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)
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If A : B = 2 : 3 and B : C = 4 : 5, then A : B : C is:
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