Ratio and Proportion — Study Notes

Overview

Ratio and Proportion forms the backbone of many SSC CGL quantitative aptitude problems. This topic directly accounts for 2–3 questions in Tier 1, but its applications extend across profit-loss, time-work, mixture-alligation, and partnership questions. 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)