Ratio, proportion and rates of change worksheet — GCSE Higher

1.A process is modelled by the recurrence P_{n+1} = 0.6P_n + 40. As n increases, P_n approaches a long-run value L, which satisfies L = 0.6L + 40. Solve this equation to find L.

2.A concrete block exerts a force of 45 N on the ground through a base with an area of 0.05 m². Work out the pressure on the ground, in N/m².

3.Write the ratio 2 : 1/4 as a ratio of whole numbers in its simplest form.

4.An online shop's total number of orders received is recorded at the end of each day for five consecutive days: Day 1, 1,284 orders; Day 2, 1,509 orders; Day 3, 1,830 orders; Day 4, 2,296 orders; Day 5, 2,510 orders. Work out between which two consecutive days the average rate of increase in orders was greatest.

5.A photo is enlarged so that its new width is 1.4 times its original width. Write the multiplier that would scale the new width back down to the original width, as a fraction in its simplest form.

6.A scale drawing of a park has a scale of 1 : 2500. On the drawing, the distance between the entrance and the lake is 4.4 cm. A jogger runs from the entrance to the lake and then back to the entrance. Work out the total distance the jogger runs, in kilometres.

7.A fixed job of fitting a solar array is shared between installers, and the time taken is inversely proportional to the number of installers working on it at once. With 4 installers the job takes 18 hours. Construct the equation connecting time T and number of installers n, then work out how long it would take with 6 installers.

8.Triangle ABC is mathematically similar to triangle PQR, with AB corresponding to PQ and BC corresponding to QR. AB = 6 cm, BC = 8 cm and PQ = 12 cm. Work out the length of QR.

9.A salt solution has a concentration of 10%. The solution contains 50 g of salt. Work out the total mass of the solution.

10.Two quantities y and z are each in direct proportion to x, and are given by y = 3x and z = 7x. Work out the difference between the value of y and the value of z when x = 4.

11.Two mathematically similar triangular flags have areas in the ratio 4 : 25. The height of the smaller flag is 6 cm. Work out the height of the larger flag.

12.A map's scale is given as '1 cm represents 2 km'. Write this scale in the form 1 : n.

13.A scale model of a bridge is built at a scale of 1 : 120. The real bridge is 84 m long. Work out the length of the model, in centimetres.

14.6 identical taps fill a paddling pool in 20 minutes. Each tap fills at the same steady rate. Work out how long 3 of these taps would take to fill the same pool.

15.The population of a town, P thousand, is plotted against time, t years, since 2020. A tangent to the graph at t = 6 has gradient 2.5. What does this gradient represent?