Ratio, Proportion & Partnership

Ratio, Proportion & Partnership

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