Ratio and Proportion — SSC GD Study Notes

Overview

Ratio and Proportion is a high-yield topic in SSC GD Elementary Mathematics. It appears directly in 2–3 questions per paper and forms the foundation for Profit-Loss, Time-Work, Mixture-Alligation and Partnership problems. Candidates often lose marks here not because the concepts are difficult, but because they rush through ratio simplification or skip unit checks.

Mastering this topic means understanding three core ideas: what ratios represent (comparison between quantities), how proportions work (equality of two ratios), and how compound ratios and partnership splits extend basic ratio logic. SSC GD expects you to solve these problems in under a minute, so fluency with the standard methods—unitary method, cross-multiplication, and ratio-share formula—is essential.

For SSC GD, problems are straightforward if you know the formula and practice mental arithmetic. The exam does not ask complex multi-step ratio chains; instead, it tests whether you can quickly spot the ratio relationship, set up the equation correctly, and calculate the answer without arithmetic mistakes.

Key Concepts

• **Ratio** is the comparison of two quantities of the same kind, written as a:b or a/b. A ratio of 3:5 means the first quantity is 3 parts and the second is 5 parts out of a total of 8 parts.

• **Proportion** states that two ratios are equal: a:b = c:d or a/b = c/d. The cross-multiplication property ad = bc is the workhorse for solving proportion problems.

• **Compound ratio** is the ratio obtained by multiplying the numerators and denominators of two or more ratios. The compound ratio of a:b and c:d is (ac):(bd).

• **Continued ratio** a:b:c means a quantity is divided into parts in the ratio a, b, and c. If the total is T, then the three parts are (a/(a+b+c))×T, (b/(a+b+c))×T, and (c/(a+b+c))×T.

• **Fourth proportional** to a, b, c is d such that a:b = c:d. Solve for d = (b×c)/a.

• **Third proportional** to a and b is c such that a:b = b:c. Solve for c = b²/a.

• **Mean proportional** (geometric mean) between a and b is √(ab). It satisfies a:m = m:b.

• **Partnership** problems distribute profit or loss in the ratio of investments. If two partners invest I₁ and I₂ for times T₁ and T₂, profit is shared in the ratio (I₁×T₁):(I₂×T₂). For equal time, profit shares are simply in the ratio of investments.

Formulas / Key Facts

• **Ratio simplification**: Divide both terms by their HCF to reduce a:b to lowest terms.
• **Proportion property**: If a:b = c:d, then ad = bc (cross-multiplication).
• **Componendo-dividendo**: If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d). Rarely needed in SSC GD but useful for faster algebra.
• **Compound ratio of a:b and c:d**: (a×c):(b×d).
• **Ratio-share formula**: If a sum S is divided in ratio a:b, the two parts are (a/(a+b))×S and (b/(a+b))×S.
• **Partnership (equal time)**: Profit ratio = Investment ratio. If A invests ₹x and B invests ₹y, profit is shared x:y.
• **Partnership (different times)**: Profit ratio = (Investment × Time) ratio. If A invests ₹x for t₁ months and B invests ₹y for t₂ months, profit is shared (x×t₁):(y×t₂).
• **Fourth proportional**: For a:b = c:d, d = (b×c)/a.
• **Third proportional**: For a:b = b:c, c = b²/a.
• **Mean proportional**: Between a and b is √(ab).

Worked Examples

**Example 1: Basic ratio division** Divide ₹8400 between A and B in the ratio 3:4.

**Solution:** Total parts = 3 + 4 = 7. A's share = (3/7) × 8400 = ₹3600. B's share = (4/7) × 8400 = ₹4800. **Answer:** A gets ₹3600, B gets ₹4800.

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**Example 2: Proportion and fourth proportional** Find the fourth proportional to 6, 8, and 9.

**Solution:** Let the fourth proportional be x. Then 6:8 = 9:x ⇒ 6x = 8 × 9 = 72 ⇒ x = 72/6 = 12. **Answer:** 12.

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**Example 3: Partnership with equal time** A and B start a business with investments ₹5000 and ₹7000. After one year, they earn ₹3600 profit. Find A's share.

**Solution:** Investment ratio = 5000:7000 = 5:7. Total parts = 5 + 7 = 12. A's share = (5/12) × 3600 = ₹1500. **Answer:** ₹1500.

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**Example 4: Partnership with different times** A invests ₹6000 for 4 months and B invests ₹8000 for 6 months. They make ₹3400 profit. Find B's share.

**Solution:** Profit ratio = (6000×4):(8000×6) = 24000:48000 = 1:2. Total parts = 1 + 2 = 3. B's share = (2/3) × 3400 = ₹2266.67 (approx ₹2267). **Answer:** ₹2267.

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**Example 5: Compound ratio** If the ratio of boys to girls in two sections is 3:2 and 5:4 respectively, find the compound ratio.

**Solution:** Compound ratio = (3×5):(2×4) = 15:8. **Answer:** 15:8.

Common Mistakes

• **Forgetting to add denominators**: When dividing a sum in ratio a:b, students often compute a/b × sum instead of a/(a+b) × sum. Always add the ratio terms first to find total parts.

• **Mixing up time and investment in partnership**: If two partners invest for different durations, profit depends on (Investment × Time), not just Investment. Read the question carefully—equal time means ignore time; different times means multiply each investment by its time.

• **Cross-multiplication errors**: In a:b = c:d, the correct equation is ad = bc, not ac = bd. Write it out explicitly to avoid flipping terms.

• **Not simplifying ratios**: Leaving an answer as 4000:6000 instead of 2:3 looks unprofessional and wastes time in multiple-choice matching. Always reduce ratios to lowest terms by dividing by the HCF.

• **Unit mismatch**: If one quantity is in kg and the other in grams, convert both to the same unit before forming a ratio. A ratio compares like quantities only; mixing units gives a meaningless number.

Quick Reference

• Ratio a:b means first part is (a/(a+b)) of the total, second part is (b/(a+b)) of the total.
• Proportion: a:b = c:d ⇒ ad = bc. Use cross-multiplication to solve for the unknown term.
• Compound ratio of a:b and c:d is (ac):(bd).
• Partnership (equal time): Profit ratio = Investment ratio.
• Partnership (unequal time): Profit ratio = (Investment × Time) ratio.
• Always reduce ratios to lowest terms by dividing by the HCF of the terms.