A invested Rs.(x+100) and B invested Rs.(x+200). After 4 months, A increased his investment by Rs.100 and after 2 more months, B increased his investment by Rs.200. If the profit share of A at the end of the year is Rs.7000 which is 2000 less than the profit share of B. Find the value of x.

To solve this problem, we need to find the value of \( x \) based on the given conditions. We will first determine the total investment and then use the profit share information to form equations that we can solve.

1. **Initial Investments and Time Periods:**
   - A invests \( Rs.(x + 100) \) initially.
   - B invests \( Rs.(x + 200) \) initially.

2. **Changes in Investments:**
   - After 4 months, A increases his investment by Rs.100. So, for the next 8 months, A's investment is \( Rs.(x + 100 + 100) = Rs.(x + 200) \).
   - After 6 months, B increases his investment by Rs.200. So, for the next 6 months, B's investment is \( Rs.(x + 200 + 200) = Rs.(x + 400) \).

3. **Calculate the Total Investment:**
   - A's total investment over 12 months:
     \[
     \text{For first 4 months:} \quad 4 \times (x + 100)
     \]
     \[
     \text{For next 8 months:} \quad 8 \times (x + 200)
     \]
     \[
     \text{Total for A:} \quad 4(x + 100) + 8(x + 200) = 4x + 400 + 8x + 1600 = 12x + 2000
     \]

   - B's total investment over 12 months:
     \[
     \text{For first 6 months:} \quad 6 \times (x + 200)
     \]
     \[
     \text{For next 6 months:} \quad 6 \times (x + 400)
     \]
     \[
     \text{Total for B:} \quad 6(x + 200) + 6(x + 400) = 6x + 1200 + 6x + 2400 = 12x + 3600
     \]

4. **Profit Shares:**
   - A's profit share is Rs.7000, and B's profit share is Rs.9000 (since B's share is Rs.2000 more than A's).

5. **Setting up the ratio:**
   \[
   \frac{\text{A's total investment}}{\text{B's total investment}} = \frac{\text{A's profit share}}{\text{B's profit share}}
   \]
   \[
   \frac{12x + 2000}{12x + 3600} = \frac{7000}{9000}
   \]
   Simplifying the ratio on the right-hand side:
   \[
   \frac{7000}{9000} = \frac{7}{9}
   \]

6. **Form the equation:**
   \[
   \frac{12x + 2000}{12x + 3600} = \frac{7}{9}
   \]

7. **Cross-multiply to solve for \( x \):**
   \[
   9(12x + 2000) = 7(12x + 3600)
   \]
   \[
   108x + 18000 = 84x + 25200
   \]
   \[
   108x - 84x = 25200 - 18000
   \]
   \[
   24x = 7200
   \]
   \[
   x = \frac{7200}{24}
   \]
   \[
   x = 300
   \]

Thus, the value of \( x \) is \( 300 \).

Would you like more details on any part of the solution or have any other questions?

---

1. How would the solution change if A increased his investment after 6 months instead of 4 months?
2. What is the total profit for A and B combined?
3. How would you set up the equation if B increased his investment by Rs.300 instead of Rs.200 after 6 months?
4. Can you derive a formula to find \( x \) if the profit shares were different?
5. What is the total amount invested by A and B over the year?
6. How would the problem change if both A and B increased their investments at the same time?
7. What if A and B had different profit ratios, say 5:6?
8. How would you verify the correctness of the calculated \( x \)?

**Tip:** Always cross-check the ratios to ensure the consistency of profit shares with the total investments.

Learn how to solve investment problems involving variable equations. This problem involves calculating investment changes and profit shares between A and B, with detailed step-by-step calculations. Suitable for students in Grades 10-12.

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