Ratio, Proportion & Partnership

Learning Objective: Apply ratio and proportion to partnership and business problems

When two friends start a business together in Pokhara -- one investing Rs. 3,00,000 and the other Rs. 2,00,000 -- how should they divide the profits? The answer lies in ratios and proportions, which are fundamental tools for solving business partnership problems and many other real-world calculations.

Ratio

A ratio compares two or more quantities of the same kind. It shows how many times one quantity contains another.

Format: a : b (read as "a is to b")

Example: If a shop has 15 shirts and 10 trousers, the ratio of shirts to trousers is 15:10 = 3:2.

Properties of Ratios:

• Ratios have no units (they are pure numbers)
• Ratios can be simplified like fractions (12:8 = 3:2)
• The order matters (3:2 is not the same as 2:3)

Proportion

A proportion states that two ratios are equal.

Format: a : b = c : d (or a/b = c/d)

Example: If 5 kg of rice costs Rs. 500, how much does 8 kg cost? 5 : 500 = 8 : x x = (500 x 8) / 5 = Rs. 800

Types of Proportion

• Direct proportion: When one quantity increases, the other also increases (more workers = more production)
• Inverse proportion: When one quantity increases, the other decreases (more workers = less time to finish)

Business Partnership Problems

When two or more people invest in a business, profits (or losses) are divided in the ratio of their investments (assuming they invest for the same time period).

Worked Example 1: Simple Partnership

Aarav and Binod start a restaurant in Kathmandu. Aarav invests Rs. 6,00,000 and Binod invests Rs. 4,00,000. After one year, the profit is Rs. 2,50,000. How do they share the profit?

Investment ratio = 6,00,000 : 4,00,000 = 3 : 2

Aarav's share = (3/5) x 2,50,000 = Rs. 1,50,000 Binod's share = (2/5) x 2,50,000 = Rs. 1,00,000

Worked Example 2: Compound Partnership (Different Time Periods)

If partners invest for different durations, we use the capital x time ratio.

Chitra invests Rs. 5,00,000 for 12 months. Deepak invests Rs. 3,00,000 for 8 months. Total profit is Rs. 1,08,000.

Chitra's investment x time = 5,00,000 x 12 = 60,00,000 Deepak's investment x time = 3,00,000 x 8 = 24,00,000

Ratio = 60,00,000 : 24,00,000 = 5 : 2

Chitra's share = (5/7) x 1,08,000 = Rs. 77,143 Deepak's share = (2/7) x 1,08,000 = Rs. 30,857

Other Ratio Applications in Business

• Profit-sharing among family members in a Nepali family business
• Mixing problems: Mixing two types of tea (Rs. 600/kg and Rs. 400/kg) in a certain ratio
• Salary ratios: If a manager earns Rs. 80,000 and an assistant earns Rs. 40,000, their salary ratio is 2:1

Key Term: A Partnership in business divides profits or losses among partners in the ratio of their capital contributions, adjusted for the time period of investment.

Summary

• A ratio compares two quantities; a proportion states that two ratios are equal.
• In simple partnerships, profits are shared in the ratio of capital invested.
• In compound partnerships, profits are shared in the ratio of (capital x time).
• Ratios are used extensively in mixing problems, salary comparisons, and financial calculations.