Ratio, Proportion & Partnership

Ratio, Proportion & Partnership

🟢 Lite

Key Rule / Formula

Ratio a:b = a/b. If a:b = c:d, then a×d = b×c (cross-multiplication). Partnership profit/loss divided in ratio of capital × time invested.

Memory Trick

“Capital × Time = Share” — for partnerships, both money AND duration matter. If A invests for 6 months and B for 4 months, their effective contribution = Capital × Months.

1-Sentence Summary

SSC tests whether you can divide quantities in given ratios and distribute partnership profits correctly based on each partner’s time-weighted capital contribution.

Ratio — The Basics

Writing a ratio: a:b means “a to b” — divide quantity a by quantity b.

• 15:20 = 3:4 (divide both by GCD = 5)
• When comparing two ratios, they must be in the same units

Types of ratios:

Type	Example	Meaning
Simple ratio	3:4	Direct comparison of two quantities
Compound ratio	(2:3) × (4:5) = 8:15	Multiply corresponding terms
Duplicate ratio	3:4 → 9:16	Square both terms
Sub-duplicate ratio	3:4 → √3:√4 = 1.73:2	Square root of both terms
Inverse ratio	3:4 → 4:3	Reverse the terms

Dividing a quantity in a ratio: Q: Divide 144 in the ratio 3:4:5. A: Sum of parts = 3+4+5 = 12. Each part = 144/12 = 12. → Parts = 3×12 = 36, 4×12 = 48, 5×12 = 60

Proportion — The Basics

Direct proportion: When one quantity increases, the other increases at the same rate.

• A ∝ B means A/B = constant
• Example: More items = more cost (at fixed price)

Inverse proportion: When one quantity increases, the other decreases.

• A ∝ 1/B means A × B = constant
• Example: More workers = less time to complete a job

Fourth proportional: If a:b = c:x, then x = (b×c)/a Mean proportional: Mean proportional between a and b = √(a×b)

Partnership — Capital × Time

Principle: Profit (or loss) is shared in the ratio of each partner’s effective contribution Effective contribution = Capital invested × Time period

Step-by-step example: A puts ₹5,000 for 12 months; B puts ₹3,000 for 8 months A’s contribution = 5000 × 12 = 60,000 B’s contribution = 3000 × 8 = 24,000 Ratio = 60000 : 24000 = 5 : 2 Total profit = ₹14,000 → A gets (5/7) × 14000 = ₹10,000, B gets (2/7) × 14000 = ₹4,000

Important: If a partner adds more money partway through, treat each period as a separate contribution and sum them up.

Must Remember

• Scale the ratio to match total parts when dividing: if ratio is 3:5 and total is 240, find the multiplier (240/8 = 30) → parts = 90 and 150
• Compound proportion: If A:B = 2:3 and C:D = 4:5, the compound ratio A×C : B×D = 8:15
• Partnerships with salary/working partner: Sometimes a working partner gets a salary before profit distribution — calculate salary first, then divide remaining profit in capital ratio
• Simple vs compound ratio: Don’t confuse them — compound multiplies, duplicate squares

⚡ SSC Exam Tip: SSC CGL Tier 2 partnership problems often involve partners joining at different times. Always calculate time-weighted capital for each partner — never just compare raw capital amounts. The partner who keeps money invested longer gets a larger share even if they invested less initially.

🟡 Standard

Concept

A ratio is a comparison of two quantities — it tells you how many times one quantity is of another. A proportion states that two ratios are equal. The most practical use in SSC is dividing things in given ratios and scaling recipes, mixtures, or profit shares. When you have a ratio a:b and the total is known, each share = (a/(a+b)) × total.

Direct Proportion: More of A means more of B (double the recipe, double the ingredients). Inverse Proportion: More of A means less of B (more workers, less time to complete a job).

Partnership is the most common application. Partners contribute capital for different time periods. Their profit/loss sharing ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂) : … This is called the “time-weighted capital” method. If a partner adds or withdraws money mid-period, calculate each period separately.

Key Points

• If a:b = c:d, then a/b = c/d and ad = bc. This is the “cross-multiply” rule.
• To divide quantity Q in ratio a:b:c: each share = Q × (individual sum) / (total sum of ratios).
• If A:B = 2:3 and B:C = 4:5, combine by making B equal: A:B:C = 8:12:15 (LCM of B denominators 3 and 4 is 12, so 2×4:3×4:3×5 = 8:12:15).
• Partnership: Active partners (who work) may get salary + profit share — treat salary as separate from profit distribution.
• If investments are for different time periods, reduce each to a “unit time equivalent” before taking ratios.

Worked Example

Q: A and B invest ₹3,000 and ₹5,000 respectively in a business. After 8 months, A adds ₹2,000 more and B withdraws ₹1,000. If the profit after a year is ₹22,000, find A’s share. Approach: A: 3000 × 8 + 5000 × 4 = 24000 + 20000 = 44000. B: 5000 × 8 + 4000 × 4 = 40000 + 16000 = 56000. Ratio = 44000:56000 = 11:14. A’s share = 11/(11+14) × 22000 = 11/25 × 22000 = ₹9,680. Answer: ₹9,680

SSC Pattern / Tips

• Always break the time period when capital changes mid-period — calculate each segment separately.
• In ratio problems, if combining two ratios where one term is common, use LCM method to harmonise.
• For three-component ratio, the sum of individual terms = (total / sum of ratio terms) × individual ratio term.

🔴 Extended

Full Concept

Ratio Advanced Concepts: The combined ratio problem is Tier 2’s most common ratio question. When you’re given A:B and B:C and need A:C, you harmonise the shared term (B) by finding the LCM of the denominators. If A:B = 2:3 and B:C = 3:4, then A:B:C = 2:3:4 directly. But if A:B = 2:5 and B:C = 4:7, then LCM of 5 and 4 is 20: multiply first by 4 → 8:20, second by 5 → 20:35. So A:B:C = 8:20:35.

Duplicate Ratio: If a:b is given, the duplicate ratio a²:b². Sub-duplicate: √a:√b. Compound ratio: a×c : b×d (for two ratios a:b and c:d).

Partnership with Salary/Interest: When partners get a salary or fixed interest on capital before profit distribution, calculate that first. Remaining profit is then distributed in capital ratio. The twist: even if a partner gets a salary, their capital contribution still matters for the remaining profit split.

Inverse Proportion: When workers increase, time decreases proportionally (more workers = less days for same work). If 10 workers do a job in 30 days, how many workers needed to finish in 15 days? 10 × 30 = 15 × x → x = 20 workers.

Continued Proportion: a, b, c are in continued proportion if a:b = b:c = k. This means b² = ac. This concept appears in proportion problems where three quantities are in a chain.

SSC CGL Deep Analysis

• Frequency: 1 question per paper. Partnership questions with capital changes appear nearly every year.
• Difficulty: Easy to medium. Ratio manipulation is straightforward; partnership with mid-period changes requires careful work.
• Recent trend: Partnership questions now include working partners who receive salary plus profit share. Sometimes three partners with staggered investments.
• Newer patterns: Questions combining ratio with profit/loss — e.g., goods bought in ratio and sold in another ratio, find overall profit/loss.
• Total weight in Tier 2: Roughly 2% of the quant paper.

High-Scoring Strategy

1. When capital changes mid-period, always make a table: Partner | Period 1 capital × time | Period 2 capital × time | Total units.
2. For ratio word problems, identify the base unit first: “if 5 mangoes cost as much as 3 oranges” → 5M = 3O → M/O = 3/5.
3. In partnership with working partners, handle salary/interest separately from profit share — add salary to their total earnings.
4. For continued proportion problems, use b² = ac to find the missing variable.
5. If goods are bought in ratio a:b and sold in ratio c:d at same price per unit, find profit/loss by equating per-unit costs.

SSC-Level Practice

Q1: Three partners A, B, C invest in a business in the ratio 3:5:7. After 5 months, A withdraws half his capital. If the profit at the end of the year is ₹76,500, find B’s share. Answer: ≈ ₹27,080 — Working: Compute each partner’s time-weighted capital in ratio units. A holds full capital 3 for the first 5 months, then half of it (1.5) for the remaining 7 months: A = 3×5 + 1.5×7 = 15 + 10.5 = 25.5. B keeps 5 for all 12 months: B = 5×12 = 60. C keeps 7 for all 12 months: C = 7×12 = 84. Total units = 25.5 + 60 + 84 = 169.5. To keep the arithmetic clean, convert to halves: total = 339/2. B’s fraction of the profit = 60 ÷ (339/2) = 120/339 = 40/113. B’s share = (40/113) × 76,500 = 3,060,000 / 113 ≈ ₹27,080.

Q2: If a:b = 2:3 and b:c = 4:5, find a:b:c. Answer: 8:12:15 — Working: LCM of b terms: 3 and 4 → 12. A:B = 2:3 → multiply by 4 → 8:12. B:C = 4:5 → multiply by 3 → 12:15. Combine: 8:12:15.

Common Traps

• Trap 1: Mixing up the base in “A is as much more than B as C is more than D” — this means (A−B) = (C−D), not A/B = C/D.
• Trap 2: Forgetting to adjust the second ratio term when harmonising ratios — always use the LCM of the shared term’s denominator/coefficient.
• Trap 3: In partnership with mid-period withdrawal/deposit, calculating time-weighted capital but forgetting that the changed amount only applies for the REMAINING period, not the whole year.

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