Free·GCSE Maths
Free GCSE Maths Questions and Answers
12 questions, one from each topic, with the answer that takes every mark and what it is worth. Nothing to sign up to and nothing to download — they are on this page.
Number, Algebra: expressions and manipulation, Equations, inequalities and graphs, Ratio, proportion and rates of change, Geometry: angles, shape and properties, Mensuration, Pythagoras and trigonometry, Probability, Statistics and Method, working and communication
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Simplify (3 × 105) × (4 × 10−2), giving your answer in standard form.
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The full-mark answer · 2 marks
Multiplying the front numbers gives 3 × 4 = 12, and adding the indices gives 105+(−2) = 103, so the product is 12 × 103. That is not standard form, because 12 is not below 10; writing 12 as 1.2 × 10 raises the index by one, giving 1.2 × 104.
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A number x is 2.4 correct to 1 decimal place. Write down the error interval for x.
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The full-mark answer · 2 marks
Half of one decimal place is 0.05, so the bounds are 2.4 − 0.05 = 2.35 and 2.4 + 0.05 = 2.45. A value of exactly 2.35 rounds up to 2.4 and is included; a value of exactly 2.45 rounds up to 2.5 and is not. The interval is 2.35 ≤ x < 2.45.
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Simplify 5a + 3b − 2a + b.
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The full-mark answer · 2 marks
Collecting the a terms gives 5a − 2a = 3a. Collecting the b terms gives 3b + b = 4b, because a term written without a number in front has a coefficient of 1. The expression simplifies to 3a + 4b and no further, since the two terms are unlike.
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Expand and simplify 3(2x − 5) − 2(x + 4).
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The full-mark answer · 3 marks
Expanding the first bracket gives 6x − 15. The second bracket is multiplied by −2, not 2, so it gives −2x − 8. Collecting like terms: 6x − 2x = 4x, and −15 − 8 = −23. The expression simplifies to 4x − 23.
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Solve 5x − 3 = 17.
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The full-mark answer · 2 marks
Adding 3 to both sides gives 5x = 20. Dividing both sides by 5 gives x = 4. Substituting back: 5 × 4 − 3 = 17, which is the right-hand side, so the solution is correct.
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Solve 4(x − 2) = 3x + 5.
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The full-mark answer · 3 marks
Expanding the bracket gives 4x − 8 = 3x + 5. Subtracting 3x from both sides gives x − 8 = 5, and adding 8 to both sides gives x = 13. Checking: 4(13 − 2) = 44 and 3 × 13 + 5 = 44, so both sides agree.
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Write the ratio 45 : 60 in its simplest form.
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The full-mark answer · 1 mark
The highest common factor of 45 and 60 is 15. Dividing both parts by 15 gives 3 : 4, and 3 and 4 have no common factor, so the ratio is in its simplest form.
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Work out the size of each interior angle of a regular decagon.
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The full-mark answer · 3 marks
The interior angles of an n-sided polygon add to (n − 2) × 180°, so a decagon has 8 × 180 = 1440°. The polygon is regular, so all ten angles are equal and each is 1440 10 = 144°. As a check, the exterior angle is 360 10 = 36°, and 144 + 36 = 180.
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Work out the circumference of a circle of radius 7 cm.
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The full-mark answer · 2 marks
The circumference of a circle is 2πr, so with r = 7 the circumference is 2 × π × 7 = 14π = 43.98 cm, which is 44.0 cm to 1 decimal place. As a check, the circumference is a little over three diameters, and three times 14 is 42.
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A bag contains 6 red, 4 blue and 2 green counters. Write down the probability of picking a red counter.
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The full-mark answer · 1 mark
There are 6 + 4 + 2 = 12 counters altogether and 6 of them are red, so the probability is 6 12 , which simplifies to 1 2 .
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The data are 4, 7, 3, 7, 9, 2, 7 and 5. Write down the mode and find the median.
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The full-mark answer · 2 marks
The mode is the most frequent value, and 7 appears three times, so the mode is 7. In order the data are 2, 3, 4, 5, 7, 7, 7, 9. There are eight values, so the median lies between the fourth and fifth: 5 + 7 2 = 6.
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Explain why a correct answer with no working can score fewer marks than a wrong one.
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The full-mark answer · 2 marks
Most questions carry more method marks than accuracy marks, and method marks are awarded for the steps rather than the result. A wrong answer supported by correct working keeps those marks; a wrong answer standing alone keeps none, because there is nothing on the page to credit. A correct answer with no working does score full marks, so writing the method down costs nothing and protects almost everything.
What the book adds
These are the answers that score everything. GCSE Mathematics Questions & Answers is the other half: the same questions answered weakly, middlingly and strongly, the mark each version scores, what separates them, and the mark scheme behind it — a hundred questions across every topic, written for the teacher marking them rather than the pupil sitting them.