A man invests 50\% of the amount invested by B. B withdraws the whole amount from the business after 4 months. C joins the business with the investment of $X$ Rs in a month after B had withdrawn from the business. At the end of the year, A and C share the same amount of profit.

If the investment of B is Rs 2400, then which of the following may be the investment of C?

(i) 1800

(ii) 3600

(iii) 2400

(iv) 7200

(v) 5400

Question

A man invests 50\% of the amount invested by B. B withdraws the whole amount from the business after 4 months. C joins the business with the investment of $X$ Rs in a month after B had withdrawn from the business. At the end of the year, A and C share the same amount of profit.

If the investment of B is Rs 2400, then which of the following may be the investment of C?

(i) 1800

(ii) 3600

(iii) 2400

(iv) 7200

(v) 5400

Accepted Answer

**Solution:**

1. Given:
   - B invests Rs 2400.
   - A invests 50\% of B's investment, so A invests Rs $1200$.
   - B withdraws after 4 months.
   - C joins after B withdraws and invests Rs $X$ for $t$ months.
   - Total duration is 12 months.

2. Let C invests Rs $X$ for $t$ months.

3. Profit share is proportional to the product of investment and time.

4. Calculate the investment-time products:
   - A's investment-time = $1200 \times 12 = 14400$
   - B's investment-time = $2400 \times 4 = 9600$
   - C's investment-time = $X \times t$

5. Since C joins after B withdraws, the time C invests is $t = 12 - 4 = 8$ months.

6. A and C share the same profit, so:
   $$1200 \times 12 = X \times 8$$
   $$14400 = 8X$$
   $$X = \frac{14400}{8} = 1800$$

7. However, the problem states that $t$ can vary from 1 month to 8 months (since C joins anytime after B withdraws).

8. For different values of $t$, calculate $X$:
   - For $t = 8$: $X = \frac{14400}{8} = 1800$
   - For $t = 6$: $X = \frac{14400}{6} = 2400$
   - For $t = 4$: $X = \frac{14400}{4} = 3600$
   - For $t = 2$: $X = \frac{14400}{2} = 7200$

9. So possible values of $X$ are 1800, 2400, 3600, and 7200.

10. From the options given, these correspond to (i) 1800, (ii) 3600, (iii) 2400, and (iv) 7200.

**Answer:** (d) (i), (ii), (iii) and (iv)

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