Ratio and Proportion
Overview
Ratio and Proportion forms the backbone of quantitative aptitude for TNPSC Group II/IIA. This topic directly tests your ability to compare quantities and establish relationships between them — skills that extend into partnership problems, mixture calculations, and even time-work scenarios.
Questions range from straightforward ratio simplification to multi-step partnership profit division and alligation-based mixture problems. Mastering this topic also accelerates your speed in Data Interpretation, where ratios frequently appear in tables and graphs.
The key to scoring here is building strong conceptual clarity on what ratios mean (relative comparison, not absolute values) and drilling the standard problem patterns until they become reflexive.
Key Concepts
- Ratio is a comparison of two quantities of the same unit, expressed as a:b or a/b. The first term is the antecedent; the second is the consequent.
- Proportion states that two ratios are equal: if a:b = c:d, then a, b, c, d are in proportion. Here, a and d are extremes; b and c are means.
- Product of extremes = Product of means: In a proportion a:b::c:d, we have a × d = b × c. This is the fundamental cross-multiplication rule.
- Continued proportion: Three quantities a, b, c are in continued proportion if a:b = b:c. Here b² = ac, and b is called the mean proportional.
- Direct proportion: When one quantity increases, the other increases proportionally (y = kx).
- Inverse proportion: When one quantity increases, the other decreases proportionally (y = k/x).
- Partnership: Profit/loss is divided in the ratio of (Capital × Time) for each partner.
- Mixtures and Alligation: Used to find the ratio in which two ingredients at different prices/concentrations must be mixed to get a desired average.
Formulas / Key Facts
| Concept | Formula/Rule |
|---|---|
| Ratio simplification | Divide both terms by their HCF |
| If a:b = c:d | a × d = b × c (cross-multiply) |
| Mean proportional of a and c | √(a × c) |
| Third proportional to a, b | b²/a |
| Fourth proportional to a, b, c | (b × c)/a |
| Componendo | If a/b = c/d, then (a+b)/b = (c+d)/d |
| Dividendo | If a/b = c/d, then (a−b)/b = (c−d)/d |
| Partnership ratio | Profit share = (Capital₁ × Time₁) : (Capital₂ × Time₂) |
| Alligation rule | Ratio = (Higher value − Mean) : (Mean − Lower value) |
| Mixture removal formula | Final concentration = Initial × (1 − R/V)ⁿ, where R = quantity removed, V = total volume, n = repetitions |
Key fact: Ratios have no units. You cannot add or subtract ratios directly — you must first convert to actual quantities using a common multiplier.
Worked Examples
Example 1: Basic Ratio The ratio of ages of A and B is 4:5. If B is 25 years old, find A's age.
Solution:
- Let ages be 4x and 5x
- Given: 5x = 25 → x = 5
- A's age = 4 × 5 = 20 years
Example 2: Partnership A invests ₹50,000 for 12 months. B invests ₹60,000 for 10 months. If total profit is ₹9,100, find B's share.
Solution:
- A's investment-time = 50,000 × 12 = 6,00,000
- B's investment-time = 60,000 × 10 = 6,00,000
- Ratio = 6,00,000 : 6,00,000 = 1:1
- B's share = 9,100 × (1/2) = ₹4,550
Example 3: Alligation (Mixture) In what ratio must rice at ₹40/kg be mixed with rice at ₹60/kg to get a mixture worth ₹45/kg?
Solution using Alligation Rule:
40 60
\ /
\ 45 /
\ /
(60−45) (45−40)
15 : 5- Ratio = 15:5 = 3:1
Verification: (40×3 + 60×1)/(3+1) = (120+60)/4 = 180/4 = 45 ✓
Example 4: Repeated Dilution A 20-litre mixture contains milk and water in ratio 3:1. If 5 litres are removed and replaced with water, find the new ratio.
Solution:
- Initial milk = 20 × (3/4) = 15 litres
- Initial water = 20 × (1/4) = 5 litres
- When 5 litres removed: milk removed = 5 × (3/4) = 3.75 litres
- Milk remaining = 15 − 3.75 = 11.25 litres
- Water remaining = 5 − 1.25 = 3.75 litres
- After adding 5 litres water: water = 3.75 + 5 = 8.75 litres
- New ratio = 11.25 : 8.75 = 1125 : 875 = 9:7
Common Mistakes
- Adding ratios directly → Wrong: 2:3 + 4:5 ≠ 6:8. Fix: Convert to actual quantities using a multiplier, then add.
- Forgetting to equalize time in partnership → Wrong: Comparing only capitals. Fix: Always multiply capital by time period for each partner.
- Confusing ratio order in alligation → Wrong: Subtracting in wrong direction. Fix: Always do (Higher − Mean) for cheaper ingredient's proportion.
- Ignoring units in ratio comparison → Wrong: Comparing ₹500 with 50 paise as 500:50. Fix: Convert to same unit first (50000 paise : 50 paise = 1000:1).
- Assuming ratio gives actual values → Wrong: If ratio is 3:4, assuming values are 3 and 4. Fix: Values are 3k and 4k; use given conditions to find k.
Quick Reference
- Ratio a:b means a/b; always simplify to lowest terms using HCF
- Proportion rule: Product of extremes = Product of means
- Mean proportional of a and c = √(ac)
- Partnership profit ratio = (Capital × Time) for each partner
- Alligation: Cheaper : Dearer = (Dearer price − Mean) : (Mean − Cheaper price)
- Mixture replacement: Final = Initial × (1 − Removed/Total)ⁿ