About this GCSE Math practice test

GCSE Maths papers test the same core topics every year across AQA, Edexcel, and OCR — the difference between grades usually comes down to accuracy and unit conversion, not obscure content. This set spans both Foundation (grades 1–5) and Higher (grades 4–9) tiers, with fully worked steps so you can see exactly where a mark is gained or lost.

GCSE Mathematics Practice Test sample questions

These starter questions help you launch a math mock test quickly. Swap them with your own worksheet, notebook, or textbook questions any time.

1. 1. Number: Work out 2/5 + 1/3, giving your answer as a fraction in its simplest form.

2. 2. Number: Write 6.7 × 10⁻⁵ as an ordinary number.

3. 3. Number: Express 90 as a product of its prime factors, writing your answer in index notation.

4. 4. Number: Work out (−3) × (−4) + (−2)³.

5. 5. Number: Convert 5/6 into a decimal, giving your answer to 3 decimal places.

6. 6. Ratio & Proportion: Divide £96 in the ratio 5:3.

7. 7. Ratio & Proportion: A recipe for 8 people needs 340 g of flour. How much flour is needed for 6 people?

8. 8. Ratio & Proportion: A car travels 175 miles in 2.5 hours. Work out its average speed in mph.

9. 9. Ratio & Proportion: The exchange rate is £1 = $1.28. Convert £65 to dollars.

10. 10. Ratio & Proportion: A photo measuring 12 cm by 8 cm is enlarged so its longer side becomes 30 cm. Work out the new length of the shorter side.

11. 11. Algebra: Solve the simultaneous equations 3x + 2y = 16 and x − y = 2.

12. 12. Algebra: Solve the inequality 4x − 3 < 2x + 9, giving your answer as an inequality.

13. 13. Algebra: Rearrange the formula v = u + at to make a the subject.

14. 14. Algebra: Find the nth term of the quadratic sequence 2, 5, 10, 17, 26, ...

15. 15. Algebra: Expand and simplify (2x − 3)(x + 4) − (x − 1)².

16. 16. Geometry & Measures: A cone has radius 4 cm and slant height 9 cm. Find its curved surface area in terms of π.

17. 17. Geometry & Measures: A ladder of length 4.5 m leans against a wall, reaching a height of 4.1 m. Calculate the angle the ladder makes with the ground, correct to 1 decimal place.

18. 18. Geometry & Measures: Find the length of an arc that subtends an angle of 60° at the centre of a circle with radius 9 cm. Give your answer in terms of π.

19. 19. Geometry & Measures: A shape is made from a rectangle measuring 12 cm by 7 cm with a right-angled triangle of base 5 cm and height 4 cm removed from one corner. Find the area of the remaining shape.

20. 20. Geometry & Measures: Triangle ABC is rotated 90° clockwise about the origin. If point A has coordinates (3, 5), find the coordinates of the image of A.

21. 21. Statistics: Find the median of this set of exam scores: 58, 71, 64, 49, 83, 76, 60.

22. 22. Statistics: The number of pets owned by 20 students is shown in a frequency table: 0 pets (4 students), 1 pet (7 students), 2 pets (6 students), 3 pets (3 students). Calculate the mean number of pets per student.

23. 23. Statistics: A cumulative frequency graph for the finishing times of 80 runners shows that 20 runners finished in under 40 minutes, and 65 runners finished in under 55 minutes. Estimate the number of runners who took between 40 and 55 minutes.

24. 24. Statistics: A histogram bar representing the age range 20–30 in a survey has width 10 and frequency density 3.5. Work out the frequency for this class interval.

25. 25. Statistics: Find the range and the interquartile range of this data set: 12, 18, 9, 25, 14, 30, 21, 16.

26. 26. Probability: A spinner has 8 equal sections numbered 1 to 8. Find the probability of spinning a number that is a multiple of 3.

27. 27. Probability: A box contains 5 green counters and 7 yellow counters. Two counters are drawn at random without replacement. Find the probability that both counters are green.

28. 28. Probability: The probability that it rains on a given day is 0.3. Find the probability that it does not rain on that day.

29. 29. Probability: A fair six-sided dice is rolled twice. Find the probability of getting a total score of 9.

30. 30. Probability: In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. Find the probability that a randomly selected student studies neither language.

Answer key & quick explanations

Short answers for the sample questions above. Use this to self-check before generating a fresh AI-built mock test.

1. 1. 2/5 + 1/3

   11/15

   Convert both fractions to a common denominator of 15: 2/5 = 6/15 and 1/3 = 5/15. Adding gives 6/15 + 5/15 = 11/15, which is already in its simplest form since 11 and 15 share no common factors.

2. 2. 6.7 × 10⁻⁵ as an ordinary number

   0.000067

   A negative power of 10 means dividing by that power of 10, so 6.7 × 10⁻⁵ = 6.7 ÷ 100000. Moving the decimal point 5 places to the left gives 0.000067.

3. 3. Prime factors of 90

   2 × 3² × 5

   Divide 90 by the smallest prime repeatedly: 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, and 5 is prime. So 90 = 2 × 3 × 3 × 5, which in index notation is 2 × 3² × 5.

4. 4. (−3) × (−4) + (−2)³

   4

   First evaluate the power: (−2)³ = −8. Then evaluate the multiplication: (−3) × (−4) = 12, since a negative times a negative is positive. Finally add: 12 + (−8) = 4.

5. 5. 5/6 as a decimal to 3 dp

   0.833

   Dividing 5 by 6 gives 0.8333... recurring. Rounding this to 3 decimal places, the fourth decimal digit (3) is less than 5, so the answer rounds down to 0.833.

6. 6. Divide £96 in ratio 5:3

   £60 and £36

   The ratio 5:3 has 5 + 3 = 8 total parts. Dividing £96 by 8 gives £12 per part. Multiplying: 5 parts = 5 × £12 = £60, and 3 parts = 3 × £12 = £36 (which checks out since £60 + £36 = £96).

7. 7. Flour for 6 people from a recipe for 8

   255 g

   First find the amount of flour needed per person: 340 g ÷ 8 = 42.5 g per person. Then multiply by 6 people: 42.5 × 6 = 255 g.

8. 8. Average speed, 175 miles in 2.5 hours

   70 mph

   Speed is calculated as distance ÷ time. Dividing 175 miles by 2.5 hours gives 175 ÷ 2.5 = 70 mph.

9. 9. Convert £65 to dollars at £1 = $1.28

   $83.20

   Since £1 is worth $1.28, multiply the pound amount by the exchange rate: 65 × 1.28 = 83.20. So £65 converts to $83.20.

10. 10. Enlarged photo shorter side

    20 cm

    The scale factor is found by dividing the new longer side by the original longer side: 30 ÷ 12 = 2.5. Applying the same scale factor to the shorter side: 8 × 2.5 = 20 cm.

11. 11. Simultaneous equations 3x+2y=16, x−y=2

    x = 4, y = 2

    From the second equation, x = y + 2. Substituting into the first equation: 3(y + 2) + 2y = 16, which simplifies to 5y + 6 = 16, so 5y = 10 and y = 2. Then x = 2 + 2 = 4, checking in the first equation: 3(4) + 2(2) = 12 + 4 = 16.

12. 12. Solve 4x−3<2x+9

    x < 6

    Subtract 2x from both sides: 2x − 3 < 9. Add 3 to both sides: 2x < 12. Divide both sides by 2: x < 6.

13. 13. Rearrange v = u + at for a

    a = (v − u) / t

    Subtract u from both sides: v − u = at. Divide both sides by t to isolate a: a = (v − u) / t.

14. 14. nth term of 2, 5, 10, 17, 26

    n² + 1

    The first differences are 3, 5, 7, 9, and the second differences are all 2, confirming a quadratic sequence with n² coefficient 2 ÷ 2 = 1. Testing n² + 1 against the sequence: 1² + 1 = 2, 2² + 1 = 5, 3² + 1 = 10, 4² + 1 = 17, 5² + 1 = 26, all of which match.

15. 15. Expand (2x−3)(x+4) − (x−1)²

    x² + 7x − 13

    Expanding (2x − 3)(x + 4) gives 2x² + 8x − 3x − 12 = 2x² + 5x − 12. Expanding (x − 1)² gives x² − 2x + 1. Subtracting: (2x² + 5x − 12) − (x² − 2x + 1) = x² + 7x − 13.

16. 16. Curved surface area of cone, r=4, l=9

    36π cm²

    The curved surface area of a cone is given by πrl. Substituting r = 4 cm and l = 9 cm: π × 4 × 9 = 36π cm².

17. 17. Angle of ladder with ground

    65.7° (1 dp)

    Using the sine ratio, sin(θ) = opposite/hypotenuse = 4.1/4.5 = 0.9111. Taking the inverse sine, θ = sin⁻¹(0.9111) ≈ 65.7°, correct to 1 decimal place.

18. 18. Arc length, 60° of circle radius 9 cm

    3π cm

    Arc length = (θ/360) × 2πr. Substituting θ = 60° and r = 9 cm: (60/360) × 2π × 9 = (1/6) × 18π = 3π cm.

19. 19. Area of rectangle minus triangle

    74 cm²

    The area of the rectangle is 12 × 7 = 84 cm². The area of the removed triangle is 0.5 × 5 × 4 = 10 cm². Subtracting: 84 − 10 = 74 cm².

20. 20. Rotate (3,5) by 90° clockwise about origin

    (5, −3)

    For a 90° clockwise rotation about the origin, the rule is (x, y) → (y, −x). Applying this to A(3, 5) gives the image point (5, −3).

21. 21. Median of 58,71,64,49,83,76,60

    64

    First arrange the data in order: 49, 58, 60, 64, 71, 76, 83. With 7 values, the median is the 4th value, which is 64.

22. 22. Mean pets from frequency table

    1.4 pets

    Total students = 4 + 7 + 6 + 3 = 20. The sum of (value × frequency) is (0×4) + (1×7) + (2×6) + (3×3) = 0 + 7 + 12 + 9 = 28. Mean = 28 ÷ 20 = 1.4 pets.

23. 23. Runners between 40 and 55 minutes

    45 runners

    The cumulative frequency at 55 minutes (65 runners) includes all runners up to that time, including the 20 runners who finished under 40 minutes. Subtracting: 65 − 20 = 45 runners finished between 40 and 55 minutes.

24. 24. Frequency from histogram bar

    35

    Frequency is calculated as class width × frequency density. Substituting width = 10 and frequency density = 3.5: 10 × 3.5 = 35.

25. 25. Range and IQR of 12,18,9,25,14,30,21,16

    Range = 21, IQR = 10

    Ordering the data: 9, 12, 14, 16, 18, 21, 25, 30. The range is the highest minus the lowest: 30 − 9 = 21. Splitting into a lower half (9, 12, 14, 16) and upper half (18, 21, 25, 30), the lower quartile is the median of the lower half = (12+14)/2 = 13, and the upper quartile is the median of the upper half = (21+25)/2 = 23, giving an IQR of 23 − 13 = 10.

26. 26. P(multiple of 3) on 8-section spinner

    1/4

    Out of the numbers 1 to 8, the multiples of 3 are 3 and 6, giving 2 favourable outcomes. The probability is 2/8, which simplifies to 1/4.

27. 27. P(both green) without replacement

    5/33

    There are 5 green and 7 yellow counters, 12 in total. The probability the first counter is green is 5/12. After removing one green counter, 4 green remain out of 11 total, so the probability the second is also green is 4/11. Multiplying: 5/12 × 4/11 = 20/132, which simplifies to 5/33.

28. 28. P(not rain)

    0.7

    Since the probability of rain and no rain must sum to 1, the probability of no rain is 1 − 0.3 = 0.7.

29. 29. P(sum of 9) on two dice

    1/9

    The pairs that give a total of 9 are (3,6), (4,5), (5,4), and (6,3), giving 4 favourable outcomes out of 36 possible outcomes. The probability is 4/36, which simplifies to 1/9.

30. 30. P(neither French nor Spanish)

    1/6

    Using the inclusion-exclusion rule, the number studying at least one language is 18 + 12 − 5 = 25. The number studying neither is 30 − 25 = 5. The probability is 5/30, which simplifies to 1/6.

Curriculum Mapping & Learning Guide

Use this breakdown to identify which skills each question tests and guide post-test review.

GCSE Math units covered

1. Chapter 1: Number (integers, fractions, decimals, standard form)
2. Chapter 2: Ratio, proportion and rates of change
3. Chapter 3: Algebra (expressions, equations, graphs, sequences)
4. Chapter 4: Geometry and Measures (angles, area, volume, transformations, trigonometry)
5. Chapter 5: Statistics (averages, charts, cumulative frequency, histograms)
6. Chapter 6: Probability (basic, combined, tree diagrams)

Math practice tests for other exams and classes

Continue the same math thread across nearby exams and school classes for year-on-year revision.