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Investments & Partnerships

Introduction

In partnership problems, two or more people invest money in a business. The profit (or loss) is shared according to the ratio of investment × time.

These questions are very common in exams and test your ability to use ratios in real-life business situations.

Pattern: Investments & Partnerships

Pattern

The key rule:

Profit share ∝ (Investment × Time).

If A invests P for T months, and B invests Q for U months, then A : B = (P × T) : (Q × U).

After finding the ratio, divide the profit in that proportion.

Step-by-Step Example

Question

A and B start a business. A invests ₹10,000 for 12 months, and B invests ₹15,000 for 8 months. If the total profit is ₹9,200, find each person’s share.

Solution

  1. Step 1: Represent investment × time.

    A’s share base = 10,000 × 12 = 120,000 B’s share base = 15,000 × 8 = 120,000

  2. Step 2: Write the ratio.

    A : B = 120,000 : 120,000 = 1 : 1

  3. Step 3: Divide the profit.

    Total profit = 9,200 Each share = 9,200 ÷ 2 = ₹4,600

  4. Step 4: Final Answer.

    A’s share = ₹4,600, B’s share = ₹4,600

  5. Step 5: Quick Check.

    Ratio = 1 : 1 → Equal share → 4,600 + 4,600 = 9,200 ✅

Question

A invests ₹8,000 for 12 months and B invests ₹12,000 for 6 months. If the total profit is ₹7,200, find each share.

Solution

  1. Step 1: Multiply investment × time.

    A’s base = 8,000 × 12 = 96,000 B’s base = 12,000 × 6 = 72,000

  2. Step 2: Write the ratio.

    A : B = 96,000 : 72,000 Simplify → 4 : 3

  3. Step 3: Divide the profit.

    Total profit = 7,200 Total parts = 4 + 3 = 7 Value of 1 part = 7,200 ÷ 7 = 1,028.57 A’s share = 4 × 1,028.57 = ₹4,114.29 B’s share = 3 × 1,028.57 = ₹3,085.71

  4. Step 4: Final Answer.

    A ≈ ₹4,114.29, B ≈ ₹3,085.71

  5. Step 5: Quick Check.

    4,114.29 + 3,085.71 ≈ 7,200 ✅ Ratio check = 4 : 3 ✅

Quick Variations

Sleeping partners: Some partners invest money but do not work. Their share is still based on investment × time.

Profit % not given: Even if profit is unknown, the ratio of shares can still be found from investment × time.

Multiple partners: Extend the ratio for 3 or more partners in the same way.

Trick to Always Use

  • Step 1: Multiply investment × time for each partner.
  • Step 2: Write the ratio of these products.
  • Step 3: Divide profit according to the ratio.
  • Step 4: Always check by adding shares to match total profit.

Summary

Summary

In partnership problems:

  • Profit share ratio = Investment × Time
  • Divide profit in this ratio
  • Works for 2 or more partners
  • Always check ratio and total sum

This simple method makes partnership questions quick and reliable in exams.

Practice

(1/5)
1. A and B invest in a business in the ratio 3 : 2. If the total profit is ₹50,000, how much is A’s share?
easy
A. ₹30,000
B. ₹20,000
C. ₹25,000
D. ₹35,000

Solution

  1. Step 1: Find total ratio parts

    Ratio = 3 : 2 → Total parts = 3 + 2 = 5.

  2. Step 2: Compute A's share fraction

    A’s share = (3/5) of total profit = (3/5) × 50,000 = ₹30,000.

  3. Final Answer:

    ₹30,000 → Option A

  4. Quick Check:

    B’s share = (2/5) × 50,000 = ₹20,000 → 30,000 + 20,000 = 50,000 ✅
Hint: Divide profit in proportion to investment parts.
Common Mistakes: Forgetting to add ratio parts before dividing the total.
2. X invests ₹8,000 for 12 months, Y invests ₹6,000 for 10 months. Find the profit-sharing ratio of X : Y.
easy
A. 3 : 2
B. 8 : 5
C. 4 : 3
D. 5 : 4

Solution

  1. Step 1: Compute investment×time for each

    X’s base = 8,000 × 12 = 96,000; Y’s base = 6,000 × 10 = 60,000.

  2. Step 2: Reduce to simplest ratio

    Ratio = 96,000 : 60,000 = 96 : 60 → divide by 12 → 8 : 5.

  3. Final Answer:

    8 : 5 → Option B

  4. Quick Check:

    96/60 = 1.6 and 8/5 = 1.6 → matches ✅
Hint: Multiply investment × time for each partner, then reduce the ratio.
Common Mistakes: Using investment amounts only and ignoring duration.
3. A invests ₹15,000 for 10 months, B invests ₹12,000 for 12 months, and C invests ₹18,000 for 8 months. If profit = ₹73,000, find C’s share.
medium
A. ₹22,000
B. ₹24,000
C. ₹26,000
D. ₹20,000

Solution

  1. Step 1: Calculate investment×time for each partner

    A = 15,000 × 10 = 150,000; B = 12,000 × 12 = 144,000; C = 18,000 × 8 = 144,000.

  2. Step 2: Simplify the ratio

    Ratio = 150,000 : 144,000 : 144,000 → divide by 6,000 → 25 : 24 : 24.

  3. Step 3: Find C's share from total profit

    Total parts = 25 + 24 + 24 = 73. C’s share = (24/73) × 73,000 = ₹24,000.

  4. Final Answer:

    ₹24,000 → Option B

  5. Quick Check:

    Each part = 73,000 ÷ 73 = 1,000 → C = 24 × 1,000 = 24,000 ✅
Hint: Compute (investment×time) for each, form ratio, then multiply fraction of profit.
Common Mistakes: Not simplifying the investment×time products before computing parts.
4. A invests ₹12,000 and B invests ₹15,000. After 6 months, C joins with ₹18,000. If total profit after 1 year is ₹26,000, find C’s share.
medium
A. ₹6,000
B. ₹7,200
C. ₹6,500
D. ₹9,000

Solution

  1. Step 1: Compute investment×duration for each partner (months participated)

    A = 12,000 × 12 = 144,000; B = 15,000 × 12 = 180,000; C = 18,000 × 6 = 108,000.

  2. Step 2: Reduce the ratio

    Ratio = 144 : 180 : 108 → divide by 12 → 12 : 15 : 9 → divide by 3 → 4 : 5 : 3.

  3. Step 3: Compute C's share from profit

    Total parts = 4 + 5 + 3 = 12. C’s share = (3/12) × 26,000 = (1/4) × 26,000 = ₹6,500.

  4. Final Answer:

    ₹6,500 → Option C

  5. Quick Check:

    One part = 26,000 ÷ 12 = 2,166.67 → C = 3 × 2,166.67 = 6,500 ✅
Hint: Late joiners count only months they participated; use investment×time for each partner.
Common Mistakes: Treating late-join investment as if it lasted whole year.
5. A, B and C invest in a business. A invests ₹10,000, B invests double of A, and C invests half of B. If profit is ₹45,000, how much does B get?
medium
A. ₹15,000
B. ₹18,000
C. ₹20,000
D. ₹22,500

Solution

  1. Step 1: Translate relative investments into numbers

    A = 10,000; B = double A = 20,000; C = half of B = 10,000.

  2. Step 2: Form the ratio

    Ratio = 10,000 : 20,000 : 10,000 = 1 : 2 : 1.

  3. Step 3: Compute B's share from profit

    Total parts = 1 + 2 + 1 = 4. One part = 45,000 ÷ 4 = 11,250. B’s share = 2 × 11,250 = ₹22,500.

  4. Final Answer:

    ₹22,500 → Option D

  5. Quick Check:

    A = 11,250, B = 22,500, C = 11,250 → sum = 45,000 ✅
Hint: Convert 'double'/'half' into numeric multiples, then split profit by total parts.
Common Mistakes: Forgetting to convert relative investment descriptions into exact numbers before forming the ratio.

Mock Test

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