Every number topic at GCSE, from the basics that appear on every paper to the Higher tier skills students most often drop marks on.
Number is the broadest topic area in GCSE Maths and appears on every single paper. It includes everything from fraction arithmetic and percentages through to surds, indices and standard form at Higher tier. This guide covers all of it — including the specific errors that examiner reports flag year after year.
Fraction errors follow students from Key Stage 3 all the way through to GCSE, and they compound badly because fraction skills underpin percentages, algebra, and ratio questions too. The most common errors:
You must find a common denominator before adding or subtracting. You cannot add numerators and denominators separately. 2/3 + 1/4 is not 3/7.
For mixed numbers: convert to improper fractions first, perform the operation, then convert back. Don't try to add whole number parts and fraction parts separately — it works for addition but fails badly for subtraction when borrowing is needed.
Multiplying: multiply numerators together, multiply denominators together. Simplify before multiplying where possible (cross-cancellation) to keep numbers manageable.
Dividing: flip the second fraction (find its reciprocal) and multiply. 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
❌ A very common error: flipping the first fraction instead of the second when dividing. Always flip the fraction after the division sign — the one you're dividing by.
Most students are comfortable with percentage increase and decrease. The skill that consistently causes difficulty is reverse percentage — finding the original value before a percentage change.
Example: A jacket costs £68 after a 15% reduction. What was the original price?
The mistake: calculating 15% of £68 and adding it. This is wrong because 15% of the reduced price is not the same as 15% of the original price.
The multiplier method works every time. A 15% reduction means you multiply by 0.85. To reverse it, divide by 0.85. A 20% increase means multiplying by 1.20 — to reverse, divide by 1.20.
Index laws are tested at both Foundation and Higher and appear in algebra, standard form, and surds questions too. There are six rules to know:
Fractional indices trip most students up. 8^(1/3) = ∛8 = 2. And 8^(2/3) = (∛8)² = 2² = 4. The denominator is the root, the numerator is the power. Always apply the root first — the numbers stay smaller and more manageable.
Standard form is used to write very large or very small numbers compactly. A number in standard form is written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
For small numbers (less than 1), the power of 10 is negative. The magnitude of the negative power tells you how many places to move the decimal point left from 1 to get to A.
When multiplying numbers in standard form: multiply the A values, add the powers of 10. Then check whether the A value in the result is still between 1 and 10 — if not, adjust.
(3 × 10⁴) × (4 × 10³) = 12 × 10⁷ = 1.2 × 10⁸
When dividing: divide the A values, subtract the powers. When adding or subtracting: convert to ordinary numbers first (or adjust so the powers match), perform the operation, then convert back.
A surd is an irrational root that cannot be simplified to a whole number or fraction. √2, √3, √5 are surds. √4 = 2 is not a surd. Surd questions at GCSE test simplification, arithmetic, and rationalising the denominator.
Find the largest perfect square factor of the number under the root, then split the surd.
Adding and subtracting: only combine like surds (same number under the root). 3√2 + 5√2 = 8√2. But 3√2 + 5√3 cannot be simplified further.
Multiplying: √a × √b = √(ab). And (√a)² = a. Use FOIL for expressions like (3 + √2)(4 − √2).
A fraction with a surd in the denominator needs to be rationalised — the surd must be removed from the denominator. For a simple surd: multiply numerator and denominator by that surd.
For a denominator of the form (a + √b): multiply by the conjugate (a − √b). This uses the difference of two squares: (a + √b)(a − √b) = a² − b, which has no surd.
Surd answers often appear in geometry questions — Pythagoras or trigonometry calculations where the answer is irrational. When a question says "give your answer in surd form" or "leave your answer in exact form", this is a signal to not press the equals button to get a decimal. Keep √ symbols in your answer throughout the working.
PaperPlus generates GCSE number questions at Foundation and Higher — fractions, percentages, surds, standard form and indices — with full mark schemes for all exam boards.
Start Practising Free →The full number content for each exam board, including which skills are Foundation-only, Higher-only or common to both, is in the AQA GCSE Maths specification and the Edexcel GCSE Maths specification.