Ratio and Proportion

Ratio and Proportion — Study Notes

Overview

Ratio and Proportion forms the backbone of many SSC CGL quantitative aptitude problems. A firm grasp of ratio manipulation, proportion solving, and compound ratio concepts is essential.

The exam tests your ability to solve equations involving ratios quickly, handle multiple-ratio problems (especially in partnership scenarios), and apply ratio principles to real-world situations like dividing profits, comparing quantities, and mixing ingredients. Mastery here means faster calculations and fewer errors across multiple QA sections. Focus on accurate fraction work, cross-multiplication speed, and recognizing when to use direct versus inverse proportion.

Key Concepts

• Ratio expresses the relationship between two quantities of the same kind as a quotient a:b = a/b. Ratios are unitless; 2 kg : 4 kg simplifies to 1:2.
• Proportion states that two ratios are equal: a:b = c:d or a/b = c/d. In a proportion, the product of extremes equals the product of means: a × d = b × c.
• Compound ratio is the ratio of products of corresponding terms from two or more ratios. If ratio of A to B is m:n and B to C is p:q, the compound ratio A:B:C is derived by multiplying. For two ratios a:b and c:d, compound ratio is (a×c):(b×d).
• Direct proportion: when one quantity increases, the other increases proportionally (x₁/y₁ = x₂/y₂). Inverse proportion: when one increases, the other decreases (x₁ × y₁ = x₂ × y₂).
• Duplicate ratio of a:b is a²:b². Triplicate ratio is a³:b³. Sub-duplicate is √a:√b. These appear in geometry and area/volume comparisons.
• Partnership problems use ratios to divide profit/loss. In simple partnership, profit is divided in the ratio of capital invested. In compound partnership, profit ratio equals (capital₁ × time₁) : (capital₂ × time₂).
• Continued ratio combines three or more quantities in a single expression like A:B:C = 2:3:5, meaning if total parts = 2+3+5 = 10, A gets 2/10 of total.
• Fourth proportional: If a:b = c:x, then x is the fourth proportional = (b × c)/a. Third proportional to a,b is x where a:b = b:x, giving x = b²/a.

Formulas / Key Facts

1. Basic proportion: If a:b = c:d, then a×d = b×c (cross-multiplication).
2. Componendo-Dividendo: If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d). Useful for quick ratio manipulation.
3. Mean proportional between a and b: √(a×b). If a, m, b are in continued proportion, m² = a×b.
4. Splitting amount in ratio m:n: First part = (m/(m+n))×Total; Second part = (n/(m+n))×Total.
5. Compound ratio of a:b and c:d: (a×c):(b×d). For three ratios a:b, c:d, e:f, compound is (a×c×e):(b×d×f).
6. Partnership formula: Profit share ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂). If time same, ratio equals capital ratio.
7. Inverse proportion formula: If x men do work in y days, then x₁×y₁ = x₂×y₂ where subscript denotes different scenarios.
8. Alligation link: When mixing two components with individual prices/strengths, the ratio of quantities mixed is inversely proportional to differences from the mean value.

Worked Examples

Example 1: If A:B = 2:3 and B:C = 4:5, find A:B:C.

Solution: Make B equal in both ratios. B appears as 3 in first and 4 in second. LCM(3,4) = 12. Multiply first ratio by 4: A:B = 8:12. Multiply second ratio by 3: B:C = 12:15. Now B = 12 in both. Combined: A:B:C = 8:12:15.

Example 2: Three partners A, B, C invest ₹4000, ₹6000, ₹8000 for 6 months, 4 months, 3 months respectively. Divide profit of ₹7400.

Solution: Profit ratio = (4000×6) : (6000×4) : (8000×3) = 24000 : 24000 : 24000 = 1:1:1. Each partner gets ₹7400/3 = ₹2466.67 (approximately ₹2467 in exam answer).

Example 3: Fourth proportional to 3, 8, 9 is?

Solution: Let fourth proportional be x. Then 3:8 = 9:x. Cross-multiply: 3x = 8×9 = 72. x = 72/3 = 24.

Example 4: The ratio of boys to girls in a class is 5:3. If 4 more boys join, ratio becomes 2:1. Find original number of students.

Solution: Let boys = 5x, girls = 3x. After 4 boys join: (5x+4)/3x = 2/1. Cross-multiply: 5x + 4 = 6x. x = 4. Original students = 5(4) + 3(4) = 20 + 12 = 32.

Common Mistakes

1. Mixing ratios without equalizing common terms → When combining A:B and B:C, students forget to make B equal in both. Always find LCM of the common term's values and scale both ratios accordingly.
2. Forgetting time factor in partnership → Students divide profit only by capital invested, ignoring duration. Always multiply capital by time period (months/years) before finding the ratio.
3. Confusing direct and inverse proportion → Using x₁/y₁ = x₂/y₂ for inverse relationships (like more workers, less time). Remember: inverse means x₁×y₁ = x₂×y₂.
4. Wrong total parts calculation → In ratio 3:4:5, total parts is 3+4+5 = 12, not 3×4×5. Each share = (ratio part / total parts) × total amount.
5. Duplicate/triplicate ratio confusion → Duplicate ratio of 2:3 is 4:9 (square each term), not 4:6 (doubling). Sub-duplicate is √2:√3, not 1:1.5.

Quick Reference

• Cross-multiplication: a:b = c:d ⇒ ad = bc (fastest solution method)
• Combining ratios: Equalize common term by LCM, then merge
• Partnership profit = (Capital × Time) ratio
• Fourth proportional to a,b,c = (b×c)/a
• Split total T in ratio m:n → Parts are mT/(m+n) and nT/(m+n)
• Compound ratio (a:b)(c:d) = ac:bd (multiply corresponding terms)