Ratio and Proportion — SSC CHSL Study Notes
Overview
This topic forms the foundation for many other areas including percentages, partnership, mixture-allegation, and time-speed-distance problems. The exam tests your ability to manipulate ratios, solve proportion equations quickly, and apply ratio logic to real-world scenarios like profit sharing and ingredient mixing.
Students must master three core areas: basic ratio operations and compound ratios, direct/inverse/continued proportions, and partnership problems (simple and compound). The questions range from straightforward ratio simplification to multi-step word problems involving age relations, salary divisions, and business profit distribution. Speed and accuracy in cross-multiplication and ratio-chain calculations directly impact your score, making this a must-master topic for the quantitative aptitude section.
Key Concepts
- Ratio expresses the relationship between two quantities of the same kind as a quotient (a:b means a/b). Ratios have no units and can be simplified like fractions by dividing by the HCF of terms.
- Proportion states that two ratios are equal (a:b = c:d, written as a:b::c:d). In any proportion, the product of extremes equals the product of means: a×d = b×c. This cross-multiplication property solves most proportion problems.
- Compound ratio is formed by multiplying corresponding terms of two or more ratios. If ratios are a:b and c:d, their compound ratio is ac:bd. Used when multiple relationships combine (like combining age ratios at different times).
- Mean proportional between two numbers a and c is the number b such that a:b = b:c, giving b² = ac, so b = √(ac). The middle term in a continued proportion.
- Third proportional to a and b is the number c where a:b = b:c, making c = b²/a. The fourth term when the first two terms repeat.
- Fourth proportional to a, b, c is the number d where a:b = c:d, giving d = bc/a. Standard proportion completion.
- Partnership applies ratio principles to divide profit or loss among partners based on their capital contributions and time periods. Simple partnership considers only capital; compound partnership accounts for different time durations.
- Direct proportion: when one quantity increases, the other increases proportionally (y ∝ x means y/x is constant). Inverse proportion: when one increases, the other decreases (y ∝ 1/x means xy is constant).
Formulas / Key Facts
- Ratio simplification: Divide both terms by their HCF. Example: 45:60 = 3:4 (dividing by 15).
- Compound ratio of a:b, c:d, e:f = ac·e : bd·f (multiply all first terms, multiply all second terms).
- Cross-multiplication rule: If a:b = c:d, then ad = bc. This solves for any unknown term.
- Mean proportional between a and c = √(ac). Example: mean proportional of 4 and 9 = √36 = 6.
- Third proportional to a, b = b²/a. If a=3, b=6, third proportional = 36/3 = 12.
- Fourth proportional to a, b, c = bc/a. If a=2, b=3, c=4, fourth proportional = 12/2 = 6.
- Simple Partnership profit division: Profit shares equal capital ratio. If A invests 5000 and B invests 3000, profit divides as 5:3.
- Compound Partnership: Profit shares = (Capital × Time) ratio. If A invests 4000 for 6 months and B invests 5000 for 8 months, ratio = (4000×6):(5000×8) = 24000:40000 = 3:5.
- Continued proportion: Three numbers a, b, c are in continued proportion if a:b = b:c, giving b² = ac.
Worked Examples
Example 1: Compound Ratio If the ratios are 2:3, 4:5, and 6:7, find their compound ratio.
Solution: Compound ratio = (2×4×6):(3×5×7) = 48:105 Simplify by dividing by HCF(48,105) = 3 = 16:35 Answer: 16:35
Example 2: Mean Proportional Find the mean proportional between 16 and 64.
Solution: Mean proportional = √(16×64) = √1024 = 32 Verify: 16:32 = 32:64 → 1:2 = 1:2 ✓ Answer: 32
Example 3: Partnership A starts a business with ₹8000. After 4 months, B joins with ₹12000. At year-end, they earn ₹9100 profit. Find each partner's share.
Solution: A's capital-months = 8000 × 12 = 96000 B's capital-months = 12000 × 8 = 96000 (B invested for 8 months) Profit ratio = 96000:96000 = 1:1 A's share = 9100 × (1/2) = ₹4550 B's share = 9100 × (1/2) = ₹4550 Answer: A gets ₹4550, B gets ₹4550
Example 4: Fourth Proportional Find the fourth proportional to 3, 7, and 9.
Solution: Let fourth proportional = x Then 3:7 = 9:x Using cross-multiplication: 3x = 7×9 3x = 63 x = 21 Answer: 21
Common Mistakes
Mistake: Forgetting to convert time to the same units in partnership problems. Fix: Always convert months to months or years to years before multiplying capital × time. Six months = 6, not 0.5 unless working in years consistently.
Mistake: Confusing third proportional and mean proportional. Fix: Mean proportional between a and c gives b where a:b = b:c (b appears twice). Third proportional to a and b gives c where a:b = b:c (a and b given, find c).
Mistake: Mixing up direct and inverse proportion. Fix: Direct proportion: both increase/decrease together (more workers → more work). Inverse proportion: one increases as other decreases (more speed → less time).
Mistake: Not simplifying compound ratios to lowest terms. Fix: After multiplying to get compound ratio, always find HCF of the two terms and divide both by it. The question expects simplified form.
Mistake: In partnership, forgetting that working partners may get salary before profit division. Fix: If a partner receives salary for work, subtract total salary from profit first, then divide remaining profit by capital-time ratio.
Quick Reference
- Ratio a:b can be written as fraction a/b; multiply/divide both terms by same number to maintain ratio.
- In proportion a:b::c:d, product of extremes = product of means → ad = bc (most useful property).
- Compound ratio of multiple ratios: multiply all numerators together, multiply all denominators together.
- Partnership profit share = (Capital invested × Time period) for each partner; express as ratio then divide total profit.
- Mean proportional between a and c is √(ac); appears as the middle term in a:b:c continued proportion.
- To compare ratios, convert to same denominator or decimal form; larger decimal means larger ratio.