A and B started a business with investments of Rs. $5X$ and Rs. $5X - 400$ respectively. After six months, B added Rs. $200$ to his investment. At the end of the year, B's share of the profit is Rs. $510$ out of a total profit of Rs. $1110$. Find the value of $X$.

Question

Accepted Answer

**Solution:**

1. Let the investments of A and B be Rs. $5X$ and Rs. $5X - 400$ respectively.

2. A invests Rs. $5X$ for the entire 12 months.

3. B invests Rs. $5X - 400$ for the first 6 months, then adds Rs. 200, making his investment Rs. $(5X - 400 + 200) = 5X - 200$ for the next 6 months.

4. Calculate the effective investment (capital × time) for both:
   - A's investment: $5X 	imes 12 = 60X$
   - B's investment: $(5X - 400) 	imes 6 + (5X - 200) 	imes 6 = 6 	imes (5X - 400 + 5X - 200) = 6 	imes (10X - 600) = 60X - 3600$

5. The profit sharing ratio of A to B is:
   $$60X : (60X - 3600)$$

6. Given B's profit is Rs. 510 out of total Rs. 1110, so the ratio of A's to B's profit is:
   $$rac{1110 - 510}{510} = rac{600}{510} = rac{20}{17}$$

7. Equate the investment ratio to profit ratio:
   $$\frac{60X}{60X - 3600} = \frac{20}{17}$$

8. Cross-multiplied:
   $$17 	imes 60X = 20 	imes (60X - 3600)$$
   $$1020X = 1200X - 72000$$

9. Rearranged:
   $$1200X - 1020X = 72000$$
   $$180X = 72000$$

10. Solving for $X$:
    $$X = \frac{72000}{180} = 400$$

**Answer:** Rs. $400$

Answer

Answer

Answer

Answer