Module 1: Number, Fractions, Percentages & Ratio
Core arithmetic, proportion, standard form & surds
Number and Ratio form the foundation of GCSE Mathematics, accounting for up to 50% of Foundation marks and 35% of Higher marks. Master these fundamental topics to build speed and accuracy across all exam papers.
1. Fractions, Decimals & Percentages
Fractions represent parts of a whole. To add or subtract fractions, you must find a common denominator.
Fraction Operations Summary
- Addition/Subtraction: a/b + c/d = (ad + bc) / bd (Convert to common denominator first).
- Multiplication: Multiply numerators together and denominators together: (a/b) × (c/d) = ac / bd.
- Division ("Keep, Change, Flip"): (a/b) ÷ (c/d) = (a/b) × (d/c) = ad / bc.
Percentage Multipliers: To increase a number by 15%, multiply by 1.15. To decrease by 15%, multiply by 0.85. For compound growth or decay over n years: Final = Initial × (Multiplier)n.
2. HCF, LCM & Prime Factorisation
Every positive integer greater than 1 can be uniquely expressed as a product of prime factors (e.g. 60 = 2² × 3 × 5).
| Concept | Definition | Method using Venn Diagram |
|---|---|---|
| HCF (Highest Common Factor) | The largest factor shared by two or more numbers. | Multiply all numbers in the intersection (middle) of the prime Venn diagram. |
| LCM (Lowest Common Multiple) | The smallest positive integer divisible by all numbers. | Multiply ALL numbers across the entire Venn diagram union. |
3. Ratio & Proportion
Ratios compare quantities of the same unit. To share an amount in a given ratio A : B : C:
- Add the parts together to get the total number of shares (A + B + C).
- Divide the total amount by the number of shares to find the value of 1 share.
- Multiply 1 share by each component of the ratio.
Direct vs Inverse Proportion
- Direct Proportion (y ∝ x): y = kx, where k is the constant of proportionality. The graph is a straight line through the origin (0,0).
- Inverse Proportion (y ∝ 1/x): y = k/x. As x increases, y decreases proportionally.
4. Standard Form & Surds (Higher Tier Focus)
Standard Form
Written as A × 10n, where 1 ≤ A < 10 and n is an integer.
- Large numbers: 450,000 = 4.5 × 105.
- Small numbers: 0.00032 = 3.2 × 10-4.
Surds
Surds are irrational square roots. Key rules include:
- √a × √b = √(a × b)
- √a / √b = √(a / b)
- √a + √b ≠ √(a + b) (Never add under roots!)
- Rationalising the Denominator: Multiply top and bottom by √b for a / √b = (a√b) / b.
Frequently Asked Questions
Use the multiplier formula: Final Amount = Initial Amount × (1 ± r)^n, where r is the percentage rate written as a decimal, and n is the number of time periods (e.g. years). For example, £500 at 3% interest for 4 years is 500 × (1.03)^4.
In direct proportion, as one quantity increases, the other increases at the exact same rate (y = kx). In inverse proportion, as one quantity doubles, the other halves (y = k/x).
Find the largest square number factor of the number under the root. For instance, to simplify √72, rewrite it as √(36 × 2) = √36 × √2 = 6√2.