GCSE Higher — Number
Every number statement a GCSE Higher student may be asked is here; together they are 15% of the assessment. There are 16 number statements to cover. Working through them one at a time beats mixed practice while a topic is still shaky — mixed practice tells you that something is wrong, not what. The list includes ordering numbers and inequality symbols, the four operations and place value, inverse operations and priority of operations, factors, multiples, primes, HCF and LCM, systematic listing and the product rule for counting and powers and roots. No statement in this area is wholly Higher, but 6 of them have a Higher-only part — systematic listing and the product rule for counting, powers and roots, calculating with roots and indices and exact calculation: fractions, surds and π. With 40% AO1 and 60% across AO2 and AO3, fluency alone is not enough — the paper repeatedly asks what your answer means. The bulk of it sits comfortably on the non-calculator paper; a calculator quietly hides the gap until the exam.
- N1 — Ordering numbers and inequality symbols
- N2 — The four operations and place value
- N3 — Inverse operations and priority of operations
- N4 — Factors, multiples, primes, HCF and LCM
- N5 — Systematic listing and the product rule for counting (part Higher)
- N6 — Powers and roots (part Higher)
- N7 — Calculating with roots and indices (part Higher)
- N8 — Exact calculation: fractions, surds and π (part Higher)
- N9 — Standard form
- N10 — Fractions and decimals, including recurring decimals (part Higher)
- …and 6 more number statements at this tier
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Number at GCSE Higher: the 16 DfE statements
Every question on this site is filed against one of these statements. Pick one to practise it on its own.
Sample number questions for GCSE Higher
- On a number line, point A is at −1. Point B is 4 units from point A. Write down the two possible positions of point B.(a)5 or −3(b)4 or −4(c)3 or −5(d)3 only
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3 or −5 — Method: a point a fixed distance from another can lie on either side of it, so move the given distance in each direction from the starting point. Working: moving 4 units to the right gives −1 + 4 = 3, and moving 4 units to the left gives −1 − 4 = −5. Answer: 3 or −5. The distractors: 5 or −3 comes from starting at 1 instead of −1, giving 1 + 4 and 1 − 4; 3 only comes from moving to the right and forgetting that the point could lie to the left as well; 4 or −4 comes from measuring the distance from zero instead of from point A, which just repeats the given distance. - Work out (−2/5) × (−10/3). Give your answer as a fraction in its simplest form.(a)−56/15(b)3/25(c)−4/3(d)4/3
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4/3 — Method: the product of two negative numbers is positive, so work with 2/5 × 10/3 and then simplify. Multiply the numerators together and the denominators together. Working: 2 × 10 = 20 and 5 × 3 = 15, giving 20/15; both 20 and 15 divide by 5, so 20/15 = 4/3. Answer: 4/3. The distractors: −4/3 has the arithmetic right but keeps a minus sign, from treating negative × negative as negative; 3/25 comes from turning the second fraction upside down and multiplying, which divides instead of multiplying and gives 2/5 × 3/10 = 6/50; −56/15 comes from adding the two fractions instead of multiplying them, giving −6/15 − 50/15. - Work out 1 − 1/2 − 1/4 − 1/8 − 1/16. Give your answer as a fraction.(a)1/16(b)1/8(c)15/16(d)3/16
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1/16 — Method: terms can only be subtracted once they share a denominator, so write every term over the largest denominator, 16, and then subtract the numerators in order from left to right. Working: 1 = 16/16, 1/2 = 8/16, 1/4 = 4/16 and 1/8 = 2/16, so the numerators give 16 − 8 − 4 − 2 − 1 = 1, over a denominator of 16. Answer: 1/16. The distractors: 1/8 comes from stopping one term early, after 16 − 8 − 4 − 2 = 2; 3/16 comes from a sign slip on the last term, adding it instead of subtracting it, which gives 2 + 1 = 3; 15/16 comes from working from the right-hand end as though the last four terms were bracketed together, so that only a single sixteenth is taken away from 1. - Work out 3/7 × 14/9. Give your answer as a fraction in its simplest form.(a)17/16(b)2/21(c)2/3(d)27/98
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2/3 — Method: multiply the numerators together and the denominators together, then divide both parts of the result by their highest common factor. Working: 3 × 14 = 42 and 7 × 9 = 63, giving 42/63; the highest common factor of 42 and 63 is 21, and 42 ÷ 21 = 2 with 63 ÷ 21 = 3. Answer: 2/3. The distractors: 17/16 comes from adding the numerators and adding the denominators, giving (3 + 14)/(7 + 9); 27/98 comes from turning the second fraction upside down and multiplying, which divides instead of multiplying and gives 3/7 × 9/14; 2/21 comes from cancelling the 7 into the 14 in the numerator but leaving the 7 in the denominator, giving 6/63. - A jug holds 3 1/3 litres of juice. Each glass holds 2/3 of a litre. Work out how many glasses can be filled from the jug.(a)5(b)20/9(c)5/3(d)2
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5 — Method: the number of glasses is the amount in the jug divided by the amount one glass holds. Write the mixed number as an improper fraction, then divide by multiplying by the reciprocal. Working: 3 1/3 = (3 × 3 + 1)/3 = 10/3, and 10/3 ÷ 2/3 = 10/3 × 3/2 = 30/6 = 5. Answer: 5. The distractors: 2 comes from writing 3 1/3 as 4/3, adding the whole number to the numerator instead of multiplying it by the denominator first, and then dividing 4/3 by 2/3; 20/9 comes from multiplying by 2/3 instead of dividing by it; 5/3 comes from dividing by 2 rather than by 2/3, as though each glass held 2 litres.
Frequently asked questions
How much of the GCSE Higher paper is number?
15% of the total marks, and the three papers each carry an equal share of the qualification, so it is spread across all of them rather than concentrated in one.
How many topics are there in number at GCSE Higher?
16 DfE content statements, coded N1 to N16 within this area. Every question in the bank is tagged to one of them.
Which number topics are Higher only?
No number statement is wholly Higher, though 6 of them have a Higher-only part. The difference between the tiers here is the demand of the questions rather than the list of topics.
Can I use a calculator for number questions?
Paper 1 is non-calculator; Papers 2 and 3 allow one. Of the 16 statements in this area, 10 are naturally non-calculator and 1 naturally calculator, with the rest workable either way — so practise both.
What kind of marks does number carry?
The GCSE Higher split is 40% AO1 (use and apply standard techniques), 30% AO2 (reason, interpret and communicate) and 30% AO3 (solve problems in and out of context). Questions in this area are set across all three.
Are these past paper questions?
No. Every question at /gcse-higher is original, written to the DfE content statements and checked before it is published. We do not host past papers or mark schemes.
Do I need an account?
No — you can start practising straight away. Progress is saved automatically in your browser.
Is it free?
Yes. The questions, worked answers and printable worksheets are all free, with no adverts.
Tips for number at GCSE Higher
- A surd is a root that cannot be written as a whole number or a fraction, such as √2 or √3. To simplify a surd, look for a square factor inside the root: √12 = √(4 × 3) = √4 × √3 = 2√3.
- √a × √b = √(ab) and √a ÷ √b = √(a ÷ b), but √(a + b) is NOT √a + √b. Roots do not split over addition or subtraction.
- You can only add surds that are 'like' surds: 2√3 + 5√3 = 7√3, but √2 + √3 cannot be simplified.
- To rationalise a denominator, multiply top and bottom by the surd: 1/√3 = √3/3. For a denominator such as 2 + √3, multiply by 2 − √3 and use the difference of two squares.
- Negative indices give reciprocals, a⁻ⁿ = 1/aⁿ, and fractional indices give roots, a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ. Paper 1 asks for exact answers — leave surds and fractions as they are.
Common mistakes (and how to avoid them)
- Splitting a root over a sum: writing √(9 + 16) = √9 + √16 = 3 + 4 = 7 when the correct answer is √25 = 5.
- Adding unlike surds: writing √2 + √3 = √5. Only identical surds can be added.
- Leaving a surd unsimplified, such as √8 instead of 2√2, and losing the accuracy mark.
- Reading a⁻ⁿ as −aⁿ. A negative index makes a fraction, not a negative number: 2⁻³ = 1/8.
Worked examples
- 50 = 25 × 2, so √50 = √25 × √2 = 5√2.
- √8 = √(4 × 2) = 2√2, so 3√2 + √8 = 3√2 + 2√2.
- Add the coefficients of the like surds: (3 + 2)√2 = 5√2.
- Multiply the numerator and the denominator by √3: (6 × √3) ÷ (√3 × √3).
- √3 × √3 = 3, so the fraction becomes 6√3 ÷ 3.
- Simplify: 6√3 ÷ 3 = 2√3.
- 8^(2/3) = (³√8)² = 2² = 4.
- 27^(−1/3) = 1 ÷ 27^(1/3) = 1 ÷ ³√27 = 1/3.
At Higher your number system grows past fractions to the irrationals — √2, π — numbers that never end and never repeat. Simplifying surds, rationalising denominators and working with negative and fractional indices are the tools you will need in the quadratic formula, in Pythagoras and in exact trigonometric values, and Paper 1 tests all of them without a calculator. Always simplify to the smallest surd, and remember the critical rule: a root does not split over a plus or a minus.
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