Sample paper · GCSE Foundation · grades 1–5
GCSE Foundation sample Paper 1 (non-calculator)
The real Paper 1 is 1.5 hour 30 minutes and 80 marks, non-calculator, and all three papers carry equal weight. This sample is 20 original questions in the same content proportions as the Foundation qualification — number 25%, algebra 20%, ratio, proportion and rates of change 25%, geometry and measures 15%, probability 7.5%, statistics 7.5% — with no calculator-only items. AO1 / AO2 / AO3 at this tier: 50% / 25% / 25%.
Non-calculatorGCSE Foundation
GCSE Foundation sample Paper 1 (non-calculator)
MathsUKwww.geekhero.co.uk
- 1.Work out 36 ÷ (2 × 3)
- 2.The diagram shows the graph of , together with the horizontal line . Use the graphs to estimate, to 1 decimal place, the two solutions of .
- 3.A spring's extension is directly proportional to the force applied to it. A force of 5 N produces an extension of 12 mm. Work out the extension produced by a force of 20 N.
- 4.A solid pyramid has a square base of side 10 cm and four identical triangular faces. Its apex is directly above the centre of the base, at a vertical height of 12 cm, and the slant height of each triangular face — the distance from the apex to the midpoint of a base edge — is 13 cm. Work out the total surface area of the pyramid.
- 5.A two-way table records 200 members of a gym. 80 of the members are women and the rest are men. 60 of the women attend yoga classes. Work out the percentage of the women who attend yoga classes.
- 6.The age, in years, and the score in a reaction test are recorded for eight members of a sports club: (14, 92), (18, 88), (23, 85), (27, 80), (31, 78), (36, 74), (42, 70), (49, 65). The eight pairs are plotted on a scatter graph. Describe the correlation between age and score.
- 7.Work out 15% of £40, using 10% and 5%.
- 8.A water butt holds 200 litres and is being drained at a steady 8 litres per minute. Write down the function for the amount of water y, in litres, left after x minutes.
- 9.A plan is drawn to a scale of 1 : 500. Work out how many metres are represented by 1 cm on the plan.
- 10.A rectangular bookmark measures 2 cm by 25 cm. Work out the area of the bookmark.
- 11.A machine produces bolts. In a sample of 250 bolts, 15 are faulty. Work out the relative frequency of a bolt being faulty, giving your answer as a fraction in its simplest form.
- 12.Write the fraction 47/50 as a decimal.
- 13.A plumber charges a call-out fee plus an hourly rate for a job. The total cost, £C, for a job lasting h hours is given by the formula C = 40 + 25h. What does the number 40 represent in this formula?
- 14.A company's profit this year is 130% of last year's profit. Last year's profit was £40,000. Work out this year's profit.
- 15.The point A(2, 5) is translated to the point B(7, 3). Write down the column vector that describes this translation.
- 16.Write down a prime number between 30 and 40.
- 17.A square tile has an area of 144 cm². Work out the side length of the tile.
- 18.Two mathematically similar oil drums have a length scale factor of 3 from the smaller drum to the larger drum. Work out the volume scale factor from the smaller drum to the larger drum.
- 19.A grain of fine sand has a mass of 0.001 grams. Write this mass in standard form.
- 20.A recipe for 4 people needs 300 g of rice. Work out how much rice is needed for 6 people, assuming the amount is in direct proportion to the number of people.
Answer key
- (c) 6 — 2 × 3 = 6, then 36 ÷ 6 = 6. Ignoring the brackets and working left to right gives 36 ÷ 2 = 18, then 18 × 3 = 54. Multiplying by the bracket instead of dividing by it gives 2 × 3 = 6, then 36 × 6 = 216. Dividing by only the 2 inside the bracket, and ignoring the × 3, gives 36 ÷ 2 = 18.
- (a) x = −1.8 and x = 2.8 — Method: the solutions of x² − x − 2 = 3 are the x-coordinates of the points where the curve y = x² − x − 2 meets the line y = 3, read off the grid to 1 decimal place. Working: the curve meets the line y = 3 at approximately x = −1.8 and at x = 2.8. Answer: x ≈ −1.8 and x ≈ 2.8. Distractor refutation: x = −1.0 and x = 2.0 comes from reading where the curve crosses the x-axis (y = 0), solving x² − x − 2 = 0, instead of where it meets the line y = 3. x = −3.0 and x = 4.0 comes from taking the two ends of the drawn curve as the intersection points, instead of finding where it actually crosses the line y = 3. x = 1.8 and x = 2.8 comes from a sign slip on the left-hand intersection, reading it as positive instead of negative.
- (c) 48 mm — Method: extension = k × force, where k = extension ÷ force. Working: k = 12 ÷ 5 = 2.4 mm per N. At 20 N: extension = 2.4 × 20 = 48 mm. Wrong options: 32 mm comes from adding the extension and force numbers instead of scaling (12 + 20); 3 mm comes from treating the relationship as inverse proportion (12 × 5 ÷ 20); 36 mm comes from using an incorrect scale factor of 3 between the forces instead of the correct factor of 4 (20 ÷ 5).
- (b) 360 cm² — Method: the total surface area is the square base plus the four triangular faces, and the height used for a triangular face is the slant height of 13 cm, not the vertical height of 12 cm. Working: the base is 10 × 10 = 100 cm²; one triangular face is (10 × 13) ÷ 2 = 65 cm², so four faces give 4 × 65 = 260 cm²; the total is 100 + 260 = 360. Answer: 360 cm². The distractors: 260 cm² comes from adding the four triangular faces and leaving out the base; 340 cm² comes from using the vertical height of 12 cm as the height of each triangle, 100 + 4 × 60; 620 cm² comes from working out each face as 10 × 13 without halving, 100 + 4 × 130.
- (a) 75% — 'Percentage of the women' restricts the group to the 80 women, of whom 60 attend yoga: 60/80 = 0.75 = 75%. Dividing by the number of men (200 − 80 = 120) instead of the number of women gives 60/120 = 0.5 = 50%. Dividing by all 200 members instead of just the 80 women gives 60/200 = 0.3 = 30%. Using the 20 women who do NOT attend yoga (80 − 60) as the numerator instead of the 60 who do gives 20/80 = 0.25 = 25%.
- (c) Strong negative correlation — Method: correlation is described by two things — the direction the points take as the graph is read from left to right, and how closely the points lie to a single straight line. Working: reading the pairs in order of age, the ages rise 14, 18, 23, 27, 31, 36, 42, 49 while the scores fall 92, 88, 85, 80, 78, 74, 70, 65; the score falls at every single step, with no reversal anywhere, so the points fall from left to right and lie close to a straight line. Answer: strong negative correlation — negative for the falling direction, strong because every point follows the pattern. The distractors: strong positive correlation comes from noticing a clear pattern and calling any clear pattern positive, without checking the direction; weak negative correlation comes from reading the direction correctly but judging points that do not lie exactly on a straight line to be only loosely related, when these eight fall without a single exception; no correlation comes from reading a falling trend as though it showed no relationship at all, when a falling trend is itself a relationship.
- (a) £6 — 10% of £40 is £4, and 5% of £40 is half of that, £2. Adding these gives 15% of £40 = £4 + £2 = £6. Finding only the 10% part and stopping there gives £4. Finding only the 5% part and stopping there gives £2. Multiplying 40 by 15 without dividing by 100 gives £600, which treats the percentage as if it were a whole number multiplier.
- (c) y = 200 − 8x — Method: in a linear model the amount present at the start is the constant term and the steady rate of change is the gradient, which is negative because the amount is falling. Working: at x = 0 minutes there are 200 litres, so the constant term is 200; 8 litres are lost every minute, so after x minutes 8x litres have gone and the amount left is y = 200 − 8x. Answer: y = 200 − 8x. The distractors: y = 8x + 200 treats the draining as filling, so the butt would gain 8 litres a minute; y = 200x − 8 swaps the two numbers over, using the starting 200 litres as the rate per minute and the 8 litres per minute as the starting amount; y = −8x − 200 makes the starting amount negative as well as the rate, so the butt would begin 200 litres in deficit.
- (b) 5 m — A scale of 1 : 500 means 1 cm on the plan represents 500 cm in real life. Converting to metres: 500 ÷ 100 = 5 m. 500 m comes from forgetting to convert the 500 cm into metres at all. 50 m comes from dividing by 10 instead of 100 when converting centimetres to metres: 500 ÷ 10 = 50. 0.5 m comes from dividing by 1000 instead of 100, as if converting to kilometres instead of metres: 500 ÷ 1000 = 0.5.
- (b) 50 cm² — Method: the area of a rectangle is length × width. Working: 2 × 25 = 50. Answer: 50 cm². The distractors: 54 cm comes from working out the perimeter, 2 × (2 + 25), which is a length and not an area; 27 cm comes from adding the two sides, 2 + 25, instead of multiplying them; 25 cm² comes from halving the product, 50 ÷ 2, which is the rule for the area of a triangle rather than of a rectangle.
- (d) 3/50 — Relative frequency is the number of faulty bolts divided by the total sample size: 15/250, which simplifies to 3/50 by dividing both the numerator and the denominator by 5. Using 235, the number of bolts that were NOT faulty, as the denominator instead of the total 250 gives 15/235, which simplifies to 3/47. Inverting the fraction, dividing the total by the number of faulty bolts instead of the other way round, gives 250/15, which simplifies to 50/3 — a value greater than 1, which cannot be a probability. Simplifying by dividing the numerator and the denominator by different numbers, 15 ÷ 15 = 1 and 250 ÷ 25 = 10, gives 1/10.
- (c) 0.94 — Method: a fraction is written as a decimal by making the denominator a power of ten, because the decimal places record tenths, hundredths and thousandths. Working: 50 × 2 = 100, so the numerator must also be multiplied by 2, giving 47 × 2 = 94 and the equivalent fraction 94/100; 94 hundredths is written with two digits after the decimal point. Answer: 0.94. The distractors: 0.47 comes from treating the denominator as though it were already 100 and writing the digits of the numerator straight after the point; 4.7 comes from dividing 47 by 10 instead of by 50; 0.094 comes from multiplying the denominator by 20 to reach 1000 but the numerator by only 2, giving 94/1000.
- (d) The fixed call-out fee, charged before the hourly rate. — Method: compare the formula with C = (fixed charge) + (rate) × h, where the fixed charge is the part that does not depend on h. Working: in C = 40 + 25h, the term 25h depends on the number of hours, h, but 40 does not change however many hours the job takes. Answer: 40 is the fixed call-out fee. 'The hourly rate' confuses the fixed term with the coefficient of h, which is actually 25. 'The total cost for a job lasting 1 hour' comes from substituting h = 1 into the formula (40 + 25 = 65) rather than reading off the constant term. 'The number of hours worked before charging starts' wrongly treats the constant, which is in pounds, as if it were measured in hours.
- (b) £52,000 — Method: convert 130% to a decimal multiplier and multiply it by last year's profit. Working: 130% = 1.3, so this year's profit is £40,000 × 1.3 = £52,000. Answer: £52,000. £12,000 comes from using only the extra 30% (130% − 100%) and forgetting to include the original 100%, £40,000 × 0.3 = £12,000. £40,130 comes from simply adding 130 onto £40,000, treating the percentage as an amount of money rather than a multiplier. £5,200 comes from misreading 130% as 13%, giving £40,000 × 0.13 = £5,200.
- (c) $\binom{5}{-2}$ — Method: the top number of a column vector is the change in the x-coordinate and the bottom number is the change in the y-coordinate, each worked out as image minus object. Working: across, 7 − 2 = 5; up, 3 − 5 = −2. So the point moves 5 to the right and 2 down. Answer: $\binom{5}{-2}$. Subtracting the other way round, object minus image, gives $\binom{-5}{2}$, which is the journey from B back to A. Recording the vertical change as 2 because the gap between 3 and 5 is 2, without noting the direction, gives $\binom{5}{2}$, a movement 2 upwards. Adding the coordinates instead of subtracting them gives $\binom{9}{8}$.
- (b) 31 — Method: a prime number has exactly two factors, 1 and itself, so check each number between 30 and 40 for other factors. Working: 3 × 11 = 33, so 33 is not prime. 2 × 17 = 34, so 34 is not prime. 4 × 9 = 36, so 36 is not prime. 31 has no factors other than 1 and 31, so it is prime. Answer: 31.
- (c) 12 cm — For a square, area = side². So side² = 144, giving side = ±12. Since a length must be positive, the side length is 12 cm. A candidate who gives both square roots without rejecting the negative one, which cannot be a length, answers 12 cm or −12 cm. A candidate who halves 144 instead of taking its square root gets 72 cm. A candidate who divides 144 by 4, confusing the area formula with a perimeter calculation, gets 36 cm.
- (d) 27 — Method: for similar shapes, the volume scale factor is the length scale factor cubed. Working: 3³ = 27. Answer: 27. 9 comes from squaring the length scale factor, which gives the area scale factor, not the volume scale factor. 3 comes from using the length scale factor itself as if it were the volume scale factor. 6 comes from doubling the length scale factor instead of cubing it.
- (d) 1 × 10⁻³ — Method: for a number smaller than 1 the index is negative, and it counts the places the decimal point moves to the right to leave a coefficient between 1 and 10. Working: the only significant digit is 1, so the coefficient is 1; the decimal point in 0.001 moves three places to the right to reach that 1, so the index is −3. Answer: 1 × 10⁻³. The distractors: 1 × 10³ comes from taking the index as positive, which describes one thousand grams rather than one thousandth of a gram; 0.1 × 10⁻² is the same mass written with a coefficient of 0.1, which is smaller than 1 and so is not standard form; 1 × 10⁻⁴ comes from counting the zero in front of the decimal point as well as the places after it.
- (d) 450 g — Find the amount of rice per person: 300 ÷ 4 = 75 g. Multiply by the new number of people: 75 × 6 = 450 g. Giving 200 g swaps which number the rice is divided and multiplied by (300 ÷ 6 × 4 = 200), scaling the wrong way. Giving 180 g uses 4 + 6 = 10 as the base instead of the original 4 people (300 × 6 ÷ 10 = 180). Giving 400 g assumes each of the 2 extra people needs 300 ÷ 6 = 50 g on top of the original 300 g (300 + 2 × 50 = 400), rather than scaling the whole amount in proportion.
How the 20 questions are shared out
- Number — 5 questions (25% of the qualification)
- Algebra — 4 questions (20% of the qualification)
- Ratio, proportion and rates of change — 5 questions (25% of the qualification)
- Geometry and measures — 3 questions (15% of the qualification)
- Probability — 2 questions (7.5% of the qualification)
- Statistics — 1 question (7.5% of the qualification)
Where an area has fewer printable questions than its share, the shortfall is filled from the other areas. These are original questions, not past papers.