Partnership with Change in Investment: Formula, Rules and Examples

Partnership with Change in Investment determines each partner’s profit or loss share when capital changes during the partnership period. The share depends on investment multiplied by the time for which it remains invested. This page explains the weighted investment formula, additions, withdrawals, multiple changes and solved numerical examples.

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What is Partnership with Change in Investment?

In a partnership with change in investment, profit or loss is divided in the ratio of each partner’s capital-time product. Capital-time product equals the amount invested multiplied by the number of months it remains invested.

If a partner invests different amounts during different periods, calculate a separate product for each period and add them. For a total period of T months, if capital changes from C₁ to C₂ after t months, the weighted investment is C₁ × t + C₂ × (T − t). The final profit-sharing ratio is the ratio of all partners’ weighted investments.

Partnership with Change in Investment Video Lessons

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Partnership Investment Withdrawal

Understand how to calculate partnership shares when a partner withdraws part of the investment after a specified period, using investment, time, and profit-sharing relationships.

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Partnership with Change in Investment Formula & Tricks

Important Formulas

Basic profit-sharing ratio
Profit share ratio = Investment × Time : Investment × Time

When investments remain unchanged, multiply each partner’s capital by the period for which it is invested.

Weighted investment after one change
Weighted investment = C₁ × t + C₂ × (T − t)

C₁ is the initial capital, C₂ is the changed capital, t is the number of months at C₁, and T is the total partnership period.

Weighted investment after multiple changes
Weighted investment = C₁t₁ + C₂t₂ + C₃t₃ + …

Multiply every capital amount by its corresponding duration and add the products.

Individual profit or loss
Individual share = Total profit or loss × Individual weighted investment ÷ Total weighted investment

Use the weighted investment ratio to calculate the exact amount received or borne by a partner.

Quick Tricks

Use capital-month units

Convert every partner’s investment into capital-month units before forming the ratio. This avoids repeatedly writing the time factor.

Example: ₹60,000 for 12 months gives 7,20,000 capital-month units. ₹80,000 for 4 months and ₹1,20,000 for 8 months gives 3,20,000 + 9,60,000 = 12,80,000 units.
Treat additions and withdrawals as separate periods

For an addition, use the higher capital only from the date of addition. For a withdrawal, use the lower capital from the date of withdrawal.

Example: If ₹50,000 becomes ₹30,000 after 5 months in a 12-month partnership, weighted investment is ₹50,000 × 5 + ₹30,000 × 7 = ₹4,60,000.
Cancel common factors in the final ratio

After calculating capital-month products, divide all terms by their common factor before finding profit shares.

Example: A:B = 7,20,000:12,80,000 becomes 9:16 after division by 80,000.

Partnership with Change in Investment Concepts

Profit sharing when one partner changes capital

Multiply each investment amount by the time it remains invested, then add the products for that partner.

Suppose a partner invests C₁ for t months and C₂ for the remaining T − t months. The partner’s weighted investment is C₁t + C₂(T − t). Compare this value with the weighted investments of the other partners.

Example: A invests ₹60,000 for 12 months. B invests ₹80,000 for 4 months and ₹1,20,000 for 8 months. A’s weighted investment is 7,20,000, while B’s is 12,80,000. Therefore, A:B = 9:16.

Addition of capital during the partnership

Capital added later contributes only for the months after the addition.

Do not multiply the added amount by the full partnership period. If initial capital is C and an additional amount D is invested after t months in a T-month partnership, the weighted investment is C × T + D × (T − t).

Example: A invests ₹40,000 for the full 12 months and adds ₹20,000 after 6 months. A’s weighted investment is ₹40,000 × 12 + ₹20,000 × 6 = ₹6,00,000.

Withdrawal or reduction of capital

After a withdrawal, use the reduced capital only for the remaining period.

If capital changes from C₁ to C₂ after t months, the weighted investment is C₁t + C₂(T − t). This rule applies whether C₂ is greater than or less than C₁.

Example: A invests ₹70,000 for 3 months and withdraws ₹20,000, leaving ₹50,000 for the next 9 months. Weighted investment = ₹70,000 × 3 + ₹50,000 × 9 = ₹6,60,000.

Multiple changes in investment

Divide the partnership period into separate intervals and calculate capital-time products for every interval.

For capital amounts C₁, C₂ and C₃ held for t₁, t₂ and t₃ months, the total weighted investment is C₁t₁ + C₂t₂ + C₃t₃. The interval durations must add up to the total partnership period.

Example: A invests ₹30,000 for 4 months, ₹50,000 for 5 months and ₹40,000 for 3 months. Weighted investment = 30,000 × 4 + 50,000 × 5 + 40,000 × 3 = ₹5,50,000.

Finding each partner’s profit share

Multiply the total profit by a partner’s weighted investment divided by the sum of all weighted investments.

First form the weighted investment ratio. If the ratio is a:b, the partners receive a/(a+b) and b/(a+b) of the total profit. The same ratio applies to loss unless the partnership agreement states otherwise.

Example: If A:B = 9:16 and total profit is ₹50,000, A receives ₹50,000 × 9/25 = ₹18,000 and B receives ₹50,000 × 16/25 = ₹32,000.

Partnership with Change in Investment Data Table

Use the structured data below while following the explanation and examples.

Capital-month calculation for a changing investment

Weighted investment is calculated separately for each period and then added.

Partner Investment and duration Weighted investment
A ₹60,000 × 12 months 7,20,000
B ₹80,000 × 4 months + ₹1,20,000 × 8 months 12,80,000
Total 20,00,000

Note: The profit-sharing ratio is A:B = 7,20,000:12,80,000 = 9:16.

Practice Partnership with Change in Investment Questions

Practise published questions related to this topic.

1A trader sold goods for ₹1250 at a profit of 25% . What was the cost price?→ 2Chitra gets a 25% discount on an oven whose marked price is ₹3000. She receives a further 20% discount on the reduced price for paying by credit card. How much does she pay (in ₹)?→ 3Nisha noted the number 76488 in her register. What is the remainder when it is divided by 8?→ 4A fruit seller has 24 kg apples worth ₹3,600. He sells 12 kg apples at a gain of 20% and the remaining at a loss of 10%. Find his gain or loss (in ₹).→ 5Find the smallest number that must be added to 1780 to make it a perfect square.→ 6The marked price of an article is 50 percent more than its cost price. If the profit earned is 25 percent, then what is the discount offered on the article?→ 7Shivam spends 50% of his income. If he saves ₹42,000, then his income (in ₹) is:→ 8Three persons P, Q and R started a business with their shares in the ratio 6 : 3 : 15. After 5 months Q withdrew his 80% share and R withdrew his 52% share 4 months prior to completion of the year. If total profit in the year is ₹29900 then find the share of R (in ₹) in the profit.→ 9Solve : 3 × (10 × ( 2² )) ÷ 20 + 13 - 3→ 10Evaluate the expression : 23 + 13 × 84 ÷ 6 + 15 - 102 ÷ 6 × 7→

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Quick Revision Notes

Partnership with Change in Investment: Revision Points

Use these formulas and rules to solve changing-capital partnership problems.

  • Profit or loss is divided in the ratio of investment × time.
  • For one change, use C₁ × t + C₂ × (T − t).
  • An addition contributes only from its date of addition.
  • A withdrawal uses the reduced capital for the remaining months.
  • For multiple changes, add all capital-time products.
  • Individual profit = Total profit × Individual weighted investment ÷ Total weighted investment.
  • Use the same weighted investment ratio for loss unless a different agreement is given.

Partnership with Change in Investment FAQs

What is the partnership with change in investment formula?

For a capital change from C₁ to C₂ after t months in a T-month period, weighted investment = C₁ × t + C₂ × (T − t).

How is profit divided when a partner adds capital midway?

Calculate the original capital for the full applicable period and the added capital only for the remaining period. For example, ₹40,000 for 12 months plus ₹20,000 added for 6 months gives 40,000 × 12 + 20,000 × 6 = ₹6,00,000 weighted units.

A invests ₹50,000 for 12 months. B invests ₹60,000 for 6 months and ₹80,000 for 6 months. What is the profit ratio?

A’s weighted investment is 6,00,000. B’s is 3,60,000 + 4,80,000 = 8,40,000. Therefore, A:B = 6,00,000:8,40,000 = 5:7.

How do you calculate a partner’s share from the total profit?

Use Individual share = Total profit × Individual weighted investment ÷ Total weighted investment. If the ratio is 5:7 and profit is ₹24,000, the first partner receives ₹10,000 and the second receives ₹14,000.

Does a withdrawn amount count for the entire partnership period?

No. The withdrawn amount stops contributing from the date of withdrawal. Use the original capital before withdrawal and the reduced capital after withdrawal.

What happens if a partner invests capital in three different stages?

Calculate a capital-time product for each stage and add them. For ₹20,000 for 3 months, ₹30,000 for 4 months and ₹50,000 for 5 months, the weighted investment is ₹60,000 + ₹1,20,000 + ₹2,50,000 = ₹4,30,000.

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