GCSE Maths revision · Ratio and proportion

Rates of change on graphs

Foundation and Higher

The idea

Two quantities are directly proportional when their graph is a straight line through the origin. The gradient of a graph is the rate of change: on a distance-time graph it is the speed.

Pick two points on the line far apart.
Gradient $= \frac{\text{change in } y}{\text{change in } x}$ , with units of $y$ per $x$ .
Say what it means in context: $4$ metres per second, or $\text{£}0.35$ per minute.

Watch out A straight line that does not pass through $(0, 0)$ is not proportional.

See it worked

Worked exampleNon-calculator3 marks

$y$ is directly proportional to $x$ . When $x = 4$ , $y = 12$ .

(a) Find the value of $y$ when $x = 10$ .

(b) Write the ratio $y : x$ in its simplest form.

(a) $y$ is always $3$ times $x$ , because $12 \div 4 = 3$ .
When $x = 10$ , $y = 3 \times 10 = 30$ .
(b) $12 : 4 = 3 : 1$ , and this ratio is the same for every pair of values.

Answer (a) $30$ ; (b) $3 : 1$

Your turn

Question 1Calculator allowed2 marks

A distance-time graph consists of straight segments joining (0, 0), (8, 480), (11, 480) and (17, 840). Time is in seconds and distance is in metres. Work out the speed during the final segment.

Question 2Non-calculator4 marks

A taxi fare graph passes through $(2,18)$ and $(6,46)$ , with distance in km and fare in pounds.

(a) Find the charge per kilometre.

(b) Find the starting charge.

Question 3Non-calculator1 mark

A straight-line graph shows water volume in cm³ against time in minutes. Its gradient is 300 cm³ per minute. What is this rate in litres per hour? Select one answer.

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