Algebra worksheet — GCSE Higher

1.A delivery van's distance-time graph shows the following: it travels 9 km in the first 15 minutes at a steady speed, stops for 45 minutes to unload, then travels a further 21 km in the next 30 minutes, also at a steady speed. Work out the average speed, in km/h, for the whole journey.

2.The mean of four numbers is 12.5. Three of the numbers are 8, 15 and 19. Form an equation using n for the fourth number, and solve it to find n.

3.Expand and simplify (x + 1)(x + 2)(x + 3).

4.A geometric sequence begins 80, 40, 20, 10, ... Work out the next term in the sequence.

5.Simplify 2/(x + 1) + 3/(x − 2), giving your answer as a single fraction.

6.A candle is 30 cm tall when lit and burns at a constant rate. After burning for 5 minutes, its height is 20 cm. The height h cm after t minutes is given by h = 30 − mt. Work out the value of m.

7.By completing the square, find the turning point of the curve y = x² − 4x + 9.

8.Tom buys 2 notebooks and 6 pencils on Monday, priced at p pence each for a notebook and q pence each for a pencil. On Tuesday he buys 3 more notebooks and 1 more pencil. Write a simplified expression, in pence, for the total amount Tom has spent on notebooks and pencils.

9.Water is poured into a container at a constant rate. A graph of the depth of the water, in cm, against the time since pouring began, in minutes, is a straight line through the origin, and it passes through the point (3, 12). Use the graph to work out the depth of the water after 7 minutes.

10.Solve the inequality 5 − x ≤ 2.

11.The equation x² − 6x + k = 0 has exactly one solution. Work out the value of k.

12.The first five terms of a quadratic sequence are 6, 11, 18, 27, 38. Work out an expression, in terms of n, for the nth term.

13.Solve the inequality 2(x − 1) ≤ 8.

14.Solve 4x − (2x − 6) = 18

15.A number machine multiplies its input by 3 and then adds 7. The output is 1. Work out the input.