Comparison of quantities of the same kind using the symbol ':' or '/' to establish relative magnitudes.
## Definition & Notation
A ratio is a comparison between two quantities of the same kind and same units. If quantity A is compared to quantity B, it is written as: - A : B (read as "A is to B") - A/B (fractional form)
The first term is called the antecedent and the second term is called the consequent.
Key principle: A ratio is a pure number with no units.
## Core Concept
Ratios express how many times one quantity is of another. They simplify relationships: - Ratio 5:3 means for every 5 units of the first quantity, there are 3 units of the second. - A ratio remains unchanged when both terms are multiplied or divided by the same non-zero number.
Equivalent ratios: - 2:3 = 4:6 = 6:9 = 10:15 (all equal)
## Formula / Rule
### Simplification of Ratios - Divide both terms by their HCF (Highest Common Factor) to reduce to simplest form.
Example: Simplify 24:36 - HCF(24, 36) = 12 - 24:36 = (24÷12):(36÷12) = 2:3
### Comparison of Ratios To compare ratios, convert to decimal or common denominator: - 3:4 = 0.75 - 5:7 ≈ 0.714 - Therefore 3:4 > 5:7
### Finding Terms When Ratio is Known If A:B = m:n and one quantity is given, find the other: - If A = x, then B = (n/m) × x - If B = y, then A = (m/n) × y
## Worked Example
Problem: Two quantities are in the ratio 5:7. If the first quantity is 150, find the second quantity.
Solution: - Let quantities be 5k and 7k (where k is the common multiplier) - First quantity: 5k = 150 - Therefore: k = 30 - Second quantity: 7k = 7 × 30 = 210 - Answer: 210
## Common Exam Applications
- Cost/Profit sharing: Dividing money in given ratio
- Speed & distance: Comparing distances or speeds as ratios
- Mixture problems: Ratio of components in a blend
- Age problems: Current age ratios to find future/past ages
- Map scale: Representing distances as ratios (1:100,000)
## Common Mistakes
- Confusing order: 5:3 is NOT the same as 3:5
- Including units: Never state a ratio as "5 metres : 3 metres" — simplify to 5:3
- Forgetting HCF: Writing 24:36 instead of reducing to 2:3 loses marks for incomplete simplification
- Applying ratio to different units: Cannot directly compare 5 kg : 3 litres without conversion
- Zero in denominator: Cannot form ratio with 0 as the second term
## Quick Check
- A ratio has two terms (antecedent and consequent)
- Both quantities must be of the same kind (same dimension)
- Ratio is a dimensionless number
- Always express in simplest form for CA Foundation exams