Types of ratios systematically categorise how two quantities relate, forming the foundation for proportion and variation problems in business mathematics.
## Core Concept
A ratio is a comparison of two quantities by division, expressing how many times one quantity is contained in another. Types of ratios classify these comparisons based on the nature of the quantities being compared and their relationship to each other.
## Main Types of Ratios
### 1. Simple Ratio - Compares two quantities directly in the form a:b - Example: Cost of materials to cost of labour = 5:3 - Exam note: Foundational; always reduce to lowest terms (use HCF)
### 2. Compound Ratio - Ratio of two or more simple ratios combined by multiplication - If ratio 1 is a:b and ratio 2 is c:d, the compound ratio is (a×c):(b×d) - Example: If profit:cost = 2:5 and cost:revenue = 3:4, then profit:revenue = (2×3):(5×4) = 6:20 = 3:10 - Exam tip: Frequently appears in business scenarios (cost-price-selling price chains)
### 3. Duplicate Ratio - Ratio of the squares of two quantities - If original ratio is a:b, duplicate ratio is a²:b² - Example: If linear dimension ratio = 2:3, then area ratio = 4:9 - Use case: Geometry-based business problems (land, materials)
### 4. Triplicate Ratio - Ratio of the cubes of two quantities - If original ratio is a:b, triplicate ratio is a³:b³ - Example: If edge ratio of two cubes = 2:3, then volume ratio = 8:27 - Exam context: Less common than duplicate; watch for 3D problems
### 5. Sub-duplicate (Square Root) Ratio - Ratio obtained by taking square roots - If original ratio is a:b, sub-duplicate ratio is √a:√b - Example: If area ratio = 16:25, then linear ratio = 4:5 - Reverse of duplicate ratio
### 6. Sub-triplicate (Cube Root) Ratio - Ratio obtained by taking cube roots - If original ratio is a:b, sub-triplicate ratio is ³√a:³√b - Reverse of triplicate ratio
### 7. Inverse (Reciprocal) Ratio - When terms of a ratio are interchanged - If ratio is a:b, inverse ratio is b:a - Example: If speed:time = 5:2, then inverse (time:speed) = 2:5 - Key relationship: Often used in variation problems (inverse proportion)
## Worked Example
Problem: A company's profit ratio to cost ratio is 1:4, and cost to revenue is 2:5. Find the compound ratio of profit to revenue.
Solution: - Profit : Cost = 1:4 - Cost : Revenue = 2:5 - Compound ratio = (1×2):(4×5) = 2:20 = 1:10 - Interpretation: For every ₹10 in revenue, profit is ₹1
## Common Exam Applications
- Costing problems: Material:labour:overhead ratios combined
- Financial analysis: Profit margin calculations using compound ratios
- Scaling: Duplicate/triplicate for area/volume scaling in production
- Inverse relationships: Speed-time, price-demand in business scenarios
- Simplification: Always reduce final answers using HCF before matching answer options
## Common Mistakes
- Not reducing to simplest form — ratios must be in lowest terms
- Confusing inverse with reciprocal — they're the same; use terminology consistently
- Incorrect compound ratio multiplication — multiply *corresponding* terms only
- Forgetting to reduce after squaring/cubing — 4:9 is the answer, not 2²:3²
- Mixing up duplicate and sub-duplicate — squares vs. square roots; the original ratio is neither pure square nor pure root