The ratio of investments of two partners A and B is 7 : 5, and the ratio of their profits is 7 : 10. If A keeps his money invested for 5 months, for how many months does B invest his money in the business?

Introduction / Context:
This partnership problem links the ratio of capital investments with the ratio of profits to deduce the time period of one partner's investment. It illustrates the fundamental principle that, in a partnership, the share of profit is proportional to capital multiplied by time. When the profit ratio does not match the simple capital ratio, it indicates different time durations of investment for the partners.

Given Data / Assumptions:

• Ratio of investments of A and B is 7 : 5.
• Ratio of profits of A and B is 7 : 10.
• A invests for 5 months.
• B invests for an unknown number of months.
• Profit shares are proportional to capital multiplied by time.

Concept / Approach:
Let the capital of A be 7x and that of B be 5x. A invests for 5 months, so his capital time product is 7x * 5. Let B invest for t months; his capital time product is 5x * t. The ratio of these capital time products must equal the ratio of profits, which is 7 : 10. From this relationship, we can form an equation involving t and solve it. This is a standard technique for questions where time and capital are both variables.

Step-by-Step Solution:
Step 1: Let A's capital be 7x and B's capital be 5x. Step 2: A invests for 5 months, so A's capital time units are 7x * 5 = 35x. Step 3: Let B invest for t months, so B's capital time units are 5x * t = 5xt. Step 4: The ratio of profits is given as 7 : 10, so 35x : 5xt = 7 : 10. Step 5: Simplify the left ratio by dividing both terms by 5x, which gives 7 : t. Step 6: Therefore, 7 : t = 7 : 10. Step 7: Comparing the terms, t must be equal to 10. Step 8: Hence B invests his money for 10 months.

Verification / Alternative check:
To verify, compute the effective contributions. For A, capital time is 7x * 5 = 35x. For B, it is 5x * 10 = 50x. The ratio 35x : 50x simplifies to 7 : 10, which matches the given profit ratio exactly. This confirms that the assumed value t = 10 months is correct and consistent with the data provided.

Why Other Options Are Wrong:
If B invested for 7 months, his capital time units would be 5x * 7 = 35x, which would give a profit ratio of 7 : 7, not 7 : 10. If B invested for 9 or 11 months, the ratios would become 35x : 45x or 35x : 55x, which simplify to 7 : 9 and 7 : 11 respectively. None of these match the given profit ratio of 7 : 10. Therefore the only correct duration among the options is 10 months.

Common Pitfalls:
A common mistake is to assume that the ratio of profit must always be the same as the ratio of capital, ignoring differences in time periods. Another error is setting up the ratio equation incorrectly, for example inverting the ratios or using 5 : 7 instead of 7 : 5. Some learners also forget to cancel the common factor x. Working symbolically with x and carefully simplifying ratios prevents these mistakes.

Final Answer:
B invests his money for 10 months, which is option D.