Mohit and Anshu invest some money in a business in the ratio 8 : 15 respectively. After 9 months Mohit withdraws his capital. If the profits are finally received in the ratio 4 : 5 (Mohit : Anshu), for how many months did Anshu invest in the business?

Introduction / Context:
This question deals with a partnership where one partner withdraws his capital after a fixed time while the other partner keeps his investment for a different period. We know the ratio of their investments and the final profit ratio. Our goal is to determine the time period for which Anshu kept his capital invested, using the concept of capital time in partnership problems.

Given Data / Assumptions:

• Mohit and Anshu invest in the ratio 8 : 15.
• Mohit withdraws his capital after 9 months.
• Profits are received in the ratio 4 : 5 (Mohit : Anshu).
• Profit is proportional to capital multiplied by time.
• We are asked to find the number of months Anshu's capital remained invested.

Concept / Approach:
Let Mohit's and Anshu's capitals be 8 units and 15 units respectively. Mohit's investment is for 9 months, while Anshu's is for t months. Profit shares will then be in the ratio 8 * 9 : 15 * t. This ratio is given to be 4 : 5. By equating these two ratios, we can solve for t, the time period for which Anshu invested his money.

Step-by-Step Solution:
Step 1: Assume Mohit's capital is 8 units and Anshu's capital is 15 units.Step 2: Mohit keeps his capital for 9 months, so his capital time is 8 * 9 = 72 unit months.Step 3: Let Anshu's time period be t months, so his capital time is 15 * t unit months.Step 4: Profit sharing ratio Mohit : Anshu = 72 : 15t.Step 5: Given profit ratio is 4 : 5, so 72 : 15t = 4 : 5.Step 6: Write as 72 / (15t) = 4 / 5.Step 7: Cross multiply: 72 * 5 = 4 * 15t.Step 8: This gives 360 = 60t.Step 9: Solve for t: t = 360 / 60 = 6 months.

Verification / Alternative check:
If Anshu invests for 6 months, his capital time is 15 * 6 = 90 unit months. Mohit's capital time is 72 unit months as calculated. The profit ratio Mohit : Anshu is then 72 : 90, which simplifies to 4 : 5 when both numbers are divided by 18. This exactly matches the given profit ratio, confirming that t = 6 months is correct.

Why Other Options Are Wrong:
If Anshu invested for 4 or 5 months, his capital time would be 15 * 4 = 60 or 15 * 5 = 75. Ratios 72 : 60 and 72 : 75 simplify to 6 : 5 and 24 : 25 respectively, not 4 : 5. If he invested for 8 months, the ratio would be 72 : 120 = 3 : 5, which is again inconsistent with the given 4 : 5. 'None of these' is not correct because 6 is one of the options and satisfies all conditions.

Common Pitfalls:
Some candidates directly compare the investment ratio 8 : 15 with the profit ratio 4 : 5 without accounting for time. Others may incorrectly assume that the time period for Anshu must be more than 9 months because Mohit withdrew, without focusing on the given profit ratio. Always use the formula profit share ratio = capital1 * time1 : capital2 * time2 and solve the resulting equation accurately.

Final Answer:
Anshu kept his capital invested in the business for 6 months.