OCR GCSE Mathematics Number: a complete overview of calculation, fractions, percentages, indices, standard form, surds and bounds

What the Number content demands

Number is the foundation the rest of the course is built on. OCR uses it in every area, from compound measures to probability, so weak arithmetic leaks marks across the whole paper. The content runs from the structure of the number system and calculation through fractions, percentages, indices and standard form to surds and bounds at Higher tier. Because OCR puts the non-calculator paper in the middle of the three (Paper 2 at Foundation, Paper 5 at Higher), fluent written and mental methods are essential.

This guide walks through the six areas of the Number content and ties together the matching dot-point pages, each of which has its own practice questions.

Structure of the number system and calculation

Order positive and negative integers, decimals and fractions by putting them in a common form, and calculate with the four operations using BIDMAS: Brackets, Indices, Division and Multiplication, then Addition and Subtraction. With directed numbers, like signs multiply to a positive and unlike signs to a negative, while a fraction bar groups its numerator and denominator. A sign or order-of-operations slip in the first line carries through, so accuracy here protects every later mark.

Factors, multiples and primes

A prime has exactly two factors. Write a number as a product of primes with a factor tree, then in index form. For the HCF, multiply the lowest power of each shared prime; for the LCM, multiply the highest power of every prime in either number. A check is that $\text{HCF} \times \text{LCM}$ equals the product of the two numbers. HCF problems ask for the largest equal grouping; LCM problems ask when repeating events coincide.

Fractions, decimals and percentages

Add and subtract fractions with a common denominator, multiply tops and bottoms, and divide by multiplying by the reciprocal. Convert a fraction to a decimal by dividing, a decimal to a percentage by multiplying by $100$, and a percentage to a fraction over $100$. To find a percentage of an amount, turn it into a decimal and multiply; to write one number as a percentage of another, divide and multiply by $100$. Chained percentage changes act on the new amount each time, so they cannot simply be added.

Standard form and indices

The index laws are $a^m \times a^n = a^{m+n}$, $a^m \div a^n = a^{m-n}$ and $(a^m)^n = a^{mn}$, with $a^0 = 1$, $a^{-n} = \tfrac{1}{a^n}$ and $a^{m/n} = \left(\sqrt[n]{a}\right)^m$ at Higher. Standard form writes a number as $a \times 10^n$ with $1 \le a < 10$. Multiply or divide standard-form numbers by combining the number parts and adding or subtracting the powers, then re-standardise so the number part lies between $1$ and $10$.

Surds (Higher)

A surd is an irrational root such as $\sqrt{2}$, kept exact rather than rounded. Simplify by taking out the largest square factor ($\sqrt{50} = 5\sqrt{2}$). Like surds add and subtract by combining coefficients; multiply using $\sqrt{a}\times\sqrt{b} = \sqrt{ab}$. Rationalise a denominator by multiplying top and bottom by the surd, or by the conjugate when the denominator is a sum, so the bottom becomes rational. Surds feed the quadratic formula and exact trigonometric values.

Rounding, estimation and bounds

Round to decimal places by counting after the point and to significant figures from the first non-zero digit, rounding up on $5$ or more. Estimate by rounding each number to one significant figure. At Higher, a value rounded to the nearest unit lies half a unit either side, giving lower and upper bounds; for the extreme value of a calculation, pick the bounds carefully, remembering that the largest quotient uses the largest numerator over the smallest denominator.

Check your knowledge

A mix of calculation, fraction, percentage, index and surd questions covering the Number content. Attempt them under timed conditions, then check against the solutions.

1. Work out $-7 + 3 \times (-4)$. (2 marks)
2. Write $84$ as a product of its prime factors in index form. (2 marks)
3. Find the LCM of $12$ and $18$. (2 marks)
4. Work out $\tfrac{2}{3} \times \tfrac{9}{10}$, giving the answer in its simplest form. (2 marks)
5. Find $35\%$ of $80$. (2 marks)
6. Write $0.00056$ in standard form. (1 mark)
7. Work out $27^{2/3}$. (2 marks)
8. Simplify $\sqrt{75}$ in the form $a\sqrt{b}$. (2 marks)