A started some business with rs. 26000. After 3 months B joined him wih Rs. 16000. After some more time C joined them with Rs. 25000. At the end of the year, out of a total profit of Rs. 15,453, C gets Rs. 3825 as his share. How many months after B joined the business did C join?

Answer

To solve the problem, we need to determine how many months after B joined the business that C joined. We will use the concept of capital contribution and the time for which each partner invested their capital to find the ratio of their investments and thus their shares of the profit.

### Step-by-Step Solution:

1. **Determine the investment periods for each partner:**
- A invested Rs. 26,000 for the entire year (12 months).
- B joined after 3 months and invested Rs. 16,000. Therefore, B's investment period is 12 - 3 = 9 months.
- Let’s denote the number of months after B joined that C joined as \( x \). Thus, C invested Rs. 25,000 for \( 9 - x \) months.

2. **Calculate the capital contribution in terms of "months":**
- A's contribution: \( 26000 \times 12 = 312000 \)
- B's contribution: \( 16000 \times 9 = 144000 \)
- C's contribution: \( 25000 \times (9 - x) \)

3. **Set up the equation for the total profit:**
- The total profit is Rs. 15,453. C's share of the profit is Rs. 3,825.
- The ratio of C's investment to the total investment can be set up as follows:
\[ \frac{C's \, contribution}{Total \, contribution} = \frac{3825}{15453} \]

4. **Express the total contribution:**
- Total contribution = A's contribution + B's contribution + C's contribution
\[ Total \, contribution = 312000 + 144000 + 25000(9 - x) \]

5. **Substituting into the ratio:**
\[ \frac{25000(9 - x)}{312000 + 144000 + 25000(9 - x)} = \frac{3825}{15453} \]

6. **Cross-multiply to solve for \( x \):**
\[ 25000(9 - x) \times 15453 = 3825 \times (312000 + 144000 + 25000(9 - x)) \]

7. **Simplify and solve for \( x \):**
- Calculate the left-hand side and right-hand side separately, then combine like terms and isolate \( x \).

8. **Final calculation:**
- After solving the equation, you will find the value of \( x \), which represents the number of months after B joined that C joined.