A conversion graph is a straight line through the origin that converts one unit into another: go up from one axis to the line, then across to the other axis. A straight line through the origin always shows direct proportion, and its gradient is the rate, such as kilometres per hour or euros per pound. A straight line that does not pass through the origin shows a fixed charge plus a rate, which is not direct proportion.
Proportion graphs
- Direct proportion: a straight line through the origin
- Gradient = change in y ÷ change in x = the rate
- Distance–time graph: gradient = speed
- A line not through the origin: not direct proportion (there is a fixed starting value)
Reading a conversion graph
To convert using a graph, start at the value on one axis, go straight to the line, then across to the other axis and read off the value. For values beyond the graph, use proportion: if 5 miles = 8 km, then 25 miles = 40 km. Read the scale on each axis carefully before reading values.
The gradient as a rate
For a proportion graph, the gradient tells you the rate: how much y changes for each unit of x. On a distance–time graph, a line through (5, 120) has gradient 120 ÷ 5 = 24, so the speed is 24 km/h. On a cost graph, the gradient is the cost per unit.
Recognising direct proportion
Two quantities are in direct proportion if doubling one doubles the other. On a graph, this is a straight line through the origin. A straight line that starts above the origin, such as a phone plan costing £10 plus 5p a minute, is linear but not direct proportion, because zero minutes still costs £10.
Comparing proportion graphs
When two lines start at the origin, the steeper line has the greater rate. For example, if one tap's graph passes through (2, 6) and another through (2, 10), the second tap fills faster, at 5 units per second compared with 3.
Using the rate in calculations
Once you know the rate, you can calculate without the graph. If £1 = €1.20, then €84 is 84 ÷ 1.2 = £70. Check that the direction of conversion makes sense: converting to the smaller unit should give a larger number.
Worked examples
Example 1. A graph shows that 5 miles = 8 km. Convert 25 miles to kilometres.
- 25 miles is 5 × 5 miles.
- 5 × 8 km.
Answer: 40 km.
Example 2. A distance–time graph is a straight line through the origin and (5, 120), with time in hours and distance in km. Find the speed.
- Gradient = 120 ÷ 5.
Answer: 24 km/h.
Example 3. The exchange rate is £1 = €1.20. Convert €84 to pounds.
- Euros to pounds: divide by 1.2.
- 84 ÷ 1.2.
Answer: £70.
Example 4. A phone plan costs £10 a month plus 5p per minute. Is the cost directly proportional to the number of minutes? Explain.
- Cost = 10 + 0.05m, so 0 minutes still costs £10.
- The graph is a straight line, but it does not pass through the origin.
Answer: No, because the graph does not pass through the origin.
Example 5. Tap A's graph passes through (2, 6) and tap B's through (2, 10), both straight lines from the origin, with time in seconds and volume in litres. Which tap fills faster, and by how much?
- Tap A: 6 ÷ 2 = 3 litres per second. Tap B: 10 ÷ 2 = 5 litres per second.
Answer: Tap B, by 2 litres per second.
Common mistakes
- Misreading the scale. Check what each square on each axis represents.
- Converting the wrong way. Euros to pounds at £1 = €1.20 means divide by 1.2.
- Calling any straight line direct proportion. It must also pass through the origin.
- Using one point not on the line. Read points exactly on the line to find the gradient.
- Forgetting units for a rate. Give units such as km/h or € per £.
How proportion graphs are examined
Proportion and conversion graphs are Foundation tier topics in the 1MA1 GCSE and the 4MA1 International GCSE, often with real-life contexts such as currency, speed or bills. Questions ask you to:
- read values from a conversion graph in both directions
- work out a rate from the gradient of a graph
- decide whether a graph shows direct proportion, with a reason
- compare two rates from their graphs
Practise with the direct and inverse proportion worksheet and the proportion revision engine.
Proportion graphs link to straight-line graphs and exchange rates.
Frequently asked questions
What does a straight line through the origin show?
That the two quantities are in direct proportion.
What does the gradient of a proportion graph tell you?
The rate, such as the speed or the cost per unit.
Is a straight line that does not go through the origin direct proportion?
No. It shows a fixed amount plus a rate.