Ratio, proportion and rates of change worksheet — GCSE Higher

1.A tangent to the graph of population against time at the point where t = 4 passes through (2, 180) and (6, 260). Work out the instantaneous rate of change of the population at t = 4, in people per year.

2.Cheng is paid an hourly rate that is proportional to the number of hours she works. She earns £52.50 for 7 hours. Work out how much she earns for 11 hours, assuming the same rate.

3.Write 75p : £1.50 as a ratio of whole numbers in its simplest form.

4.A car travels for 3 hours at an average speed of 80 km/h and then for 1 hour at an average speed of 40 km/h. Work out the average speed for the whole journey.

5.A shop's number of online orders is proportional to its number of in-store orders. Last month there were 45 in-store orders and 180 online orders. Write the ratio of online orders to in-store orders, in its simplest form.

6.A straight line passes through the points (1, 20) and (5, 8). Work out the gradient of the line.

7.Line P passes through the origin and the point (3, 21). Line Q passes through the origin and the point (5, 45). Both lines represent the distance, in metres, run by an athlete against the time, in seconds. Work out the gradient of the steeper line.

8.Two scale plans of the same football pitch are drawn: Plan A has a scale of 1 : 1000 and Plan B has a scale of 1 : 200. A pitch length is 8 cm on Plan A. Work out the length of the same pitch on Plan B, in centimetres.

9.y is directly proportional to x. When x = 5, y = 18. Work out the value of y when x = 15.

10.y is inversely proportional to x, so y = k ÷ x. When x = 5, the value of y is 8. Work out the value of k.

11.Two numbers are in the ratio 3 : 7. Their sum is 60. Work out the positive difference between the two numbers.

12.On a straight-line graph the volume of water, V litres, in a tank is plotted on the vertical axis and the time, t minutes, on the horizontal axis. The line passes through (2, 50) and (6, 130). Work out the gradient of the line and give its units.

13.A candidate wants to estimate the instantaneous rate of change of a reservoir's water level, in metres, at t = 5 days after heavy rain began. The reservoir's water level is plotted against time, in days, since the rain began. The candidate uses the chord joining the points at t = 0 days and t = 20 days to estimate the rate of change at t = 5 days. Give a reason why this is likely to be a poor estimate.

14.A machine fills bottles at a constant rate. It fills 18 bottles in 3 minutes. Working at the same rate, work out how many bottles the machine fills in 8 minutes.

15.An amount of money is shared in the ratio 1:2:3. The largest share is £90 more than the smallest share. Work out the total amount that was shared.