Ravi and Ramu are partners in a business. Ravi contributes one sixth of the total capital for 16 months. Ramu receives two thirds of the total profit. Assuming profit is shared in proportion to capital multiplied by time, for how many months was Ramu's money used in the business?

Introduction / Context:
This question tests a deeper understanding of partnership problems where one partner's share of profit is given as a fraction of the total profit. We know the capital share and time for Ravi, and the fraction of total profit received by Ramu. Our goal is to use the proportionality of profit to capital time and find for how long Ramu's capital remained invested.

Given Data / Assumptions:

• Ravi contributes one sixth of the total capital.
• Ravi's capital remains invested for 16 months.
• Ramu receives two thirds of the total profit.
• Total capital is shared between Ravi and Ramu.
• Profit shares are proportional to capital multiplied by time.

Concept / Approach:
If the total capital is C, Ravi's capital is C/6 and Ramu's capital is 5C/6. Profit is shared in the ratio of their capital time contributions. Since Ramu receives two thirds of the total profit, the ratio of Ramu's share to Ravi's share is 2 : 1. By equating this profit ratio to the ratio of their capital time products, we can solve for the unknown time period for which Ramu's money was used.

Step-by-Step Solution:
Step 1: Let total capital be C. Then Ravi's capital is C/6 and Ramu's capital is 5C/6.Step 2: Ravi's capital time = (C/6) * 16 = (16C) / 6.Step 3: Let t be the number of months Ramu's money was used. Then Ramu's capital time = (5C/6) * t = (5Ct) / 6.Step 4: Profit ratio Ramu : Ravi = (5Ct / 6) : (16C / 6) = 5t : 16.Step 5: Ramu receives two thirds of the total profit, while Ravi receives one third, so Ramu : Ravi = 2 : 1.Step 6: Equate the ratios: 5t : 16 = 2 : 1.Step 7: From 5t / 16 = 2 / 1, cross multiply to get 5t = 32.Step 8: Solve for t: t = 32 / 5 = 6.4 months.

Verification / Alternative check:
If t = 6.4 months, Ramu's capital time is (5C/6) * 6.4 = (32C) / 6 and Ravi's is (16C) / 6. The ratio is (32C / 6) : (16C / 6) = 32 : 16 = 2 : 1. Thus Ramu receives twice the share of Ravi, that is two thirds of the total profit, which matches the given condition. This confirms that t = 6.4 months is correct.

Why Other Options Are Wrong:
Time periods of 4 months, 5 months or 8 months do not satisfy the ratio 5t : 16 = 2 : 1. For example, if t = 8, the ratio becomes 40 : 16 = 5 : 2, which corresponds to Ramu receiving five sevenths of the profit, not two thirds. Therefore those values are inconsistent with the given fraction of profit. 'None of these' is not correct because one of the options, 6.4 months, satisfies the partnership equations exactly.

Common Pitfalls:
Many students assume the time period must be an integer and try to force one of the neat integer options instead of solving the equation properly. Others confuse the fraction of total profit (two thirds) with a direct ratio of capital amounts. Always set up the profit ratio correctly and equate it with the capital time ratio to obtain the correct time period, even if the answer is a decimal.

Final Answer:
The number of months for which Ramu's money was used is 6.4 months.