Printable · GCSE Foundation · ages 14-16
Algebra worksheet — GCSE Foundation
Fifteen questions across the algebra statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
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Algebra worksheet — GCSE Foundation
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- 1.y = x + 4. Work out the value of y when x = 3.y = x + 4
- 2.Priya has a budget of £50 for a school trip. The coach costs £14 and each student ticket costs £4. Using the inequality 14 + 4s ≤ 50, work out the greatest number of student tickets, s, she can buy.
- 3.A pattern is made from tiles. Pattern 1 has 7 tiles, pattern 2 has 11 tiles, pattern 3 has 15 tiles, and pattern 4 has 19 tiles, with each new pattern having 4 more tiles than the one before it. Work out an expression, in terms of n, for the number of tiles in pattern n.
- 4.Write down the gradient of the line with equation y = 7x − 2.y = 7x − 2
- 5.A phone plan costs a fixed £15 per month plus 8p per minute of calls. In one month, Jamal's bill was £23.80. Form an equation using m for the number of minutes, and solve it to find how many minutes Jamal used that month.
- 6.To rearrange y = 2x − 5 to make x the subject, Sam writes: 'Add 5 to both sides to get y + 5 = 2x, then divide both sides by 2 to get x = (y + 5)/2.' Which statement about Sam's method is correct?y = 2x − 5
- 7.The triangular numbers begin 1, 3, 6, 10, 15, ... Work out the next term in the sequence.
- 8.A rectangle has length (2x + 5) cm and width (x − 2) cm. Work out an expression, in terms of x, for the perimeter of the rectangle. Give your answer in its simplest form.
- 9.Isla says that the lines y = 2x + 1 and y = −2x + 3 are parallel. Write down the statement that gives the correct answer for the correct reason.y = 2x + 1y = -2x + 3
- 10.A gardener charges a fixed fee of £15 plus £8 for each hour worked. The total charge, £T, for a job lasting h hours is given by T = 15 + 8h. Work out the charge for a job lasting 4 hours, and identify what kind of statement T = 15 + 8h is.
- 11.A ball is thrown in the air. Its height, h metres, above the ground after t seconds is given in this table: when t = 0, h = 0; when t = 1, h = 15; when t = 2, h = 20; when t = 3, h = 15; when t = 4, h = 0. Use the table to find the two times, in seconds, at which the ball is at ground level.
- 12.A car's fuel tank contains 50 litres when it starts a journey. After travelling for 2 hours, it contains 38 litres, and fuel is used at a constant rate. Work out the rate at which fuel is used, in litres per hour.
- 13.Which of these is an identity?
- 14.The solution to an inequality is n ≤ 5. Write down the largest integer value of n that satisfies this inequality.
- 15.Given that (x − 4)(x + 9) = 0, write down the two solutions of x.
Answer key
- (a) 7 — Method: substitute the given value of x into the rule and carry out the operation written in it. Working: y = x + 4 with x = 3 gives y = 3 + 4, an addition of two whole numbers. Answer: y = 7. The distractors: 12 comes from multiplying by 4 instead of adding 4; 4 comes from writing down the number in the rule and never using the input at all; 1 comes from subtracting, 4 − 3, instead of adding.
- (b) 9 — Subtract 14 from both sides: 4s ≤ 36. Divide both sides by 4: s ≤ 9, so the greatest number of tickets is 9. A candidate who forgets the £14 coach cost solves 4s ≤ 50, getting s ≤ 12.5, rounded down to 12. A candidate who adds the £14 instead of subtracting it solves 4s ≤ 64, getting s = 16. A candidate who miscalculates 50 − 14 as 32 solves 4s ≤ 32, getting s = 8.
- (c) 4n + 3 — Method: use the given number of extra tiles per pattern as the coefficient of n, then find the constant that fits pattern 1. Working: each pattern adds 4 tiles, so the expression has the form 4n + c. Substituting n = 1: 4(1) + c = 7, so c = 3. Answer: the number of tiles in pattern n is 4n + 3. The value 4n comes from leaving out the constant c altogether. The value 4n + 7 comes from using pattern 1's total, 7, as the constant directly instead of subtracting the difference first. The value 7n + 4 comes from swapping the roles of the number of tiles in pattern 1 and the number added each time.
- (b) 7 — Method: compare the equation with y = mx + c, where m is the gradient. Working: in y = 7x − 2, the coefficient of x is 7. Answer: the gradient is 7. −2 comes from confusing the gradient with the y-intercept. −7 comes from a sign error, treating the coefficient of x as negative. 2 comes from using the y-intercept but dropping its sign.
- (d) 110 — Method: form the equation 15 + 0.08m = 23.80, where m is the number of minutes, then solve for m. Working: subtract the fixed fee: 0.08m = 23.80 − 15 = 8.80. Divide by the cost per minute: m = 8.80 ÷ 0.08 = 110. Answer: 110 minutes. 1.1 comes from using 8 instead of 0.08 as the cost per minute, forgetting to convert pence to pounds. 297.5 comes from dividing the whole bill by the cost per minute without subtracting the fixed fee first, 23.80 ÷ 0.08. 485 comes from adding the fixed fee to the bill instead of subtracting it, before dividing by the cost per minute, (23.80 + 15) ÷ 0.08.
- (c) Yes; x = (y + 5)/2 is right — Sam's method is correct throughout: adding 5 to both sides gives y + 5 = 2x, and dividing both sides by 2 gives x = (y + 5)/2, so Sam is right. The option giving x = (y − 5)/2 assumes 5 should be subtracted again, but 2x − 5 = y means 5 has already been subtracted, so it must be added back, not taken away a second time. The option giving x = y/2 + 5 divides only the y term by 2 and leaves the 5 unhalved, which is not a valid rearrangement. The option agreeing Sam is correct but changing step 2 to x = 2(y + 5) confuses '2x' with 'x divided by 2' — since x is multiplied by 2, the inverse is division, not multiplication. Sam's working, and his final formula x = (y + 5)/2, are both correct.
- (d) 21 — The differences between consecutive triangular numbers increase by 1 each time: 3−1=2, 6−3=3, 10−6=4, 15−10=5. So the next difference is 6, giving 15+6=21. A candidate who keeps the difference the same as the previous step (adding 5 again) would reach 20. A candidate who uses a constant difference of 3 throughout would reach 18. A candidate who instead finds the square of the term number (5²=25, since 15 is the 5th term) would reach 25 — that is the rule for square numbers, not triangular numbers.
- (b) 6x + 6 — Perimeter = 2[(2x + 5) + (x − 2)] = 2(3x + 3) = 6x + 6. A candidate who adds the length and width but forgets to double for the perimeter gets 3x + 3. A candidate who makes a sign error and adds 2 instead of subtracting it before doubling gets 2[(2x + 5) + (x + 2)] = 6x + 14. A candidate who multiplies the length and width instead of adding them, confusing the perimeter formula with the area formula, and then doubles that product, gets 2(2x + 5)(x − 2) = 4x² + 2x − 20.
- (b) No, because their gradients are 2 and −2 — Method: two lines are parallel exactly when their gradients are equal as signed numbers, so m is read from each equation written in the form y = mx + c and the two are compared. Working: y = 2x + 1 has gradient 2 and y = −2x + 3 has gradient −2; those are not equal, so the lines are not parallel, and indeed one slopes upwards while the other slopes downwards. Answer: No, because their gradients are 2 and −2. The distractors: saying yes because both gradients have size 2 comes from comparing the sizes of the gradients and ignoring their signs; saying yes because the gradients add to 0 comes from using a sum of zero as the test for parallel lines instead of equality of gradients; saying no because the y-intercepts are 1 and 3 reaches the right verdict by the wrong route, since the intercepts decide where the lines sit rather than whether they are parallel.
- (b) £47, and T=15+8h is a formula (relates T and h). — The charge is £15 fixed plus £8 for each hour: T=15+8h, so for h=4, T=15+8×4=15+32=47. Because T=15+8h relates two different quantities, the total charge T and the number of hours h, it is a formula, not an equation — an equation is solved for one particular value of an unknown, but this relationship holds for every value of h a job might last. A candidate who forgets to include the fixed £15 fee would compute only 8×4=32. A candidate who adds the three numbers in the formula together instead of multiplying the hourly rate by the number of hours would compute 15+8+4=27. A candidate who correctly finds the charge but mistakes the formula for an equation, treating it as something to be solved for one specific value of h rather than a general relationship between T and h, would pick the correct charge with the wrong classification.
- (a) t = 0 or t = 4 — The ball is at ground level exactly when h = 0. From the table, h = 0 at t = 0 and at t = 4, so those are the two times. Distractor origins: t = 1 or t = 3 picks the times with equal (but non-zero) height instead of ground level; t = 2 picks the time of maximum height instead of ground level; t = 0 finds only the starting time and misses the second one.
- (c) 6 — Method: rate = amount of fuel used ÷ time taken. Working: fuel used = 50 − 38 = 12 litres, time taken = 2 hours, so rate = 12 ÷ 2 = 6 litres per hour. Answer: the rate is 6 litres per hour. 44 comes from finding the average of the two fuel amounts, (50 + 38) ÷ 2, instead of the fuel used. 12 comes from working out the fuel used but forgetting to divide by the time taken. 19 comes from dividing the remaining fuel, 38, by the time taken instead of the fuel used.
- (a) 2(3x + 1) = 6x + 2 — Expanding 2(3x + 1) = 6x + 2 gives an expression that matches the right-hand side exactly for every value of x — it is an identity. 4x − 3 = 3x + 5 is an ordinary equation with one solution, x = 8. 7 − x = x − 7 is also an ordinary equation with one solution, x = 7. 5x + 1 = 5(x + 1) never holds for any value of x at all, since expanding the right-hand side gives 5x + 5, and 5x + 1 = 5x + 5 would require 1 = 5, which is impossible.
- (a) 5 — The symbol ≤ means n can equal 5 or any number less than 5, so 5 is included and is the largest integer value. A candidate who treats the inequality as strict, as if it were n < 5, answers 4. A candidate who confuses ≤ with ≥ and looks for a value just above the boundary answers 6. A candidate who makes a sign error and reads the inequality as n ≤ −5 answers −5.
- (d) x = 4 or x = −9 — Since (x − 4)(x + 9) = 0, either x − 4 = 0, giving x = 4, or x + 9 = 0, giving x = −9. A candidate who takes the sign shown in each bracket rather than its opposite gets x = −4 or x = 9. A candidate who makes both solutions negative gets x = −4 or x = −9. A candidate who drops both signs and makes both solutions positive gets x = 4 or x = 9.
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