Ratio and Proportion
Overview
Ratio and Proportion is a foundational topic in the TNPSC Group IV Aptitude section, appearing consistently across papers. This topic tests your ability to compare quantities, distribute amounts, and solve real-world problems involving mixtures, partnerships, and scaling.
Mastering this topic is essential because it forms the base for other aptitude areas like percentages, time-work, and mixtures. Questions are typically straightforward at the Group IV level (SSLC standard), but speed and accuracy matter.
Students must develop fluency in converting ratios, finding unknowns using proportionality, and recognizing when to apply direct versus inverse relationships.
Key Concepts
- Ratio is a comparison of two quantities of the same unit, written as a:b or a/b. The first term (a) is the antecedent, the second (b) is the consequent.
- Proportion means 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:b::c:d, we have a × d = b × c. This is the fundamental rule for solving proportion problems.
- Direct Proportion: When one quantity increases, the other increases proportionally. Example: More items purchased → More money spent.
- Inverse Proportion: When one quantity increases, the other decreases proportionally. Example: More workers → Less time to complete work.
- Compound Ratio: The ratio of products of corresponding terms. Compound ratio of a:b and c:d is ac:bd.
- Duplicate Ratio: The ratio of squares. Duplicate ratio of a:b is a²:b².
- Sub-duplicate Ratio: The ratio of square roots. Sub-duplicate ratio of a:b is √a:√b.
Formulas / Key Facts
| Concept | Formula/Rule |
|---|---|
| Ratio a:b in fraction form | a/b |
| Proportion rule | If a:b = c:d, then ad = bc |
| Dividing N in ratio a:b | First part = N × a/(a+b), Second part = N × b/(a+b) |
| Dividing N in ratio a:b:c | Parts are Na/(a+b+c), Nb/(a+b+c), Nc/(a+b+c) |
| Compound ratio of a:b and c:d | ac : bd |
| Duplicate ratio of a:b | a² : b² |
| Sub-duplicate ratio of a:b | √a : √b |
| Triplicate ratio of a:b | a³ : b³ |
| Direct proportion | x₁/y₁ = x₂/y₂ |
| Inverse proportion | x₁ × y₁ = x₂ × y₂ |
Must-remember facts:
- Always express ratios in lowest terms (divide by HCF)
- Ratios have no units — quantities must be in same unit before comparing
- If a:b = 2:3, then a = 2k and b = 3k for some constant k
Worked Examples
Example 1: Basic Division Divide ₹630 between A and B in the ratio 4:5.
Solution:
- Sum of ratio parts = 4 + 5 = 9
- A's share = 630 × 4/9 = 280
- B's share = 630 × 5/9 = 350
Answer: A gets ₹280, B gets ₹350
Example 2: Finding Unknown in Proportion If 3:x = x:12, find x.
Solution:
- Using product rule: 3 × 12 = x × x
- 36 = x²
- x = 6
Answer: x = 6
Example 3: Compound Ratio Find the compound ratio of 2:3, 4:5, and 6:7.
Solution:
- Multiply all antecedents: 2 × 4 × 6 = 48
- Multiply all consequents: 3 × 5 × 7 = 105
- Compound ratio = 48:105 = 16:35 (dividing by HCF 3)
Answer: 16:35
Example 4: Inverse Proportion If 15 workers complete a job in 12 days, how many days will 20 workers take?
Solution:
- More workers → Less days (inverse proportion)
- Workers × Days = Constant
- 15 × 12 = 20 × d
- 180 = 20d
- d = 9
Answer: 9 days
Example 5: Three-part Division A sum of ₹1800 is divided among P, Q, R in the ratio 2:3:4. Find each share.
Solution:
- Total parts = 2 + 3 + 4 = 9
- P = 1800 × 2/9 = 400
- Q = 1800 × 3/9 = 600
- R = 1800 × 4/9 = 800
Answer: P = ₹400, Q = ₹600, R = ₹800
Common Mistakes
- Forgetting to convert units → If comparing 2 metres and 50 cm, first convert to same unit: 200 cm : 50 cm = 4:1. Always check units before forming ratios.
- Not simplifying to lowest terms → Writing 6:9 instead of 2:3. Always divide both terms by their HCF for the final answer.
- Confusing direct and inverse proportion → Thinking more workers means more days. Ask yourself: "If one increases, does the other increase (direct) or decrease (inverse)?"
- Adding ratios incorrectly → When finding compound ratio, multiply the terms, don't add them. Compound of 2:3 and 4:5 is 8:15, not 6:8.
- Misapplying proportion rule → In a:b::c:d, writing a×c = b×d. The correct rule is: product of extremes = product of means, so a×d = b×c.
Quick Reference
- Ratio a:b means a/b; always simplify to lowest terms
- Proportion rule: a:b::c:d means a×d = b×c
- To divide N in ratio a:b:c → parts are Na/(a+b+c), Nb/(a+b+c), Nc/(a+b+c)
- Direct: both increase/decrease together; Inverse: one up, other down
- Compound ratio = product of antecedents : product of consequents
- Mean proportional of a and b is √(ab)