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Quantitative Aptitude

Partnership

Linear Equations7 / 51Profit Loss & Discount
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Concepts (2)

In compound partnership, profit is shared in the ratio of equivalent capital, calculated as (Investment × Time) for each partner. Master ratios for quick calculations.

Core Formula

In a compound partnership, partners invest capital for different time periods. The profit (or loss) is shared in the ratio of their equivalent capital, which is the product of their investment and the duration for which it was invested.

If P1, P2, P3 are the profits of partners A, B, C respectively, and I1, I2, I3 are their investments for time periods T1, T2, T3 respectively (all time periods in the same unit, e.g., months or years):

P1 : P2 : P3 = (I1 × T1) : (I2 × T2) : (I3 × T3)

This is also known as the ratio of their 'equivalent capital' or 'capital-time product'.

Worked Example 1

A and B start a business. A invests ₹50,000 for 8 months, and B invests ₹60,000 for 10 months. If the total profit at the end of the year is ₹94,000, find the share of each.

Solution:

  1. Calculate Equivalent Capital Ratio:
    • A's Equivalent Capital = Investment × Time = ₹50,000 × 8 months = ₹400,000
    • B's Equivalent Capital = Investment × Time = ₹60,000 × 10 months = ₹600,000
  2. Form the Ratio:
    • Ratio of Equivalent Capital (A:B) = 400,000 : 600,000
    • Simplify the ratio: 4 : 6 = 2 : 3
  3. Calculate Shares:
    • Total ratio parts = 2 + 3 = 5
    • A's Share = (2/5) × ₹94,000 = ₹37,600
    • B's Share = (3/5) × ₹94,000 = ₹56,400

Worked Example 2

Ram, Shyam, and Mohan started a business. Ram invested ₹15,000 for 12 months. Shyam joined after 3 months with ₹20,000. Mohan joined after 6 months with ₹25,000. If the total profit at the end of the year was ₹28,000, what is Mohan's share?

Solution:

  1. Determine Investment Periods:
    • Ram's investment period = 12 months
    • Shyam joined after 3 months, so his investment period = 12 - 3 = 9 months
    • Mohan joined after 6 months, so his investment period = 12 - 6 = 6 months
  2. Calculate Equivalent Capital Ratio:
    • Ram's (I × T) = 15,000 × 12 = 180,000
    • Shyam's (I × T) = 20,000 × 9 = 180,000
    • Mohan's (I × T) = 25,000 × 6 = 150,000
  3. Form and Simplify the Ratio (Ram : Shyam : Mohan):
    • 180,000 : 180,000 : 150,000
    • Divide by 10,000: 18 : 18 : 15
    • Divide by 3: 6 : 6 : 5
  4. Calculate Mohan's Share:
    • Total ratio parts = 6 + 6 + 5 = 17
    • Mohan's Share = (5/17) × ₹28,000 = ₹8,235.29 (approx)

Shortcuts & Tricks

  1. Simplify Early: When forming the (I × T) ratio, cancel out common zeros or common factors from the investments and times as early as possible. For example, in Ex. 1, instead of (500008) : (6000010), you can directly use (58) : (610) = 40:60 = 2:3. This saves calculation time and reduces error.
  2. Focus on Relative Time: If one partner invests for 12 months, and another joins after 3 months, their relative time is 12 and 9. Don't overthink the 'year-end' part; just find the actual duration of investment.
  3. Unit Consistency: Always ensure all time periods are in the same unit (e.g., all in months, or all in years). If an investment is for 1.5 years, convert it to 18 months if other investments are in months.

Common Mistakes

  1. Incorrect Time Calculation: Students often confuse 'joined after X months' with 'invested for X months'. If a business runs for 12 months and a partner joins after 3 months, their investment is for 12-3 = 9 months, not 3 months.
  2. Ignoring Simplification: Multiplying large numbers (I × T) first and then simplifying the ratio leads to unnecessary large calculations and higher chances of error. Always simplify the ratio of (I × T) as much as possible before performing final calculations.
  3. Unit Inconsistency: Mixing months and years for time periods without conversion will lead to incorrect ratios and profit shares.

Derivation (brief)

The fundamental principle of partnership is that profit is directly proportional to the capital invested. When the capital is invested for different durations, the 'effective' or 'equivalent' capital for each partner needs to be considered. If you invest ₹100 for 12 months, it's equivalent to investing ₹1200 for 1 month. Similarly, if you invest ₹100 for 6 months, it's equivalent to ₹600 for 1 month. Thus, the profit share is proportional to the product of the capital and the time it was invested (I × T). This ensures fairness, as a larger investment for a shorter time can be equivalent to a smaller investment for a longer time.

Advanced Examples

Example: A, B, and C start a business. A invests ₹20,000 for the first 4 months, then increases his investment to ₹25,000 for the next 6 months, and finally withdraws ₹5,000 for the remaining 2 months of the year. B invests ₹30,000 for the entire year. C invests ₹10,000 for the first 6 months, then withdraws his entire capital. If the total profit is ₹54,000, find A's share.

Solution:

  1. Calculate A's Equivalent Capital:
    • (₹20,000 × 4 months) + (₹25,000 × 6 months) + (₹20,000 × 2 months)
    • = 80,000 + 150,000 + 40,000 = ₹270,000
  2. Calculate B's Equivalent Capital:
    • (₹30,000 × 12 months) = ₹360,000
  3. Calculate C's Equivalent Capital:
    • (₹10,000 × 6 months) = ₹60,000
  4. Form and Simplify the Ratio (A : B : C):
    • 270,000 : 360,000 : 60,000
    • Divide by 10,000: 27 : 36 : 6
    • Divide by 3: 9 : 12 : 2
  5. Calculate A's Share:
    • Total ratio parts = 9 + 12 + 2 = 23
    • A's Share = (9/23) × ₹54,000 = ₹21,130.43 (approx)

Variation Types

  1. Partners Joining/Leaving: As seen in examples, calculate the exact duration each partner's capital was in the business.
  2. Varying Investments: A partner might change their investment during the period (add or withdraw capital). In such cases, calculate the (Investment × Time) for each distinct phase and sum them up to get the total equivalent capital for that partner.
  3. Monthly/Regular Contributions: If partners contribute a fixed amount monthly, calculate the sum of (Investment × Time) for each month's contribution. For example, if ₹1000 is invested monthly for 6 months, the first ₹1000 is for 6 months, the second for 5 months, and so on. This forms an arithmetic progression.

Time-Saving Methods

  • Ratio of Ratios: If you have multiple partners and their investments and times are given, write down the (I × T) for each. Before multiplying, look for common factors across all terms. For instance, if investments are 20k, 30k, 40k, and times are 8, 10, 6 months, you can write (20k8) : (30k10) : (40k6). Cancel 'k' from all. Then (208) : (3010) : (406). Cancel 10 from all: (28) : (310) : (4*6) = 16 : 30 : 24. Then divide by 2: 8 : 15 : 12. This step-by-step simplification prevents large numbers and reduces calculation burden.
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Partnership profit is shared based on the ratio of (Capital × Time) invested by each partner. Understand this core concept for quick problem-solving for SSC CGL.

Core Formula

In a partnership, profit is shared among partners in proportion to the product of their invested capital and the time for which it was invested.

For two partners, A and B: Profit_A : Profit_B = (Capital_A × Time_A) : (Capital_B × Time_B)

For multiple partners, the same principle applies: Profit_1 : Profit_2 : ... : Profit_n = (C1 × T1) : (C2 × T2) : ... : (Cn × Tn)

  • If the time period of investment is the same for all partners, then the profit is shared in the ratio of their capitals: Profit Ratio = Capital Ratio.
  • If the capital invested is the same for all partners, then the profit is shared in the ratio of their time periods: Profit Ratio = Time Ratio.

Worked Example 1

A and B start a business by investing Rs. 8000 and Rs. 12000 respectively. If the total profit at the end of the year is Rs. 5000, find the share of each partner.

Solution:

  1. Identify Capital and Time:
    • Capital_A = Rs. 8000, Capital_B = Rs. 12000
    • Time_A = 1 year (12 months), Time_B = 1 year (12 months). Since time is constant, we can use the capital ratio directly.
  2. Calculate Profit Ratio:
    • Profit_A : Profit_B = Capital_A : Capital_B = 8000 : 12000
    • Simplify the ratio: 8 : 12 = 2 : 3
  3. Calculate Shares:
    • Total ratio parts = 2 + 3 = 5
    • A's share = (2/5) × 5000 = Rs. 2000
    • B's share = (3/5) × 5000 = Rs. 3000

Worked Example 2

P invests Rs. 15000 for 8 months and Q invests Rs. 10000 for 12 months. If the total profit is Rs. 13500, what is Q's share?

Solution:

  1. Identify Capital and Time:
    • Capital_P = Rs. 15000, Time_P = 8 months
    • Capital_Q = Rs. 10000, Time_Q = 12 months
  2. Calculate (Capital × Time) product for each:
    • P's equivalent investment = 15000 × 8 = 120000
    • Q's equivalent investment = 10000 × 12 = 120000
  3. Calculate Profit Ratio:
    • Profit_P : Profit_Q = 120000 : 120000
    • Simplify the ratio: 1 : 1
  4. Calculate Q's Share:
    • Total ratio parts = 1 + 1 = 2
    • Q's share = (1/2) × 13500 = Rs. 6750

Shortcuts & Tricks

  • Simplify Ratios Early: Always simplify the capital and time values before multiplying, if possible, to work with smaller numbers. For example, if C1=8000, C2=12000, T1=12, T2=12, then (800012):(1200012) can be simplified to 8:12 or 2:3 directly.
  • Consistent Units: Ensure all time periods are in the same units (e.g., all months or all years). If a partner joins/leaves, calculate their exact duration of investment.
  • Mental Math for Common Ratios: Practice quickly identifying common ratios like 1:1, 1:2, 2:3, etc., to speed up calculations.

Common Mistakes

  1. Ignoring Time: The most frequent error is assuming profit is always shared solely based on capital, forgetting to factor in the time duration for which the capital was invested.
  2. Incorrect Time Calculation: Miscalculating the duration for which a partner's capital was active, especially when partners join or leave mid-year. Always count months accurately.
  3. Arithmetic Errors: Simple calculation mistakes during multiplication or ratio simplification can lead to incorrect answers. Double-check your basic arithmetic.

Derivation (brief)

The concept of profit sharing in partnership is rooted in fairness and proportionality. Profit is essentially the reward for the resources (capital) employed over a period (time). Therefore, it's logical that someone who invests more capital or keeps their capital invested for a longer duration should receive a proportionally higher share of the profit. The product of Capital (C) and Time (T) effectively represents the 'total investment effort' or 'capital-months/years' contributed by each partner. Thus, the profit is distributed in the ratio of these C×T products.

Advanced Examples

1. Working/Sleeping Partner: A and B start a business. A invests Rs. 20000 and B invests Rs. 30000. A is an active partner and receives 10% of the total profit as salary, and the remaining profit is distributed in their capital ratio. If the total profit is Rs. 15000, find B's share.

Solution:

  • Total Profit = Rs. 15000
  • A's Salary = 10% of 15000 = Rs. 1500
  • Remaining Profit = 15000 - 1500 = Rs. 13500
  • Capital Ratio (A:B) = 20000 : 30000 = 2 : 3
  • A's share from remaining profit = (2/5) × 13500 = Rs. 5400
  • B's share from remaining profit = (3/5) × 13500 = Rs. 8100
  • A's Total Share = Salary + Share from remaining = 1500 + 5400 = Rs. 6900
  • B's Total Share = Rs. 8100

2. Partner Joins Mid-Year: X, Y, and Z start a business. X invests Rs. 40000. Y joins after 3 months with Rs. 50000. Z joins after another 2 months (i.e., 5 months from start) with Rs. 60000. If the total profit at the end of the year is Rs. 28500, find Z's share.

Solution:

  • Total time = 1 year = 12 months.
  • X's investment: C_X = 40000, T_X = 12 months. (C_X × T_X) = 40000 × 12 = 480000
  • Y's investment: C_Y = 50000, T_Y = 12 - 3 = 9 months. (C_Y × T_Y) = 50000 × 9 = 450000
  • Z's investment: C_Z = 60000, T_Z = 12 - 5 = 7 months. (C_Z × T_Z) = 60000 × 7 = 420000
  • Profit Ratio (X:Y:Z) = 480000 : 450000 : 420000
  • Simplify by dividing by 10000: 48 : 45 : 42
  • Simplify by dividing by 3: 16 : 15 : 14
  • Total ratio parts = 16 + 15 + 14 = 45
  • Z's share = (14/45) × 28500 = 14 × 633.33 (approx) = Rs. 8866.67. (For SSC, numbers are usually cleaner. Let's assume total profit was 27000 for cleaner division: Z's share = (14/45) * 27000 = 14 * 600 = Rs. 8400).

Variation Types

  1. Simple Partnership: All partners invest for the same duration. Profit is directly proportional to capital.
  2. Compound Partnership: Partners invest for different durations. Profit is proportional to (Capital × Time).
  3. Working/Sleeping Partners: One or more partners receive a fixed salary or commission before the remaining profit is distributed based on C×T ratio.
  4. Capital Change Mid-Term: A partner increases or decreases their capital during the year. In such cases, calculate the (Capital × Time) for each distinct period and sum them up for that partner's total equivalent investment.

Time-Saving Methods

  • Ratio Simplification at Each Step: Instead of multiplying large numbers and then simplifying, simplify the capital and time values first. For example, if C1=20000, C2=30000, T1=12, T2=9, then C1:C2 = 2:3 and T1:T2 = 4:3. So (C1T1):(C2T2) = (24):(33) = 8:9. This avoids 2000012 and 300009.
  • Fractional Approach: If profits are given as fractions or percentages, work with them directly. E.g., if A gets 1/4th of profit, then remaining 3/4th is shared by others.
  • Option Elimination: In MCQs, sometimes you can eliminate options based on the ratio. If A:B is 2:3, A's share must be an even number and B's share a multiple of 3, and B's share must be 1.5 times A's share.
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