Ratio and Proportion — Railway Group D Study Notes

Overview

Ratio and Proportion is a fundamental topic in Railway Group D Mathematics, appearing in 3–5 questions every exam. It forms the backbone for solving problems in partnership, mixing solutions, time-work relationships, and even profit-loss scenarios. The exam tests your ability to quickly set up ratio equations, manipulate them algebraically, and apply them to word problems involving money distribution, ingredient mixing, or proportional division of quantities.

Mastery of this topic requires fluency in three core areas: basic ratio operations (simplification, comparison, compounding), direct and inverse proportion identification, and partnership calculations where profit or capital is divided in given ratios. Most questions are application-based rather than theoretical—you'll rarely see "define ratio" but frequently encounter "three partners invest in ratio 2:3:5 and total profit is ₹40,000; find each share." Speed and accuracy in ratio manipulation directly impact your score, since these problems typically take 45–90 seconds when solved efficiently.

The topic connects naturally with percentage (converting ratios to percentages), fractions (ratios are fractions in disguise), and algebra (solving for unknown quantities in proportions). Understanding the difference between direct proportion (both quantities increase together) and inverse proportion (one increases as the other decreases) prevents the most common conceptual errors on exam day.

Key Concepts

• **Ratio** expresses the relationship between two quantities of the same kind as a:b or a/b. A ratio of 3:4 means the first quantity is 3 parts for every 4 parts of the second. Ratios have no units and can be simplified like fractions.

• **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. This cross-multiplication property solves most proportion problems in one step.

• **Compound ratio** is the ratio of products of corresponding terms from two or more ratios. The compound ratio of a:b and c:d is (a×c):(b×d). If ratios are 2:3 and 4:5, compound ratio is (2×4):(3×5) = 8:15.

• **Direct proportion** (x ∝ y) means if one quantity doubles, the other doubles too. Cost ∝ quantity purchased. Use the relationship x₁/y₁ = x₂/y₂ to solve.

• **Inverse proportion** (x ∝ 1/y) means if one quantity doubles, the other halves. Workers ∝ 1/time to complete work. Use x₁×y₁ = x₂×y₂.

• **Duplicate and triplicate ratios** mean squaring or cubing both terms. Duplicate ratio of a:b is a²:b². Triplicate ratio is a³:b³. Sub-duplicate (square root) and sub-triplicate (cube root) ratios reverse this operation.

• **Partnership** divides profit or loss among partners in the ratio of their (capital × time). If capitals are equal, profit ratio equals time ratio. If time is equal, profit ratio equals capital ratio. For varying capital and time, multiply each partner's contribution to get the ratio.

• **Continued ratio** a:b:c means a/b and b/c are both maintained. If a:b = 2:3 and b:c = 4:5, convert to common b: a:b:c = 8:12:15.

Formulas / Key Facts

**Basic ratio operations:**

• Simplification: Divide both terms by their HCF. 12:18 = 2:3 after dividing by 6.
• Comparison: Convert ratios to same second term or compare decimal values. To compare 2:3 and 3:5, use 10:15 vs 9:15.

**Proportion formulas:**

• Fourth proportional to a, b, c is d where a:b = c:d, so d = (b×c)/a.
• Third proportional to a, b is c where a:b = b:c, so c = b²/a.
• Mean proportional between a and b is √(a×b).

**Compound ratio formula:**

• Compound ratio of a:b, c:d, e:f is (a×c×e):(b×d×f).

**Partnership profit formula:**

• Profit share = (Individual capital × time) / (Sum of all capital×time products) × Total profit.
• If A invests ₹x for t₁ months and B invests ₹y for t₂ months, profit ratio = (x×t₁):(y×t₂).

**Mixture ratio formula:**

• When mixing two quantities at prices p and q to get average price m, ratio = (m−q):(p−m).
• This is the alligation rule derived from weighted average.

**Proportion test:**

• Four numbers a, b, c, d are in proportion if a×d = b×c (product of extremes = product of means).

Worked Examples

**Example 1: Basic ratio division** Divide ₹850 among A, B, C in the ratio 5:8:7.

*Solution:* Total parts = 5 + 8 + 7 = 20 A's share = (5/20) × 850 = 5 × 42.5 = ₹212.50 B's share = (8/20) × 850 = 8 × 42.5 = ₹340 C's share = (7/20) × 850 = 7 × 42.5 = ₹297.50 Check: 212.50 + 340 + 297.50 = ₹850 ✓

**Example 2: Partnership with time variation** A starts a business with ₹5000. After 4 months, B joins with ₹6000. At year-end profit is ₹9200. Find each partner's share.

*Solution:* A's capital×time = 5000 × 12 = 60,000 B's capital×time = 6000 × 8 = 48,000 Profit ratio = 60,000:48,000 = 5:4 Total parts = 5 + 4 = 9 A's share = (5/9) × 9200 = ₹5111.11 B's share = (4/9) × 9200 = ₹4088.89

**Example 3: Fourth proportional** Find the fourth proportional to 3, 7, and 9.

*Solution:* Let fourth proportional be x. Then 3:7 = 9:x Using a×d = b×c: 3×x = 7×9 3x = 63 x = 21 The fourth proportional is 21.

**Example 4: Inverse proportion** 15 workers complete a task in 20 days. How many days will 25 workers take?

*Solution:* More workers mean less time (inverse proportion). Workers₁ × Days₁ = Workers₂ × Days₂ 15 × 20 = 25 × Days₂ 300 = 25 × Days₂ Days₂ = 300/25 = 12 days

Common Mistakes

**Mistake 1: Adding ratios incorrectly** Wrong: "If ratio is 2:3, then adding 5 to both gives new ratio 7:8." Correct: Ratios represent multiplicative relationships, not additive. Adding the same number to both terms changes the ratio unpredictably. To increase a ratio proportionally, multiply both terms by the same factor.

**Mistake 2: Confusing direct and inverse proportion** Wrong: "More workers means more time" or solving inverse proportion problems with x₁/y₁ = x₂/y₂. Correct: Identify the relationship first. If both increase together, use direct proportion (division). If one increases as other decreases, use inverse proportion (multiplication). Workers and time are inversely proportional.

**Mistake 3: Forgetting to account for time in partnership** Wrong: Three partners invest ₹2000, ₹3000, ₹4000 for different months but dividing profit as 2:3:4. Correct: Multiply each capital by the number of months invested. If A invests for 12 months, B for 8, C for 6, ratio is (2000×12):(3000×8):(4000×6) = 24000:24000:24000 = 1:1:1, not 2:3:4.

**Mistake 4: Not simplifying ratios before calculation** Wrong: Working with ratio 36:48:60 directly in division problems leads to calculation errors. Correct: Always simplify first by dividing by HCF. 36:48:60 = 3:4:5 after dividing by 12. Simpler numbers reduce arithmetic mistakes and save time.

**Mistake 5: Cross-multiplying incorrectly in continued ratios** Wrong: If a:b = 2:3 and b:c = 4:5, concluding a:b:c = 2:3:4:5. Correct: Make the common term (b) equal in both ratios. Multiply first ratio by 4 and second by 3: a:b = 8:12 and b:c = 12:15, so a:b:c = 8:12:15.

Quick Reference

• **Ratio a:b in fraction form** = a/b. Always simplify by dividing by HCF.
• **Four numbers in proportion** when a×d = b×c (extremes product = means product).
• **Compound ratio of multiple ratios** = multiply all first terms : multiply all second terms.
• **Partnership profit ratio** = (Capital₁ × Time₁):(Capital₂ × Time₂):...
• **Direct proportion:** Both quantities move together. Use x₁/y₁ = x₂/y₂.
• **Inverse proportion:** Quantities move opposite. Use x₁×y₁ = x₂×y₂.
• **To divide amount in ratio a:b:c:** Each share = (ratio part / sum of parts) × total amount.
• **Mean proportional between a and b** = √(a×b); useful in geometry problems.