Ratio, proportion and rates of change worksheet — GCSE Higher

1.A shop's daily profit, in pounds, is plotted against the price charged for an item, in pounds. A tangent to the profit graph at a price of £15 passes through the points (12, 540) and (18, 480). Work out the instantaneous rate of change of profit with respect to price at £15, and state whether profit is increasing or decreasing there.

2.Two mathematically similar logos are printed on a poster. They have areas 20 cm² and 45 cm². Nadia says that the length scale factor from the smaller logo to the larger logo is 45 ÷ 20 = 2.25. Give a reason why Nadia is incorrect, and work out the correct length scale factor.

3.In a class, 50% of the students study French, 30% study Spanish and the rest study German. Write down the ratio of French : Spanish : German students, in its simplest form.

4.A coach journey of 240 km takes 3 hours. For this fixed distance the average speed needed is inversely proportional to the time taken. Work out the average speed needed to complete the same journey in 2 hours.

5.The height of a ball, in metres, above the ground is plotted against time, in seconds, after it is thrown. The tangent to the graph at t = 1.5 seconds has gradient 2. Which of these four statements about the ball at t = 1.5 seconds is correct?

6.Simplify the ratio 12 : 18 : 30 to its simplest form.

7.A map has a scale of 1:25000. A road on the map is 6 cm long. Work out the real-life length of the road, giving your answer in kilometres.

8.Leah puts £4000 into a savings account paying 3% compound interest each year. At the end of 2 years she takes out all of the money and spends £1500 of it on a laptop. Work out how much of the money she has left.

9.A hiker's distance from the start of the trail, in kilometres, is plotted against time, in minutes, since she set off. At t = 45 minutes she stops to rest, so her distance from the start is neither increasing nor decreasing at that instant. Which statement about the tangent to the distance–time graph at t = 45 minutes is correct?

10.Work out the value of x when 4/x = 2/5.

11.A room is drawn on a plan with a scale of 1 : 200. On the plan the room has an area of 12 cm². Work out the real area of the room, in square metres.

12.A company's turnover this year is £180,000. Last year's turnover was £120,000. Write down this year's turnover as a percentage of last year's turnover.

13.A scale drawing of a garden uses a scale of 1 : 20. Write down a formula for the real length, L metres, in terms of the length on the drawing, d centimetres, given that 1 metre = 100 centimetres.

14.Two mathematically similar circles have radii 4 cm and 20 cm. Write the ratio of the area of the smaller circle to the area of the larger circle in its simplest form.

15.At an animal shelter, the ratio of cats to dogs is 2:5. There are 18 more dogs than cats. Work out the number of cats.