11. Partnership problems

 11. Partnership problems

Home page of Arithmetic Problems 

Below are 10 partnership problems, each with a step-by-step solution procedure and detailed explanation. Partnership problems in arithmetic typically involve dividing profits (or losses) among partners based on their investments, time periods, or work contributions. The key concept is that profit is proportional to the capital contribution × time invested (often called the investment ratio). The standard approach is:
  • Investment ratio = (Capital of partner 1 × Time) : (Capital of partner 2 × Time) : ...
  • Profit share = Total profit × (Partner’s investment / Total investment)
Problems may involve equal or different time periods, additional investments, or withdrawals. Units are typically currency (e.g., dollars, rupees) for capital and profit, and months or years for time. Each problem covers different scenarios to provide a comprehensive understanding.
Problem 1: Basic Profit Sharing
A and B invest $6000 and $8000 in a business. They agree to share profits in the ratio of their investments. If the total profit is $7000, find each partner’s share.
Step-by-Step Solution:
  1. Identify given values:
    • A’s investment = $6000
    • B’s investment = $8000
    • Total profit = $7000
  2. Calculate investment ratio: A : B = 6000 : 8000 = 3 : 4 (simplified by dividing by 2000)
  3. Calculate total parts: 3 + 4 = 7 parts
  4. Calculate profit per part: $7000 / 7 = $1000
  5. Calculate each share:
    • A’s share = 3 × 1000 = $3000
    • B’s share = 4 × 1000 = $4000
  6. State result: A gets $3000, B gets $4000.
Detailed Explanation: Profit is divided proportionally to investments since time is not specified (assumed equal). The ratio of capitals (3:4) determines how the profit is split. Each part of the ratio represents a fraction of the total profit, calculated by dividing the profit by the sum of ratio parts. Verification: $3000 + $4000 = $7000.
Answer: A: $3000, B: $4000
Problem 2: Different Time Periods
A invests $5000 for 12 months, and B invests $7000 for 8 months. They share profits in the ratio of their investments. If the profit is $3600, find each share.
Step-by-Step Solution:
  1. Identify given values:
    • A: $5000, 12 months
    • B: $7000, 8 months
    • Total profit = $3600
  2. Calculate investment contributions:
    • A’s contribution = 5000 × 12 = 60,000
    • B’s contribution = 7000 × 8 = 56,000
  3. Calculate ratio: A : B = 60,000 : 56,000 = 15 : 14 (simplified by dividing by 4000)
  4. Calculate total parts: 15 + 14 = 29 parts
  5. Calculate profit per part: $3600 / 29 ≈ $124.14
  6. Calculate each share:
    • A’s share = 15 × 124.14 ≈ $1862.10
    • B’s share = 14 × 124.14 ≈ $1737.90
  7. State result: A gets ≈ $1862.10, B gets ≈ $1737.90.
Detailed Explanation: Profit is proportional to capital × time. Multiplying each partner’s capital by their time gives their contribution. The ratio (15:14) accounts for A’s longer investment period, giving A a slightly larger share despite B’s larger capital. Verification: $1862.10 + $1737.90 ≈ $3600 (minor rounding).
Answer: A: ≈ $1862.10, B: ≈ $1737.90
Problem 3: Three Partners
A, B, and C invest $4000, $6000, and $8000 in a business. They share profits in the ratio of their investments. If the profit is $9000, find each share.
Step-by-Step Solution:
  1. Identify given values:
    • A: $4000
    • B: $6000
    • C: $8000
    • Total profit = $9000
  2. Calculate ratio: A : B : C = 4000 : 6000 : 8000 = 2 : 3 : 4 (simplified by dividing by 2000)
  3. Calculate total parts: 2 + 3 + 4 = 9 parts
  4. Calculate profit per part: $9000 / 9 = $1000
  5. Calculate each share:
    • A’s share = 2 × 1000 = $2000
    • B’s share = 3 × 1000 = $3000
    • C’s share = 4 × 1000 = $4000
  6. State result: A gets $2000, B gets $3000, C gets $4000.
Detailed Explanation: With three partners, the process is the same: profit is divided by the investment ratio. C’s larger investment yields the highest share. The sum of shares ($2000 + $3000 + $4000 = $9000) confirms correctness. This extends the basic partnership to multiple partners.
Answer: A: $2000, B: $3000, C: $4000
Problem 4: Profit After Deduction
A and B invest $10,000 and $15,000. They share profits in the ratio of their investments after deducting $3000 for expenses from a total profit of $8000. Find each share.
Step-by-Step Solution:
  1. Identify given values:
    • A: $10,000
    • B: $15,000
    • Total profit = $8000
    • Expenses = $3000
  2. Calculate profit after expenses: $8000 - $3000 = $5000
  3. Calculate ratio: A : B = 10,000 : 15,000 = 2 : 3
  4. Calculate total parts: 2 + 3 = 5 parts
  5. Calculate profit per part: $5000 / 5 = $1000
  6. Calculate each share:
    • A’s share = 2 × 1000 = $2000
    • B’s share = 3 × 1000 = $3000
  7. State result: A gets $2000, B gets $3000.
Detailed Explanation: Expenses are deducted before profit distribution, reducing the profit to $5000. The investment ratio (2:3) determines the split. This problem introduces the concept of net profit, common in real-world partnerships where costs are subtracted first.
Answer: A: $2000, B: $3000
Problem 5: Different Start Times
A invests $12,000 for 6 months, and B joins after 2 months with $18,000 for the remaining 4 months. If the profit is $5000, find each share.
Step-by-Step Solution:
  1. Identify given values:
    • A: $12,000, 6 months
    • B: $18,000, 4 months
    • Total profit = $5000
  2. Calculate contributions:
    • A: 12,000 × 6 = 72,000
    • B: 18,000 × 4 = 72,000
  3. Calculate ratio: A : B = 72,000 : 72,000 = 1 : 1
  4. Calculate total parts: 1 + 1 = 2 parts
  5. Calculate profit per part: $5000 / 2 = $2500
  6. Calculate each share: A = $2500, B = $2500
  7. State result: A gets $2500, B gets $2500.
Detailed Explanation: B joins later, so their time is shorter, but the contributions balance out (72,000 each), leading to an equal ratio. The profit splits evenly, showing how time differences can offset capital differences. Verification: $2500 + $2500 = $5000.
Answer: A: $2500, B: $2500
Problem 6: Additional Investment
A invests $8000, and B invests $10,000. After 4 months, A invests an additional $4000. If the profit after 12 months is $6600, find each share.
Step-by-Step Solution:
  1. Identify given values:
    • A: $8000 for 12 months, +$4000 for 8 months
    • B: $10,000 for 12 months
    • Total profit = $6600
  2. Calculate contributions:
    • A: (8000 × 12) + (4000 × 8) = 96,000 + 32,000 = 128,000
    • B: 10,000 × 12 = 120,000
  3. Calculate ratio: A : B = 128,000 : 120,000 = 16 : 15
  4. Calculate total parts: 16 + 15 = 31 parts
  5. Calculate profit per part: $6600 / 31 ≈ $212.90
  6. Calculate each share:
    • A’s share = 16 × 212.90 ≈ $3406.40
    • B’s share = 15 × 212.90 ≈ $3193.50
  7. State result: A gets ≈ $3406.40, B gets ≈ $3193.50.
Detailed Explanation: A’s additional investment after 4 months increases their contribution, calculated separately for each period. The ratio reflects this, giving A a slightly larger share. The sum ($3406.40 + $3193.50 ≈ $6600) confirms accuracy, accounting for rounding.
Answer: A: ≈ $3406.40, B: ≈ $3193.50
Problem 7: Withdrawal of Capital
A and B invest $20,000 and $25,000 for 12 months. After 6 months, A withdraws $5000. If the profit is $7200, find each share.
Step-by-Step Solution:
  1. Identify given values:
    • A: $20,000 for 6 months, $15,000 for 6 months
    • B: $25,000 for 12 months
    • Total profit = $7200
  2. Calculate contributions:
    • A: (20,000 × 6) + (15,000 × 6) = 120,000 + 90,000 = 210,000
    • B: 25,000 × 12 = 300,000
  3. Calculate ratio: A : B = 210,000 : 300,000 = 7 : 10
  4. Calculate total parts: 7 + 10 = 17 parts
  5. Calculate profit per part: $7200 / 17 ≈ $423.53
  6. Calculate each share:
    • A’s share = 7 × 423.53 ≈ $2964.71
    • B’s share = 10 × 423.53 ≈ $4235.30
  7. State result: A gets ≈ $2964.70, B gets ≈ $4235.30.
Detailed Explanation: A’s withdrawal reduces their contribution for the second half, lowering their share. The contributions are calculated for each period, and the ratio determines the profit split. B’s larger share reflects their consistent investment. Verification: $2964.70 + $4235.30 ≈ $7200.
Answer: A: ≈ $2964.70, B: ≈ $4235.30
Problem 8: Loss Sharing
A and B invest $9000 and $12,000. They share losses in the ratio of their investments. If the business incurs a loss of $4200, find each partner’s share of the loss.
Step-by-Step Solution:
  1. Identify given values:
    • A: $9000
    • B: $12,000
    • Total loss = $4200
  2. Calculate ratio: A : B = 9000 : 12,000 = 3 : 4
  3. Calculate total parts: 3 + 4 = 7 parts
  4. Calculate loss per part: $4200 / 7 = $600
  5. Calculate each share:
    • A’s share = 3 × 600 = $1800
    • B’s share = 4 × 600 = $2400
  6. State result: A’s loss is $1800, B’s loss is $2400.
Detailed Explanation: Losses are shared like profits, proportional to investments. The ratio (3:4) splits the loss, with B bearing more due to the larger investment. The sum ($1800 + $2400 = $4200) confirms correctness. This shows losses follow the same logic as profits.
Answer: A: $1800, B: $2400
Problem 9: Profit with Fixed Share
A and B invest $15,000 and $20,000. They agree B gets $2000 from the profit for managing, and the rest is shared by their investments. If the profit is $6500, find each share.
Step-by-Step Solution:
  1. Identify given values:
    • A: $15,000
    • B: $20,000
    • Total profit = $6500
    • B’s fixed share = $2000
  2. Calculate remaining profit: $6500 - $2000 = $4500
  3. Calculate ratio: A : B = 15,000 : 20,000 = 3 : 4
  4. Calculate total parts: 3 + 4 = 7 parts
  5. Calculate profit per part: $4500 / 7 ≈ $642.86
  6. Calculate each share of remaining profit:
    • A’s share = 3 × 642.86 ≈ $1928.58
    • B’s share = 4 × 642.86 ≈ $2571.44
  7. Add B’s fixed share: B’s total = $2571.44 + $2000 ≈ $4571.44
  8. State result: A gets ≈ $1928.60, B gets ≈ $4571.40.
Detailed Explanation: B’s fixed share is deducted first, and the remaining profit is split by the investment ratio. B’s total includes both the ratio-based share and the fixed amount, significantly increasing their portion. Verification: $1928.60 + $4571.40 ≈ $6500.
Answer: A: ≈ $1928.60, B: ≈ $4571.40
Problem 10: Finding Investment
A and B share a profit of $4000 in the ratio 5:3. If A invested for 12 months and B for 8 months, and their capital ratio is proportional to the profit ratio, find their investments.
Step-by-Step Solution:
  1. Identify given values:
    • Profit ratio = 5 : 3
    • Total profit = $4000
    • A’s time = 12 months
    • B’s time = 8 months
  2. Calculate profit shares:
    • Total parts = 5 + 3 = 8
    • Profit per part = $4000 / 8 = $500
    • A’s profit = 5 × 500 = $2500
    • B’s profit = 3 × 500 = $1500
  3. Set up contribution ratio:
    • Let A’s capital = x, B’s capital = y
    • Contribution ratio = (x × 12) : (y × 8) = 5 : 3
    • 12x / 8y = 5/3
    • 12x = (5/3) × 8y
    • 12x = 40/3 y
    • x/y = 40/36 = 10/9
  4. Assume y = 9000 (to simplify):
    • x = (10/9) × 9000 = 10,000
  5. Verify:
    • A’s contribution = 10,000 × 12 = 120,000
    • B’s contribution = 9000 × 8 = 72,000
    • Ratio = 120,000 : 72,000 = 5 : 3 (matches profit ratio)
  6. State result: A’s investment = $10,000, B’s investment = $9000.
Detailed Explanation: The profit ratio equals the contribution ratio (capital × time). Solving the ratio equation gives the capital ratio (10:9). Choosing a value for B’s capital simplifies to whole numbers, and verification confirms the profit ratio matches. This reverses typical problems to find investments.
Answer: A: $10,000, B: $9000
Summary
These 10 partnership problems cover basic profit sharing, different time periods, multiple partners, expenses, staggered starts, additional investments, withdrawals, losses, fixed shares, and finding investments. Each uses the principle that profit is proportional to capital × time, with ratios determining shares. Step-by-step solutions ensure clarity, and detailed explanations highlight how time, additional capital, or deductions affect the outcome. Verification confirms accuracy, and the problems build a strong understanding of partnership calculations. 

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