Three partners Rahul, Sonu and Rishab-invested their amounts in ratio of `2:5:7`. At the end of 6 months, 'Rahul' added some more amount such that his investment becomes equal to `1/3`rd of sum of 'Sonu' and 'Risabh' initial investment. If at the end of the year, Sonu's share in profit is ₹320, then find the total profit.

Options

A. ₹960
B. ₹920
C. ₹1050
D. ₹1450

Answer

To solve the problem step by step, we will follow the given information and calculations:

### Step 1: Define the Initial Investments
Let the initial investments of Rahul, Sonu, and Rishab be represented as:
- Rahul = 2x
- Sonu = 5x
- Rishab = 7x

### Step 2: Calculate the Total Initial Investment
The total initial investment can be calculated as:
\[ \text{Total Initial Investment} = 2x + 5x + 7x = 14x \]

### Step 3: Determine the Investment After 6 Months
After 6 months, Rahul adds an additional amount (let's call it y) such that his total investment becomes equal to one-third of the sum of Sonu and Rishab's initial investments.

The sum of Sonu and Rishab's initial investments:
\[ 5x + 7x = 12x \]

### Step 4: Set Up the Equation for Rahul's Investment
According to the problem:
\[ 2x + y = \frac{1}{3} \times 12x \]
This simplifies to:
\[ 2x + y = 4x \]

### Step 5: Solve for y
Rearranging the equation gives:
\[ y = 4x - 2x = 2x \]

### Step 6: Calculate Rahul's Total Investment
Now, Rahul's total investment after adding y is:
\[ \text{Rahul's Total Investment} = 2x + 2x = 4x \]

### Step 7: Calculate the Investment for the Next 6 Months
- For the next 6 months, the investments will be:
- Rahul: \( 4x \times 6 = 24x \)
- Sonu: \( 5x \times 6 = 30x \)
- Rishab: \( 7x \times 6 = 42x \)

### Step 8: Calculate Total Investment Over the Year
Now, we can calculate the total investment for the year:
- Rahul: \( 12x + 24x = 36x \)
- Sonu: \( 30x + 30x = 60x \)
- Rishab: \( 42x + 42x = 84x \)

### Step 9: Calculate the Ratio of Investments
The total investments are:
- Rahul: 36x
- Sonu: 60x
- Rishab: 84x

The ratio of their investments is:
\[ 36 : 60 : 84 \]

### Step 10: Simplify the Ratio
To simplify the ratio, we can divide each term by 12:
\[ 3 : 5 : 7 \]

### Step 11: Determine Sonu's Share in Profit
Given that Sonu's share in profit is ₹320 and corresponds to the middle term (5 parts) of the ratio, we can find the value of one part:
\[ \text{Value of 1 part} = \frac{320}{5} = 64 \]

### Step 12: Calculate Total Profit
Now, we can calculate the total profit:
The total number of parts in the ratio is:
\[ 3 + 5 + 7 = 15 \]

Thus, the total profit is:
\[ \text{Total Profit} = 15 \times 64 = 960 \]

### Final Answer
The total profit is ₹960.

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