GCSE Mathematics 03 — Ratio, Proportion and Rates of Change

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Topics include Simplifying and Writing Ratios, Sharing in a Ratio, Algebraic Ratios and Combining Ratios, Direct Proportion and Recipes, Map Scales and Scale Drawings, Algebraic Direct and Inverse Proportion, Proportion Graphs and Reasoning, and Percentages: Growth, Decay, and Finance.

Mathematics EN

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Simplifying and Writing Ratios

Expressing comparisons between quantities in their simplest form and unitary format.

Key points

A ratio $a:b$ compares $a$ parts of one quantity to $b$ parts of another.
Ratios must compare quantities in the same units (e.g., convert cm to m before simplifying).
To simplify, divide all parts by their Highest Common Factor (HCF).
Two ratios are equivalent if they represent the same fraction: $\frac{a}{b}=\frac{c}{d}$ . Quick check: cross-multiply ( $a\times d$ and $b\times c$ ) — if equal, the ratios are equivalent.
To remove fractions or decimals, multiply all parts by a suitable power of 10 or the Lowest Common Multiple (LCM) of denominators.

Worked example

Question

Simplify the ratio

450 \text{ g} : 1.2 \text{ kg}

and write your answer in the form

1:n

Solution

1. Convert to the same units:

1.2 \text{ kg} = 1200 \text{ g}

.
2. Write ratio:

450 : 1200

.
3. Simplify by dividing by 10:

45 : 120

.
4. Divide by 15 (HCF):

3 : 8

.
5. To write as

1:n

, divide both sides by 3:

1 : \frac{8}{3}

1 : 2.67

(approx).
Answer:

1 : 2.6\dot{6}

Common pitfalls

Forgetting to convert units before simplifying (e.g., treating $100\text{cm} : 1\text{m}$ as $100:1$ ).
Rounding errors when writing in the form $1:n$ if the decimal is recurring.
Confusing ratio order; $a:b$ is not the same as $b:a$ .
Checking equivalence by comparing the parts separately (e.g., 3 vs 12) instead of simplifying or using cross-multiplication.

Prerequisites

Factors and Multiples
Unit Conversions
Fractions and Decimals

Further resources

Simplifying Ratios
Clear introduction to simplifying ratios with integer and decimal examples.

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