GCSE Maths Number: Fractions, Percentages, and Ratios Guide

Number topics appear throughout GCSE Maths papers for all exam boards. This guide is part of our complete GCSE Maths revision guide — visit there for an overview of all topics. These fundamental skills underpin success across the entire specification, from basic calculations through to complex problem-solving. This guide covers the essential techniques you need to master, whether you’re studying Foundation or Higher tier.

Working with Fractions

Understanding fraction operations is crucial. To add or subtract fractions, find a common denominator by identifying the lowest common multiple of the denominators. For 1/3 + 1/4, the common denominator is 12, so convert to 4/12 + 3/12 = 7/12.

Multiplying fractions is more straightforward: multiply numerators together and denominators together, then simplify. For 2/3 × 3/5, multiply to get 6/15, which simplifies to 2/5 by dividing both top and bottom by 3.

Dividing fractions uses the “keep, change, flip” method: keep the first fraction, change division to multiplication, flip the second fraction. For 2/3 ÷ 3/4, this becomes 2/3 × 4/3 = 8/9.

Converting between mixed numbers and improper fractions is essential. To convert 2 3/4 to an improper fraction, multiply the whole number by the denominator and add the numerator: (2 × 4) + 3 = 11, giving 11/4. To convert back, divide the numerator by denominator: 11 ÷ 4 = 2 remainder 3, so 2 3/4.

Decimals and Place Value

Understanding place value is fundamental to all number work. In 3.472, the 4 represents 4 tenths, 7 represents 7 hundredths, and 2 represents 2 thousandths. This understanding is crucial for ordering decimals, rounding, and estimation.

When multiplying decimals, ignore decimal points initially, multiply the whole numbers, then count total decimal places in the question and apply them to your answer. For 0.3 × 0.4, multiply 3 × 4 = 12, then apply two decimal places to get 0.12.

Division by decimals can be tricky. Convert to whole numbers by multiplying both numbers by the same power of 10. For 2.4 ÷ 0.6, multiply both by 10 to get 24 ÷ 6 = 4.

Percentages

Three key percentage calculations appear frequently. To find a percentage of an amount, convert the percentage to a decimal by dividing by 100, then multiply. For 15% of £40, calculate 0.15 × 40 = £6.

To express one number as a percentage of another, divide the first by the second and multiply by 100. For example, expressing 12 out of 80 as a percentage: (12 ÷ 80) × 100 = 15%.

Percentage change questions use the formula: (change ÷ original) × 100. If a price increases from £40 to £48, the change is £8, so percentage increase = (8 ÷ 40) × 100 = 20%.

Multipliers make percentage problems more efficient. To increase by 15%, multiply by 1.15. To decrease by 15%, multiply by 0.85. For repeated percentage changes, raise the multiplier to a power. If an investment grows by 5% annually for 3 years, multiply by 1.05³.

Reverse percentage problems are Higher tier. If £45 represents 120% of the original price (after a 20% increase), divide by the multiplier: £45 ÷ 1.2 = £37.50.

Ratio and Proportion

Ratios compare quantities. To simplify ratios, divide all parts by their highest common factor. The ratio 12:18 simplifies to 2:3 by dividing both by 6.

Sharing in a ratio requires adding ratio parts to find total parts, then dividing the amount by total parts to find the value of one part. To share £150 in the ratio 2:3, total parts = 5, so one part = £150 ÷ 5 = £30. Therefore, shares are £60 and £90.

Direct proportion means as one quantity increases, another increases proportionally. If 3 pens cost £2.40, 5 pens cost £2.40 ÷ 3 × 5 = £4.00. The unitary method (finding the cost of one) is the most reliable approach.

Inverse proportion means as one quantity increases, another decreases proportionally. If 4 people take 6 hours to complete a job, 2 people would take 12 hours because there are half as many people, so it takes twice as long.

Standard Form

Standard form expresses very large or small numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. For example, 3,000,000 = 3 × 10⁶ and 0.00004 = 4 × 10⁻⁵.

To multiply numbers in standard form, multiply the front numbers and add the powers. (2 × 10³) × (3 × 10⁴) = 6 × 10⁷. If the result doesn’t follow standard form rules, adjust: 12 × 10⁵ = 1.2 × 10⁶.

For division, divide front numbers and subtract powers. (6 × 10⁵) ÷ (2 × 10²) = 3 × 10³. Again, adjust if necessary to maintain proper standard form.

Powers and Roots

Index laws are essential. When multiplying powers with the same base, add the indices: x³ × x⁴ = x⁷. When dividing, subtract indices: x⁵ ÷ x² = x³. When raising a power to another power, multiply indices: (x²)³ = x⁶.

Negative indices represent reciprocals: x⁻² = 1/x². Fractional indices represent roots: x^(1/2) = √x and x^(1/3) = ∛x. Combined fractional indices both root and power: x^(2/3) = (∛x)².

Surds

Surds are irrational roots like √2 that cannot be simplified to whole numbers. Simplify surds by finding square factors: √12 = √(4 × 3) = 2√3. Rationalising denominators removes surds from denominators by multiplying top and bottom by the surd: 1/√2 = 1/√2 × √2/√2 = √2/2.

Estimation and Bounds

Round to one significant figure when estimating. For 3.8 × 21.2 ÷ 0.47, round to 4 × 20 ÷ 0.5 = 80 ÷ 0.5 = 160. This provides a quick reasonableness check for calculator answers.

Upper and lower bounds recognise measurement limitations. A length given as 5m to the nearest metre could be anything from 4.5m up to but not including 5.5m. These are the lower and upper bounds.

Related GCSE Maths Guides

• Algebra revision guide — expressions, equations, and sequences
• Statistics and probability guide — averages, charts, and probability trees
• GCSE Maths formula sheet guide — formulas provided in the exam vs those you must memorise

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