MATHS / QUANTATIVE ABILITY: Partnership NOTES & PRACTICE QUESTIONS

What is Partnership?

A partnership is a business where two or more people invest money and share profits (or losses) based on their investments.

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Types of Partnership Questions in Exams

1. Simple Partnership (Same Time Period)

In this type, all partners invest their money for the same time period. The profit is divided in proportion to their investments.

Formula:

$$\text{Profit share of a partner} = \left( \frac{\text{Partner's Investment}}{\text{Total Investment}} \right) \times \text{Total Profit}$$

Problem:
A and B invest ₹10,000 and ₹15,000 in a business. If the total profit is ₹5,000, how much profit does A get?

Solution:

• Total investment = 10,000 + 15,000 = ₹25,000

• A’s share = $\frac{10,000}{25,000}\times 5000$

• A’s share = ₹2,000

Thus, A gets ₹2,000, and B gets ₹3,000.

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2. Time-Based Partnership (Different Time Periods)

If partners invest money for different time durations, the profit is divided based on (Investment × Time).

Formula:

$$\text{Profit share} = \left( \frac{\text{Investment} \times \text{Time}}{\text{Total (Investment × Time)}} \right) \times \text{Total Profit}$$

Example:

Problem:
A invests ₹8,000 for 6 months, and B invests ₹6,000 for 8 months. If the total profit is ₹7,000, how much profit does B get?

Solution:

• A’s effective investment = 8,000 × 6 = 48,000
• B’s effective investment = 6,000 × 8 = 48,000
• Total = 48,000 + 48,000 = 96,000

• B’s share = $\frac{48,000}{96,000}\times 7,000$

• B’s share = ₹3,500

Thus, A and B each get ₹3,500.

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3. Partner Joins Late or Leaves Early

Sometimes, a partner may join after a few months or leave before the year ends. In such cases, we calculate their profit share based on the time they stayed in the business.

Example:

Problem:
A starts a business with ₹12,000. After 4 months, B joins with ₹8,000. If the total profit after 1 year is ₹10,000, how much profit does B get?

Solution:

• A’s effective investment = 12,000 × 12 = 1,44,000
• B’s effective investment = 8,000 × 8 = 64,000
• Total = 1,44,000 + 64,000 = 2,08,000

• B’s share = $\frac{64,000}{2,08,000}\times 10,000$

• B’s share = ₹3,076.92

Thus, A gets ₹6,923.08 and B gets ₹3,076.92.

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4. Change in Investment Amount

Sometimes, a partner increases or decreases their investment after a few months.

Example:

Problem:
A invests ₹5,000. After 3 months, A adds ₹2,000 more. B invests ₹7,000 for the full year. If the total profit is ₹6,000, how much profit does A get?

Solution:

• A’s first investment = 5,000 × 3 = 15,000
• A adds ₹2000 means= 5000+2000=7000 
• So, A’s second investment = 7,000 × 9 = 63,000
• Total A’s effective investment = 15,000 + 63,000 = 78,000
• B’s effective investment = 7,000 × 12 = 84,000
• Total = 78,000 + 84,000 = 1,62,000

• A’s share = $\frac{78,000}{1,62,000}\times 6,000$

• A’s share = ₹2,888.89

Thus, A gets ₹2,888.89 and B gets ₹3,111.11.

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5. Salary-Based Partnership

If a partner takes a fixed salary in addition to the profit, we first subtract the salary from the total profit, then divide the remaining profit.

Example:

Problem:
A and B invest ₹20,000 and ₹30,000 respectively. A gets a salary of ₹500 per month. If the total profit is ₹12,000, what is B’s share?

Solution:

• A’s total salary for 12 months = 500 × 12 = ₹6,000
• Remaining profit = 12,000 - 6,000 = ₹6,000

• A:B investment ratio = 20,000:30,000 = 2:3
• B’s share = $\frac{3}{5} \times 6,000$
  
• B’s share = ₹3,600

Thus, A gets ₹2,400 from the remaining profit + ₹6,000 salary = ₹8,400, and B gets ₹3,600.

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6. Charity Deduction Before Profit Sharing

If some profit is given to charity, we first subtract that amount before distributing the remaining profit.

Example:

Problem:
A and B invest in the ratio 3:2. If 10% of the total profit of ₹5,000 is donated to charity, how much does A get?

Solution:

• Charity amount = 10% of 5,000 = ₹500
• Remaining profit = 5,000 - 500 = ₹4,500

• A’s share = $\frac{3}{5}\times 4,500$

• A’s share = ₹2,700

Thus, A gets ₹2,700, and B gets ₹1,800.

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Practice Questions for You

Try solving these and let me know your answers!

1. A, B, and C invest ₹10,000, ₹15,000, and ₹20,000. After 6 months, A withdraws ₹5,000, and C adds ₹5,000. If the total profit at the end of the year is ₹25,000, what is B’s share?

2. X and Y start a business with ₹12,000 and ₹18,000. After 4 months, Z joins with ₹24,000. If the total profit at the end of the year is ₹36,000, what is Z’s share?

3. P invests ₹8,000 for 9 months, and Q invests ₹6,000 for 12 months. If the total profit is ₹7,200, what is P’s share?

Summary of Key Formulas