Ratio, Proportion & Partnership

Ratio, Proportion & Partnership

🟢 Lite

Key Formula/Rule

Ratio a:b = a ÷ b. Proportion a:b = c:d means a/b = c/d. Partnership profit share = Capital × Time invested.

Memory Trick

A ratio is just a division problem in disguise. When you see 3:7, think “3 parts out of 7 total” or “3/7ths of the total.” For partnership, the share is proportional to capital × time — money invested for longer gets a bigger slice.

1-Sentence Summary

Tests your ability to compare quantities fairly and divide things in correct proportions — essential for profit-sharing, mixing, and age problems.

Ratio — Core Skills

Writing and simplifying ratios:

• 15:20 = divide both by GCD (5) → 3:4
• Convert to same units: 2 hours : 45 minutes = 120 min : 45 min = 8:3

Dividing a quantity in a given ratio: Q: Divide ₹500 in the ratio 3:2. A:

• Sum = 3 + 2 = 5 parts
• Each part = 500 ÷ 5 = ₹100
• Parts = 3×100 = ₹300, 2×100 = ₹200

Compound ratio: (2:3) × (4:5) = (2×4) : (3×5) = 8:15

Special ratios:

• Duplicate: 3:4 → 3²:4² = 9:16
• Sub-duplicate: 3:4 → √3:√4 = 1.73:2
• Inverse: 3:4 → 4:3 (reverse)

Proportion — Direct and Inverse

Direct proportion: a/b = c/d — when one increases, the other increases at the same rate.

• “If 5 pens cost ₹50, how much do 8 pens cost?”
• 5:50 = 8:x → x = (50×8)/5 = ₹80

Inverse proportion: a × b = constant — when one increases, the other decreases.

• “If 10 workers complete a job in 6 days, how many days for 15 workers?”
• 10 workers → 6 days. More workers = fewer days.
• 10 × 6 = 15 × x → x = (10×6)/15 = 4 days

Partnership — Capital × Time Method

Core principle: Each partner’s share = (their capital) × (time invested), divided proportionally.

Step-by-step:

Example: A puts ₹5,000 for 12 months; B puts ₹3,000 for 8 months.

• A’s effective contribution = 5000 × 12 = 60,000
• B’s effective contribution = 3000 × 8 = 24,000
• Total = 84,000 → Ratio = 60,000 : 24,000 = 5 : 2

If total profit = ₹14,000:

• A’s share = (5/7) × 14000 = ₹10,000
• B’s share = (2/7) × 14000 = ₹4,000

⚡ Key rule: Only the product of capital × time matters. A partner who invests less but for longer may get more. Also note: “month” is the usual time unit in partnership problems.

Partnership Variations to Watch

1. Partner joins partway through the year: A puts ₹4,000 for full year; B puts ₹5,000 but joins after 6 months.

• A = 4000 × 12 = 48,000
• B = 5000 × 6 = 30,000
• Ratio = 48,000 : 30,000 = 8 : 5

2. Partner adds more money: A puts ₹3,000 for 8 months, then adds ₹2,000 more for 4 months.

• First period: 3000 × 8 = 24,000
• Second period: 5000 × 4 = 20,000 (capital is now 5000, not 3000)
• Total for A = 44,000

3. Working partner gets salary + share: A manages business (salary ₹500/month) + capital ₹3,000; B is silent partner ₹5,000. Step 1: Deduct salary from total profit Step 2: Divide remaining profit in capital × time ratio

Must Remember

• Scale the ratio to match total: if ratio 3:7 and total is 200, multiplier = 200/10 = 20 → parts = 60 and 140
• When dividing a number in a ratio, always sum the ratio parts first
• “Dividing in the ratio of 2:3” means first part / second part = 2/3

⚡ CUET Exam Tip: CUET ratio questions often involve mixing (alligation) and age problems. The key skill is identifying whether the proportion is direct (same direction) or inverse (opposite direction). Practice mixing problems where you find the mean price or mean quantity — these appear frequently and are very formula-driven.

🟡 Standard

Concept

A ratio is simply a way of comparing two quantities by division. If there are 15 boys and 10 girls in a class, the ratio of boys to girls is 15:10, which simplifies to 3:2. Simplifying ratios works exactly like simplifying fractions — divide both sides by their HCF. Always give ratios in their simplest form.

Proportion is when two ratios are equal. If 3:2 = 6:4, we say they are in proportion. This is the basis of the “cross-multiplication” rule you learned: a/b = c/d means a × d = b × c. This is incredibly useful for solving problems where you need to find a missing quantity.

Now here’s where it gets practical: direct vs. inverse proportion. In direct proportion, when one thing increases, the other increases too. More items cost more money (assuming unit price is constant). In inverse proportion, when one thing increases, the other decreases. More workers finish a job faster. CUET loves asking questions about both — make sure you know which type you’re dealing with before you set up the equation.

Partnership is just ratio applied to business. When two or more people invest money in a business, they share profits according to how much they invested and for how long. The key formula is: Share ∝ Capital × Time. If A puts in ₹10,000 for a full year and B puts in ₹5,000 for the same year, their profit-sharing ratio is 10,000:5,000 = 2:1. Simple enough.

But what if B joined halfway through the year? Then A’s investment = 10,000 × 12 = 1,20,000 (rupee-months) and B’s = 5,000 × 6 = 30,000. Ratio becomes 1,20,000:30,000 = 4:1. The person who kept money in the business longer gets a proportionally bigger share — even though B’s capital was smaller.

Key Formulas

Formula	Use
a:b = c:d ⟹ a×d = b×c	Cross-multiplication in proportion
Divide N in ratio a:b → aN/(a+b), bN/(a+b)	Splitting a quantity into ratio
Direct: a/b = constant → x₁/x₂ = y₁/y₂	Direct proportion setup
Inverse: a×b = constant → x₁/x₂ = y₂/y₁	Inverse proportion setup
Partnership share = Capital × Time	Profit division in business

Worked Example

Q: A and B start a business with ₹30,000 and ₹20,000 respectively. After 8 months, A adds another ₹10,000. If they make a profit of ₹46,000 in a year, how is it divided?

Step 1: Calculate rupee-months for A and B

• A: ₹30,000 for 8 months + ₹40,000 for 4 months = (30,000 × 8) + (40,000 × 4) = 2,40,000 + 1,60,000 = 4,00,000
• B: ₹20,000 for 12 months = 20,000 × 12 = 2,40,000

Step 2: Ratio = 4,00,000 : 2,40,000 = 5 : 3

Step 3: Total parts = 8. Profit per part = 46,000 ÷ 8 = 5,750

• A’s share = 5 × 5,750 = ₹28,750
• B’s share = 3 × 5,750 = ₹17,250

Answer: A = ₹28,750, B = ₹17,250

Common Errors

• Assuming equal time when capital differs → Always compute Capital × Time as rupee-months or rupee-years before comparing shares
• Mixing up direct and inverse proportion → Direct: more of A means more of B. Inverse: more of A means less of B. Wrong setup gives completely wrong answers.
• Not simplifying the ratio before dividing profit → Always reduce a:b to simplest form (divide by HCF) before using it to split anything

🔴 Extended

Full Concept

Ratio of Three Quantities

Ratios can compare more than two things. If the ratio of flour:water:salt in a recipe is 5:3:1 and the total flour is 500g, each “unit” = 500 ÷ 5 = 100g. So water = 3 × 100 = 300g and salt = 1 × 100 = 100g. The key is: find the value of one unit first, then multiply.

For three quantities a:b:c divided in ratio a:b:c, individual shares = (a×Total)/(a+b+c), etc.

Compound Ratio

This is when you multiply two ratios together. (a:b) × (c:d) = (ac : bd). For example, (2:3) × (4:5) = (8:15). This shows up in problems about pipes filling tanks, workers doing jobs, or combinations of rates. If 4 taps of equal flow rate fill a tank in 6 hours, and we want to find how long 6 taps take, we’re really comparing rates: 4 taps × 6 hrs = total work. So 6 taps = (4 × 6) ÷ 6 = 4 hours. This is also an inverse proportion problem in disguise.

Continued Proportion

When a:b = b:c = c:d, the three quantities a, b, c, d are in continued proportion. Here, b is the mean proportional between a and c, and c is the mean proportional between b and d. The constant ratio is called the common ratio. If a, b, c are in continued proportion, then b² = ac. This property is extremely useful for checking answers and solving for unknowns.

Why Capital × Time = Share

This is fundamental to partnership and worth understanding deeply. Imagine A puts ₹100 in for 1 year and B puts ₹100 in for 1 year. Both contributed equally for equal time, so they split profits 50:50. Now if A puts ₹100 for 1 year and B puts ₹100 for only 6 months, B’s money did “half a year’s worth of work” compared to A’s, so B deserves half of A’s share. The “work” done by money is: Amount × Duration.

Think of it like this: ₹100 for 12 months = ₹1,200 “rupee-months” of contribution. If B only had ₹100 for 6 months, that’s ₹600 rupee-months. The ratio of their contributions is 1,200:600 = 2:1. The share is always proportional to the total capital-time contribution.

Age Ratio Problems

These are a CUET favourite. The trick is that the ratio between two people’s ages stays constant — if A is twice as old as B today, A will always be twice as old (the gap stays the same). But the difference between their ages is what stays fixed in absolute terms. So if A is 30 and B is 15 (ratio 2:1), after 10 years A is 40 and B is 25 — still ratio 8:5, not 2:1.

Watch out for problems like: “4 years ago, the ratio of ages was 3:1. 4 years hence, it will be 5:2. Find their present ages.” Set up two equations using the given ratios and solve simultaneously.

Multiple Approaches

Standard — Algebraic Setup: For ratio problems, assign variables to each part: if a:b = 3:5 and a + b = 80, then a = 3k, b = 5k, so 8k = 80 → k = 10 → a = 30, b = 50.

Shortcut — Proportion Cross-Multiplication: When given a:b = c:d and three values, cross-multiply to find the fourth: a/b = c/d ⟹ d = (b×c)/a.

Shortcut — Partnership Share: Share = (Individual’s Capital × Time) ÷ (Sum of all Capital × Time contributions) × Total Profit.

CUET-Level Problems

Q1: If a:b = 2:3, b:c = 4:5, find a:b:c. Working: Make b the same in both ratios. LCM of 3 and 4 = 12. a:b = 2:3 = 8:12 b:c = 4:5 = 12:15 So a:b:c = 8 : 12 : 15 Answer: 8 : 12 : 15

Q2: A, B, and C started a business. A invested ₹25,000 for 2 years. B invested ₹30,000 for the first year and then withdrew ₹10,000. C invested ₹20,000 for the entire period. If the profit after 2 years is ₹1,40,000, find each person’s share. Working:

• A: 25,000 × 24 = 6,00,000 rupee-months
• B: 30,000 × 12 + 20,000 × 12 = 3,60,000 + 2,40,000 = 6,00,000 rupee-months
• C: 20,000 × 24 = 4,80,000 rupee-months
• Total = 6,00,000 + 6,00,000 + 4,80,000 = 16,80,000
• Ratio = 6,00,000 : 6,00,000 : 4,80,000 = 6 : 6 : 4.8 = 5 : 5 : 4 (multiplying by 10, dividing by 12,000)
• Parts = 14. Profit per part = 1,40,000 ÷ 14 = 10,000
• A = 5 × 10,000 = ₹50,000
• B = 5 × 10,000 = ₹50,000
• C = 4 × 10,000 = ₹40,000 Answer: A = ₹50,000, B = ₹50,000, C = ₹40,000

Tricky Cases

• Ratio changing over time in age problems — The absolute difference stays the same (A − B is constant), but the ratio A/B changes. Don’t confuse the two.
• Compound ratio with zero terms — (0:5) × (3:4) = (0:20) — a zero in any term makes that ratio’s product term zero. Sometimes used as a trick.
• Continued proportion with more than 3 terms — If a:b = b:c = c:d, then a, b, c, d are in continued proportion. b³ = a×c²? No — actually a/b = b/c = c/d = r, so a = ar³, b = ar², c = ar, d = a/r. The middle terms are geometric means of their neighbours.
• “Working partner” vs “Sleeping partner” — A working partner may get a salary on top of their profit share. In such problems, first deduct the salary from total profit, then divide the remaining in the capital ratio.

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