Number

What you'll learn

This revision guide covers all Number topics in the OCR GCSE Mathematics specification. You'll master fundamental numerical skills including operations with integers, fractions, decimals and percentages, as well as ratio, proportion and standard form. These foundational topics appear throughout both Foundation and Higher tier papers and underpin success across all areas of GCSE Mathematics.

Key terms and definitions

Integer — A whole number that can be positive, negative or zero (e.g. -3, 0, 17)

Prime number — A natural number greater than 1 that has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7, 11)

Reciprocal — The multiplicative inverse of a number; for any non-zero number n, its reciprocal is 1/n

Standard form — A way of writing very large or very small numbers as A × 10ⁿ where 1 ≤ A < 10 and n is an integer

Rational number — Any number that can be expressed as a fraction p/q where p and q are integers and q ≠ 0

Irrational number — A number that cannot be expressed as a fraction; its decimal expansion is non-terminating and non-repeating (e.g. π, √2)

Product of prime factors — Any integer written as a multiplication of prime numbers only (e.g. 60 = 2² × 3 × 5)

Significant figures — Digits in a number that contribute to its precision, counting from the first non-zero digit

Core concepts

Place value and ordering

Place value determines the value of each digit in a number based on its position. Understanding this system is essential for all calculations and rounding.

Decimal places:

• The first decimal place represents tenths (0.1)
• The second represents hundredths (0.01)
• The third represents thousandths (0.001)

Ordering numbers:

• For positive numbers, compare digits from left to right
• For negative numbers, remember that -10 < -5 (closer to zero is larger)
• Convert all numbers to the same form (all decimals or all fractions) before comparing

Rounding:

• To round to n decimal places, look at the (n+1)th decimal place
• If this digit is 5 or more, round up; if less than 5, round down
• To round to n significant figures, count n digits from the first non-zero digit
• Example: 0.004567 rounded to 2 s.f. = 0.0046

Operations with fractions, decimals and percentages

All three representations describe parts of a whole and are interchangeable.

Converting between forms:

• Fraction to decimal: divide numerator by denominator
• Decimal to percentage: multiply by 100
• Percentage to fraction: write as a fraction over 100, then simplify
• Example: 3/8 = 0.375 = 37.5%

Fraction calculations:

• Addition/subtraction: find a common denominator, then add/subtract numerators
• Multiplication: multiply numerators together and denominators together, then simplify
• Division: multiply by the reciprocal of the second fraction
• Example: 2/3 ÷ 5/6 = 2/3 × 6/5 = 12/15 = 4/5

Percentage calculations:

• Finding a percentage of an amount: convert to decimal and multiply
• Percentage increase/decrease: multiply by (100 ± percentage)/100
• Finding the original amount: divide by the multiplier
• Example: After a 15% increase, a price is £46. Original price = 46 ÷ 1.15 = £40

Reverse percentages: When given a value after a percentage change, divide by the decimal multiplier.

• Price after 20% increase = original × 1.20
• Price after 35% decrease = original × 0.65

Ratio and proportion

Ratio expresses how many times one value contains another. Proportion describes the relationship between parts and the whole.

Simplifying ratios:

• Divide all parts by their highest common factor (HCF)
• Example: 18:24:30 = 3:4:5 (dividing by 6)

Sharing in a ratio:

1. Add the ratio parts to find total shares
2. Divide the amount by total shares to find one share
3. Multiply each ratio part by the value of one share

• Example: Share £240 in ratio 3:5
• Total shares = 8, one share = £30, amounts = £90 and £150

Direct proportion:

• As one quantity increases, the other increases at the same rate
• y ∝ x means y = kx for some constant k
• Use: find k, then apply to new values

Inverse proportion:

• As one quantity increases, the other decreases
• y ∝ 1/x means y = k/x
• Example: 6 workers take 10 days. How long for 15 workers?
• k = 6 × 10 = 60, so 15 workers take 60/15 = 4 days

Powers, roots and indices

Index laws (same base):

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1 (for any a ≠ 0)
• a⁻ⁿ = 1/aⁿ
• a^(1/n) = ⁿ√a
• a^(m/n) = (ⁿ√a)ᵐ

Fractional and negative indices:

• 16^(3/4) = (⁴√16)³ = 2³ = 8
• 25^(-1/2) = 1/√25 = 1/5

Surds:

• √a × √b = √(ab)
• √a ÷ √b = √(a/b)
• Simplify by finding perfect square factors: √72 = √(36 × 2) = 6√2

Rationalising denominators:

• Multiply numerator and denominator by the surd
• Example: 3/√5 = 3√5/5

Prime factors, HCF and LCM

Prime factor decomposition: Use a factor tree or repeated division to express numbers as products of primes.

• 84 = 2² × 3 × 7

Highest Common Factor (HCF):

• Write each number as a product of primes
• Take the lowest power of each common prime factor
• Example: HCF of 60 and 84
• 60 = 2² × 3 × 5
• 84 = 2² × 3 × 7
• HCF = 2² × 3 = 12

Lowest Common Multiple (LCM):

• Write each number as a product of primes
• Take the highest power of each prime factor that appears
• Example: LCM of 60 and 84
• LCM = 2² × 3 × 5 × 7 = 420

Standard form and estimation

Standard form expresses numbers as A × 10ⁿ where 1 ≤ A < 10.

Writing numbers in standard form:

• Large numbers: count how many places the decimal point moves left (positive power)
• Small numbers: count how many places it moves right (negative power)
• 45,000,000 = 4.5 × 10⁷
• 0.00023 = 2.3 × 10⁻⁴

Calculations with standard form:

• Multiplication: multiply A values, add powers of 10
• Division: divide A values, subtract powers of 10
• Example: (3 × 10⁵) × (4 × 10⁻²) = 12 × 10³ = 1.2 × 10⁴

Estimation: Round each number to 1 significant figure before calculating.

• Example: Estimate (31.7 × 489)/(0.52 × 19.1)
• ≈ (30 × 500)/(0.5 × 20) = 15,000/10 = 1,500

Worked examples

Example 1: Percentage change and reverse percentages (Foundation/Higher)

Question: A shop increases all prices by 12%. A television now costs £313.60. (a) Calculate the original price. [2 marks] (b) Calculate the price after a further 8% increase. [2 marks]

Solution: (a) After 12% increase, price = original × 1.12 ✓ Original price = 313.60 ÷ 1.12 = £280 ✓

(b) Further 8% increase: 313.60 × 1.08 ✓ = £338.69 (to nearest penny) ✓

Mark scheme notes:

• (a): 1 mark for method (dividing by 1.12), 1 mark for correct answer
• (b): 1 mark for multiplying by 1.08, 1 mark for correct final answer
• Common error: multiplying £280 by 1.08 instead of £313.60

Example 2: HCF, LCM and prime factors (Foundation/Higher)

Question: (a) Express 252 as a product of its prime factors. [2 marks] (b) Find the HCF and LCM of 252 and 180. [3 marks]

Solution: (a) 252 = 2 × 126 = 2 × 2 × 63 = 2 × 2 × 9 × 7 = 2² × 3² × 7 ✓✓

(b) 180 = 2² × 3² × 5 ✓

HCF = 2² × 3² = 36 ✓ LCM = 2² × 3² × 5 × 7 = 1260 ✓

Mark scheme notes:

• (a): 1 mark for correct method, 1 mark for fully correct expression
• (b): 1 mark for 180 in prime factors, 1 mark each for HCF and LCM
• Must use prime factor form to gain method marks

Example 3: Ratio and proportion (Higher)

Question: y is inversely proportional to the square of x. When x = 2, y = 12. Find y when x = 6. [3 marks]

Solution: y ∝ 1/x² means y = k/x² ✓

When x = 2, y = 12: 12 = k/4 k = 48 ✓

When x = 6: y = 48/36 = 4/3 or 1.33̇ ✓

Mark scheme notes:

• 1 mark for correct proportionality equation
• 1 mark for finding k = 48
• 1 mark for correct final answer (accept equivalent forms)

Common mistakes and how to avoid them

• Confusion with negative numbers: Remember that -5 - (-3) = -5 + 3 = -2. The minus of a negative becomes addition. Always write it out as addition/subtraction of a positive to avoid errors.

• Incorrect percentage multipliers: For a 30% decrease, multiply by 0.70, not 0.30. The multiplier represents what remains, not what's removed. Create a quick reference: increase by n% → multiply by (100+n)/100, decrease by n% → multiply by (100-n)/100.

• Adding fractions without common denominators: You cannot add 1/3 + 1/4 to get 2/7. Always find a common denominator first (in this case 12), giving 4/12 + 3/12 = 7/12.

• Misapplying index laws: The law aᵐ × aⁿ = aᵐ⁺ⁿ only works with the same base. You cannot simplify 2³ × 3² using index laws. Calculate each part separately: 8 × 9 = 72.

• Rounding too early: When performing multi-step calculations, keep full accuracy until the final answer. Rounding intermediate steps introduces cumulative errors that lose marks.

• Standard form format errors: Writing 23 × 10⁴ is not standard form because the first number must be between 1 and 10. Convert to 2.3 × 10⁵. Always check: is your A value in the range 1 ≤ A < 10?

Exam technique for "Number"

• Show all working clearly: Even if you use a calculator, write down the calculation you're entering. This earns method marks if your final answer is wrong. For a 3-mark question, expect to show at least 2 clear steps.

• Command word awareness: "Calculate" requires a numerical answer with working. "Estimate" means round to 1 significant figure first. "Express" (for prime factors) requires index notation, not just multiplication.

• Check answers are sensible: If you calculate that a television originally cost £3,000 before a 12% increase, but now costs £313, immediately recheck. Use estimation to verify magnitude.

• Manage calculator use: Know how to enter fractions and standard form on your calculator model. For OCR papers, ensure you can access fraction, power and root functions efficiently to save time.

Quick revision summary

Number forms the foundation of GCSE Mathematics. Master the four operations with integers, fractions, decimals and percentages, ensuring you can convert fluently between forms. Understand ratio and proportion, both direct and inverse. Apply index laws confidently, including fractional and negative indices. Use prime factorisation to find HCF and LCM. Work competently with standard form for large and small numbers. Always show clear working, check answers are reasonable, and round only at the final step. These skills underpin success across all GCSE mathematics topics.