Ratio and Proportion
Overview
Ratio and Proportion forms a foundational pillar of quantitative aptitude in MPESB exams. Mastering it gives you a toolkit that speeds up seemingly unrelated calculations.
The core skill here is recognizing relationships between quantities. When the question says "A is twice B," your mind should instantly see the ratio 2:1. When it says "if one increases, the other decreases proportionally," you must identify inverse proportion. MPESB often tests your ability to set up the correct proportion type rather than merely compute.
Students who score well treat ratios as a language—a shorthand for comparing magnitudes. Once fluent, you'll solve problems mentally that others struggle with on paper.
Key Concepts
- Ratio compares two quantities of the same unit. If A:B = 3:5, it means for every 3 parts of A, there are 5 parts of B. The actual values could be 6 and 10, or 300 and 500—the ratio remains unchanged.
- Proportion states that two ratios are equal. If A:B = C:D, then A, B, C, D are in proportion. The middle terms (B and C) are called means; the outer terms (A and D) are called extremes.
- Direct Proportion: When one quantity increases, the other increases at the same rate. Example: More workers → more output (if efficiency is constant). Mathematically, x/y = constant.
- Inverse Proportion: When one quantity increases, the other decreases proportionally. Example: More workers → less time to finish the same work. Mathematically, x × y = constant.
- Compound Ratio: Multiply corresponding terms of two or more ratios. If ratios are a:b and c:d, the compound ratio is ac:bd.
- Duplicate and Triplicate Ratios: The duplicate ratio of a:b is a²:b². The triplicate ratio is a³:b³. These appear in problems involving areas and volumes.
- Componendo-Dividendo: If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d). This shortcut dramatically speeds certain calculations.
Formulas / Key Facts
| Formula / Rule | Context |
|---|---|
| If A:B = m:n, then A = mk and B = nk for some constant k | Converting ratio to actual values |
| Product of means = Product of extremes (B × C = A × D) | Testing whether four numbers are in proportion |
| Direct Proportion: x₁/y₁ = x₂/y₂ | Use when both quantities move in the same direction |
| Inverse Proportion: x₁ × y₁ = x₂ × y₂ | Use when quantities move in opposite directions |
| Compound Ratio of (a:b), (c:d), (e:f) = ace : bdf | Combining multiple ratios |
| Sub-duplicate ratio of a:b = √a : √b | Related to square roots |
| If A:B = 2:3 and B:C = 4:5, make B common → A:B:C = 8:12:15 | Linking two ratios through a common term |
| Mean proportional of a and c = √(ac) | The middle term in a:x = x:c |
Worked Examples
Example 1: Basic Ratio Problem
Q: The ratio of ages of Ram and Shyam is 4:5. If Ram is 24 years old, find Shyam's age.
Solution: Let the ages be 4k and 5k. Given: 4k = 24 → k = 6 Shyam's age = 5k = 5 × 6 = 30 years
Example 2: Inverse Proportion
Q: 12 workers can complete a task in 20 days. How many days will 15 workers take to complete the same task?
Solution: More workers → fewer days (inverse proportion). Workers × Days = constant 12 × 20 = 15 × D 240 = 15D D = 240/15 = 16 days
Example 3: Combining Ratios
Q: A:B = 2:3 and B:C = 5:7. Find A:B:C.
Solution: Make B the same in both ratios. In first ratio, B = 3. In second ratio, B = 5. LCM of 3 and 5 = 15.
Multiply first ratio by 5: A:B = 10:15 Multiply second ratio by 3: B:C = 15:21
Now B is common. A:B:C = 10:15:21
Example 4: Compound Ratio
Q: Find the compound ratio of 2:3, 5:7, and 4:9.
Solution: Compound ratio = (2 × 5 × 4) : (3 × 7 × 9) = 40 : 189 Answer: 40:189
Example 5: Direct Proportion with Cost
Q: If 15 books cost ₹675, what is the cost of 25 books?
Solution: More books → more cost (direct proportion). 15/675 = 25/x is incorrect setup. Correct: 15/25 = 675/x? No. Actually: Cost per book = 675/15 = 45 Cost of 25 books = 25 × 45 = ₹1125
Or using proportion: 15:25 = 675:x 15x = 25 × 675 x = 16875/15 = ₹1125
Common Mistakes
- Confusing direct and inverse proportion → Before calculating, ask: "If one goes up, does the other go up or down?" Up-up or down-down means direct. Up-down or down-up means inverse.
- Forgetting to make the common term equal when linking ratios → If A:B = 2:3 and B:C = 4:5, you cannot simply write A:B:C = 2:3:5. You must equalize B first using LCM.
- Adding ratios instead of multiplying for compound ratio → Compound ratio of 2:3 and 4:5 is 8:15, not 6:8. Always multiply corresponding terms.
- Using actual values when only ratio is needed → If asked "what is the ratio of their speeds," don't waste time finding individual speeds. Work with the ratio throughout.
- Ignoring units → Ratios compare same-unit quantities. If one is in hours and another in minutes, convert first.
- Misapplying componendo-dividendo → This shortcut works only when you have a/b = c/d format. Verify the setup before applying.
Quick Reference
- Ratio = parts, not actuals. 3:5 means 3 parts and 5 parts; use k to find values.
- Direct: same direction → x/y = constant. Inverse: opposite direction → xy = constant.
- Compound ratio: multiply across (a:b) × (c:d) = ac:bd.
- Link ratios via LCM of the common term.
- Mean × Mean = Extreme × Extreme (product rule for proportion).
- Quick check: In a valid proportion, cross-multiplication gives equal products.