Ratio and Proportion — SSC MTS Study Notes
Overview
The topic tests your ability to compare quantities, scale them proportionally, and solve real-world problems involving partnerships and resource distribution. Mastering this area is essential because ratio concepts underpin several other topics like percentage, mixture problems, time-work relationships, and profit-loss calculations.
For SSC MTS, you must be comfortable with three key areas: basic ratio simplification and manipulation, compound ratios (ratios of ratios), and partnership problems where profit is divided according to capital and time contributions. The questions are usually straightforward but require careful attention to whether ratios are direct or inverse, and whether you're dealing with two quantities or three or more. Most problems can be solved in 60–90 seconds if you know the standard patterns.
Expect word problems involving age ratios, ingredient mixing, salary divisions, or business partnerships. The exam typically avoids overly complex algebraic manipulations, focusing instead on your ability to set up proportions correctly and perform basic arithmetic quickly and accurately.
Key Concepts
- Ratio expresses how many times one quantity contains another. Written as a:b, it means for every 'a' units of the first quantity, there are 'b' units of the second. Ratios are fractions in disguise: a:b = a/b.
- Proportion states that two ratios are equal: a:b = c:d, or equivalently a/b = c/d. Cross-multiplication gives ad = bc, the fundamental property used to solve proportion problems.
- Direct Proportion: When one quantity increases, the other increases proportionally (y = kx). Example: more workers complete more work in the same time.
- Inverse Proportion: When one quantity increases, the other decreases proportionally (xy = k). Example: more workers complete the same work in less time.
- Compound Ratio: The ratio obtained by multiplying corresponding terms of two or more ratios. If ratios are a:b and c:d, the compound ratio is ac:bd. Used when multiple factors affect a comparison simultaneously.
- Partnership: Business profit or loss is divided among partners in the ratio of their effective investments, which equals (Capital × Time). Equal time investments simplify to capital ratio; equal capital investments simplify to time ratio.
- Componendo-Dividendo: A shortcut for proportion problems. If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). Rarely needed for SSC MTS but useful for speed.
- Ratio manipulation rules: Multiplying or dividing all terms by the same number doesn't change the ratio. To compare or combine ratios with different terms, find the LCM of unlike terms and scale accordingly.
Formulas / Key Facts
- Basic proportion: If a:b = c:d, then ad = bc (cross-multiplication property).
- Direct variation: y = kx, so y₁/x₁ = y₂/x₂ when quantity varies directly.
- Inverse variation: xy = k, so x₁y₁ = x₂y₂ when quantity varies inversely.
- Compound ratio of a:b and c:d is ac:bd. Of three ratios a:b, c:d, e:f is ace:bdf.
- Partnership profit division: Profit ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂) : ...
- Third proportional to a and b is x such that a:b = b:x, giving x = b²/a.
- Fourth proportional to a, b, c is x such that a:b = c:x, giving x = bc/a.
- Mean proportional between a and b is √(ab), satisfying a:x = x:b.
- Ratio to actual values: If ratio is a:b and sum is S, first part = a/(a+b) × S, second part = b/(a+b) × S.
- Combining ratios: If A:B = 2:3 and B:C = 4:5, make B equal by LCM(3,4)=12. Then A:B:C = 8:12:15.
Worked Examples
Example 1: Basic Ratio Division Divide ₹1800 among A, B, and C in the ratio 2:3:4.
Solution: Sum of ratio terms = 2 + 3 + 4 = 9 A's share = (2/9) × 1800 = ₹400 B's share = (3/9) × 1800 = ₹600 C's share = (4/9) × 1800 = ₹800
Answer: ₹400, ₹600, ₹800
Example 2: Compound Ratio The compound ratio of 3:4 and 5:7 is what?
Solution: Multiply corresponding terms: First terms: 3 × 5 = 15 Second terms: 4 × 7 = 28 Compound ratio = 15:28
Answer: 15:28
Example 3: Partnership Problem A invests ₹5000 for 6 months and B invests ₹6000 for 4 months. How should they divide a profit of ₹2200?
Solution: Effective investment ratio = (Capital × Time) ratio A's effective investment = 5000 × 6 = 30000 B's effective investment = 6000 × 4 = 24000 Ratio = 30000:24000 = 5:4
Sum of ratio = 5 + 4 = 9 A's profit = (5/9) × 2200 = ₹1222.22 (approximately ₹1222) B's profit = (4/9) × 2200 = ₹977.78 (approximately ₹978)
Answer: A gets ₹1222, B gets ₹978
Example 4: Inverse Proportion If 12 workers can complete a work in 8 days, how many workers are needed to complete it in 6 days?
Solution: Workers and days are inversely proportional (more workers, fewer days). Workers₁ × Days₁ = Workers₂ × Days₂ 12 × 8 = x × 6 96 = 6x x = 16
Answer: 16 workers
Common Mistakes
Mistake 1: Confusing direct and inverse proportion Wrong thinking: More workers means more days (direct proportion). Correct fix: More workers complete the same work in fewer days (inverse proportion). Use x₁y₁ = x₂y₂, not x₁/y₁ = x₂/y₂.
Mistake 2: Adding ratios incorrectly Wrong thinking: If A:B = 2:3 and B:C = 3:4, then A:B:C = 2:3:4. Correct fix: The middle term B must be equal in both ratios. Scale them: A:B = 2:3 and B:C = 3:4 gives A:B:C = 2:3:4 only if both B's are the same value. Here they already match, so 2:3:4 is correct. If B values differ, find LCM and scale.
Mistake 3: Forgetting time in partnership problems Wrong thinking: Divide profit in ratio of capitals only. Correct fix: Profit divides in ratio of (Capital × Time). If time differs, you must multiply each partner's capital by their investment duration.
Mistake 4: Not simplifying ratios before calculation Wrong thinking: Working with 150:225:300 as-is. Correct fix: Simplify to 2:3:4 by dividing by GCD (75). Smaller numbers reduce arithmetic errors and save time.
Mistake 5: Misreading "in proportion" vs "in ratio" Wrong thinking: Treating a/b = c/d problems like simple ratio divisions. Correct fix: Proportion equations need cross-multiplication (ad = bc). Set up the equation correctly based on what varies with what.
Quick Reference
- Ratio a:b means a/b. To find actual values from ratio and sum, use [a/(a+b)]×Sum.
- Proportion a:b = c:d gives ad = bc. Use cross-multiplication to solve for unknowns.
- Compound ratio of a:b and c:d is ac:bd. Multiply numerators, multiply denominators.
- Partnership profit = (Capital × Time) ratio. Same time → use capital ratio; same capital → use time ratio.
- Direct proportion: y₁/x₁ = y₂/x₂. Inverse proportion: x₁y₁ = x₂y₂.
- Always simplify ratios to smallest integers by dividing by GCD before further calculations.