amount of space a solid takes up, measured in cubic units.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Convert 2 marks Easier

(a) Convert 0.45 km to metres. [1 mark] (b) Convert 5.2 kg to grams. [1 mark]

Mark scheme — 2 marks available

  • (a) 450 m — B1
  • (b) 5200 g — B1

Model answer

(a) \(0.45 \times 1000 = 450\) m. (b) \(5.2 \times 1000 = 5200\) g.

2. Exam question Calculate 2 marks Easier

A bag of rice has a mass of 1.5 kg. Calculate how many 60 g portions can be made from the bag.

Mark scheme — 2 marks available

  • 1500 g seen, or \(1500 \div 60\) — M1
  • 25 — A1

Model answer

\(1.5\text{ kg} = 1500\) g, and \(1500 \div 60 = 25\) portions.

3. Exam question Calculate 5 marks Core

The diagram shows a scale drawing of a living room. The scale is 1 cm represents 2 m. (a) Calculate the real length of the room. [1 mark] (b) Calculate the real width of the room. [1 mark] A rug has an area of 21 m\(^2\). (c) What percentage of the floor of the room is covered by the rug? [3 marks]

A floor plan of a living room measuring 6 cm by 3.5 cm on the drawing, with the scale 1 cm represents 2 m.

Mark scheme — 5 marks available

  • (a) 12 m — B1
  • (b) 7 m — B1
  • (c) \(12 \times 7 = 84\) — M1
  • (c) \(\dfrac{21}{84}\) or \(21 \div 84\) — M1
  • (c) 25% — A1

Model answer

(a) \(6 \times 2 = 12\) m. (b) \(3.5 \times 2 = 7\) m. (c) The floor area is \(12 \times 7 = 84\) m\(^2\), so the rug covers \(\dfrac{21}{84} = \dfrac{1}{4} = 25\%\).

4. Exam question Calculate 3 marks Core

A plan of a field has a scale of \(1 : 5000\). The field is 4.5 cm long on the plan. Calculate the real length of the field in metres.

Mark scheme — 3 marks available

  • \(4.5 \times 5000\) — M1
  • \(22\,500\) cm — M1
  • 225 m — A1

Model answer

\(4.5 \times 5000 = 22\,500\) cm, which is 225 m.

5. Exam question Calculate 3 marks Core

A water tank has a volume of 2.4 m\(^3\). 1 m\(^3\) is 1000 litres. Calculate the volume of the tank in litres, and the number of 20-litre buckets needed to fill it.

Mark scheme — 3 marks available

  • \(2.4 \times 1000\) — M1
  • 2400 litres — A1
  • 120 buckets — B1

Model answer

\(2.4 \times 1000 = 2400\) litres. The number of buckets is \(2400 \div 20 = 120\).

6. Exam question Convert 4 marks Stretch

(a) Convert 5 m\(^2\) to cm\(^2\). [2 marks] (b) Convert \(15\,000\) mm\(^2\) to cm\(^2\). [2 marks]

Mark scheme — 4 marks available

  • (a) \(5 \times 10\,000\) — M1
  • (a) \(50\,000\) — A1
  • (b) \(15\,000 \div 100\) — M1
  • (b) 150 — A1

Model answer

(a) \(1\text{ m}^2 = 10\,000\) cm\(^2\), so \(5 \times 10\,000 = 50\,000\) cm\(^2\). (b) \(1\text{ cm}^2 = 100\) mm\(^2\), so \(15\,000 \div 100 = 150\) cm\(^2\).

7. Multiple choice 1 mark Easier

Change 3.6 km to metres.

  1. A 3600 m Correct
  2. B 360 m
  3. C 36 m
  4. D 36 000 m

Why: Multiply by 1000: \(3.6 \times 1000 = 3600\).

8. Multiple choice 1 mark Easier

Change 2450 g to kilograms.

  1. A 24.5 kg
  2. B 0.245 kg
  3. C 2450 kg
  4. D 2.45 kg Correct

Why: Divide by 1000: \(2450 \div 1000 = 2.45\).

9. Multiple choice 1 mark Easier

Change 0.75 litres to millilitres.

  1. A 75 ml
  2. B 7.5 ml
  3. C 750 ml Correct
  4. D 7500 ml

Why: Multiply by 1000: \(0.75 \times 1000 = 750\).

10. Multiple choice 1 mark Easier

Change 250 cm to metres.

  1. A 25 m
  2. B 2.5 m Correct
  3. C 0.25 m
  4. D 2500 m

Why: Divide by 100: \(250 \div 100 = 2.5\).

11. Multiple choice 1 mark Core

A map has a scale of \(1 : 50\,000\). Two towns are 4 cm apart on the map. What is the real distance?

  1. A 2 km Correct
  2. B 200 km
  3. C 20 km
  4. D 0.2 km

Why: \(4 \times 50\,000 = 200\,000\) cm. Divide by 100 to get 2000 m, then by 1000 to get 2 km.

12. Multiple choice 1 mark Easier

A scale drawing uses 1 cm to represent 2 m. A line on the drawing is 7 cm. How long is it in real life?

  1. A 3.5 m
  2. B 9 m
  3. C 140 m
  4. D 14 m Correct

Why: Multiply by the scale: \(7 \times 2 = 14\) m.

13. Multiple choice 1 mark Core

A cake uses 250 g of flour. How many cakes can be made from a 2 kg bag?

  1. A 80
  2. B 125
  3. C 8 Correct
  4. D 500

Why: \(2\text{ kg} = 2000\) g, and \(2000 \div 250 = 8\).

14. Multiple choice 1 mark Stretch

How many cm\(^2\) are in 1 m\(^2\)?

  1. A 100
  2. B 10 000 Correct
  3. C 1000
  4. D 1 000 000

Why: A square metre is a square of side 100 cm, so its area is \(100 \times 100 = 10\,000\) cm\(^2\).

15. Multiple choice 1 mark Stretch

Change 5 m\(^3\) to cm\(^3\).

  1. A 5 000 000 cm\(^3\) Correct
  2. B 500 000 cm\(^3\)
  3. C 500 cm\(^3\)
  4. D 5000 cm\(^3\)

Why: \(1\text{ m}^3 = 100^3 = 1\,000\,000\) cm\(^3\), so \(5\text{ m}^3 = 5\,000\,000\) cm\(^3\).