A started some business with Rs. 26000. After 3 months B joined him with Rs. 16,000. After some more time C joined them with Rs. 25,000. At the end of the year, out of a total profit of Rs. 15,453, C gets Rs 3825 as his share. How many month after B joined the business. did C join?

Answer

To solve the problem, let's break it down step by step.

### Step 1: Understand the Investment Periods
- A starts the business with Rs. 26,000 for the entire year (12 months).
- B joins after 3 months with Rs. 16,000, so B's investment period is 12 - 3 = 9 months.
- C joins after B, and we need to find out how many months after B joined C joined. Let’s denote the number of months after B joined as \( x \). Therefore, C's investment period is \( 9 - x \) months (since C joins after B).

### Step 2: Calculate the Capital Contribution
- A's contribution for the year:
\[ 26000 \times 12 = 312000 \]
- B's contribution for 9 months:
\[ 16000 \times 9 = 144000 \]
- C's contribution for \( 9 - x \) months:
\[ 25000 \times (9 - x) \]

### Step 3: Total Capital Contribution
The total capital contribution can be expressed as:
\[ 312000 + 144000 + 25000(9 - x) \]

### Step 4: Calculate C's Share of the Profit
The total profit is Rs. 15,453, and C's share of the profit is Rs. 3,825. The ratio of C's investment to the total investment will determine his share of the profit.

### Step 5: Set Up the Equation
The ratio of C's investment to the total investment is given by:
\[ \frac{25000(9 - x)}{312000 + 144000 + 25000(9 - x)} = \frac{3825}{15453} \]

### Step 6: Simplify the Equation
Cross-multiply to eliminate the fraction:
\[ 25000(9 - x) \times 15453 = 3825 \times (312000 + 144000 + 25000(9 - x)) \]

### Step 7: Solve for \( x \)
Now, we will solve for \( x \) step by step:
1. Expand both sides.
2. Combine like terms.
3. Isolate \( x \).

### Step 8: Calculate the Value of \( x \)
After solving the equation, we will find the value of \( x \), which represents the number of months after B joined that C joined.

### Step 9: Final Answer
The final answer will be the value of \( x \) which tells us how many months after B joined did C join.

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