Partnership

Partnership problems are really just ratio problems in a business context. The core idea is simple: profit is shared in the ratio of each partner’s effective investment (capital multiplied by time). Once we get this one concept, partnership problems become straightforward.

What Is a Partnership?

When two or more people invest money together in a business and share profits (or losses), it’s called a partnership.

• Active / Working Partner: Manages the business day-to-day. May receive a salary or commission before profit is distributed.
• Sleeping / Silent Partner: Invests money but doesn’t participate in running the business.

Types of Partnership

Simple Partnership

All partners invest for the same duration. Profit is shared in the ratio of their investments.

Example: A invests Rs 30,000 and B invests Rs 50,000. They run the business for 1 year. Profit = Rs 24,000.

Ratio = 30000 : 50000 = 3 : 5

• A’s share = 24000 × 3/8 = Rs 9,000
• B’s share = 24000 × 5/8 = Rs 15,000

Compound Partnership

Partners invest for different durations. Profit is shared in the ratio of (Capital × Time) for each partner.

In simple language, the Capital × Time product represents the “weight” of each partner’s contribution. Someone who invests Rs 10,000 for 12 months has the same weight as someone who invests Rs 20,000 for 6 months.

How Compound Partnership Works

Example: A starts a business with Rs 40,000. After 3 months, B joins with Rs 60,000. At the end of the year, the total profit is Rs 28,000. How is it divided?

• A’s investment = 40,000 × 12 = 4,80,000 (invested for full year)
• B’s investment = 60,000 × 9 = 5,40,000 (invested for 9 months)
• Ratio = 480000 : 540000 = 8 : 9

A’s share = 28000 × 8/17 = Rs 13,176.47 B’s share = 28000 × 9/17 = Rs 14,823.53

Simplification trick: We don’t need to keep the full numbers. Just work with simplified units.

• A: 40 × 12 = 480
• B: 60 × 9 = 540
• Ratio = 480:540 = 8:9

Working Partner Problems

When one partner actively manages the business, they often get a salary or commission BEFORE the profit is split.

Example: A and B start a business investing Rs 50,000 and Rs 30,000 respectively. A manages the business and receives 10% of the total profit as salary. The remaining profit is divided in the ratio of investments. If total profit is Rs 40,000, find each person’s share.

1. A’s salary = 10% of 40,000 = Rs 4,000
2. Remaining profit = 40,000 - 4,000 = Rs 36,000
3. Investment ratio = 50000 : 30000 = 5 : 3
4. A’s share from remaining = 36,000 × 5/8 = Rs 22,500
5. B’s share = 36,000 × 3/8 = Rs 13,500

Total for A = 4,000 + 22,500 = Rs 26,500 Total for B = Rs 13,500

Partners Joining and Leaving

When partners join at different times or leave mid-way, calculate Capital × Time for each partner based on when they were actually in the business.

Example: A starts with Rs 20,000. B joins after 4 months with Rs 25,000. C joins after 6 months with Rs 30,000. Profit after 12 months?

• A: 20,000 × 12 = 240,000
• B: 25,000 × 8 = 200,000
• C: 30,000 × 6 = 180,000

Ratio = 240 : 200 : 180 = 12 : 10 : 9

Changing Investment Mid-Year

Sometimes a partner increases or decreases their investment during the year. We treat each period separately.

Example: A starts with Rs 30,000. After 6 months, A adds Rs 20,000 more. B invests Rs 40,000 for the full year. Find the profit ratio.

• A: 30,000 × 6 + 50,000 × 6 = 180,000 + 300,000 = 480,000
• B: 40,000 × 12 = 480,000

Ratio = 480000 : 480000 = 1 : 1

Even though A invested less initially, adding more mid-year equalized it.

Worked Examples

Example 1: A and B partner in a business. A invests Rs 40,000 for 8 months and B invests Rs 35,000 for 10 months. If the total profit is Rs 31,500, find each person’s share.

• A’s contribution = 40 × 8 = 320
• B’s contribution = 35 × 10 = 350
• Ratio = 320 : 350 = 32 : 35

A’s share = 31,500 × 32/67 = Rs 15,074 (approximately) B’s share = 31,500 × 35/67 = Rs 16,426 (approximately)

Example 2: A, B, and C start a business. A invests 3 times as much as B, and B invests 2/3 of C’s investment. If the annual profit is Rs 66,000, find C’s share.

Let C’s investment = x. Then B = 2x/3. And A = 3 × 2x/3 = 2x. Ratio = A : B : C = 2x : 2x/3 : x = 6 : 2 : 3

C’s share = 66,000 × 3/11 = Rs 18,000

Example 3: A starts a business with Rs 10,000. B joins after 3 months with Rs 16,000. C joins after 6 months with Rs 20,000. B withdraws Rs 6,000 after 9 months. Find the profit-sharing ratio at the end of the year.

• A: 10,000 × 12 = 120,000
• B: 16,000 × 6 + 10,000 × 3 = 96,000 + 30,000 = 126,000 (B was in from month 4-12, but withdrew 6000 after month 9, so 16000 × 6 months from month 4-9, then 10000 × 3 months from month 10-12)
• C: 20,000 × 6 = 120,000

Ratio = 120 : 126 : 120 = 20 : 21 : 20

Example 4: Two partners A and B invest Rs 1,00,000 and Rs 1,50,000 respectively. A is a working partner and gets 15% of the profit as management fee. If A’s total income is Rs 30,000, what is the total profit?

Let total profit = P. A’s management fee = 0.15P Remaining = 0.85P, split in 100:150 = 2:3. A’s share of remaining = 0.85P × 2/5 = 0.34P. A’s total = 0.15P + 0.34P = 0.49P = 30,000. P = 30,000/0.49 = Rs 61,224.49 (approximately)

Example 5: A and B are partners. A receives 1/3 of the profit and B receives the rest. If A’s income is increased by Rs 600 when the profit rises from 5% to 7%, find the total capital invested.

A gets 1/3 of the profit. Increase in A’s share = 1/3 × increase in total profit. Increase in total profit = (7% - 5%) × Total capital = 2% × Total capital. A’s increase = 1/3 × 2% × Total capital = 600. Total capital = 600 × 3 × 100/2 = Rs 90,000

Common Exam Patterns

1. “Invest different amounts, same time” → Simple ratio of capitals
2. “Join at different times” → Capital × Time for each partner
3. “Working partner gets salary/commission” → Deduct salary first, then divide remainder in investment ratio
4. “Investment changes mid-year” → Split into periods, calculate C×T for each period, sum them
5. “Find investment given profit share” → Set up ratio equation, solve for unknown
6. “Difference between two partners’ shares” → Express difference in terms of total, solve

Practice Problems

Q1: A starts a business with Rs 25,000. After 4 months, B joins with Rs 30,000. At the end of the year, the profit is Rs 47,000. Find the share of each.

Q2: A, B, and C invest Rs 40,000, Rs 50,000, and Rs 60,000 respectively. A is a working partner and receives 12.5% of the profit as salary. If the total profit is Rs 56,000, find A’s total earnings.

Q3: A starts a business with Rs 4,500. After 6 months, B joins with some capital. If the profit at the end of the year is divided between A and B in the ratio 2:1, how much did B invest?

Answers:

A1: A’s contribution = 25 × 12 = 300. B’s contribution = 30 × 8 = 240. Ratio = 300:240 = 5:4. A’s share = 47,000 × 5/9 ≈ Rs 26,111. B’s share = 47,000 × 4/9 ≈ Rs 20,889.

A2: A’s salary = 12.5% of 56,000 = Rs 7,000. Remaining = Rs 49,000. Ratio = 40:50:60 = 4:5:6. A’s share from remaining = 49,000 × 4/15 ≈ Rs 13,067. A’s total = 7,000 + 13,067 = Rs 20,067 (approximately).

A3: A’s contribution = 4500 × 12 = 54,000. B’s contribution = B × 6. Ratio: 54000 : 6B = 2 : 1. So 54000/6B = 2/1 → 54000 = 12B → B = Rs 4,500.