Arun , Bhavya & Ashu entered into a partnership , ratio if investment of Arun & Bhavya is `4:x` and ratio of investment of Bhavya & Ashu is `3:4` .If at the end of two years Ashu receives Rs 1850 as profit out of total profit Rs 3700 . Then find the value of x ?

Options

A. A)12
B. B)14
C. C)16
D. D)8

Answer

To solve the problem step by step, we will follow the given information and use the concept of ratios and profits in partnerships.

### Step 1: Define the Ratios
We are given:
- The investment ratio of Arun and Bhavya is \(4:x\).
- The investment ratio of Bhavya and Ashu is \(3:4\).

### Step 2: Express All Investments in Terms of a Common Variable
Let’s express the investments of Arun, Bhavya, and Ashu in terms of a common variable.

From the first ratio:
- Let Arun's investment = \(4k\)
- Let Bhavya's investment = \(xk\)

From the second ratio:
- Let Bhavya's investment = \(3m\)
- Let Ashu's investment = \(4m\)

Since both expressions represent Bhavya's investment, we can set them equal:
\[ xk = 3m \]
This gives us:
\[ m = \frac{xk}{3} \]

Now substituting \(m\) in Ashu's investment:
\[ \text{Ashu's investment} = 4m = 4 \left(\frac{xk}{3}\right) = \frac{4xk}{3} \]

### Step 3: Write the Combined Investment Ratios
Now we have the investments:
- Arun = \(4k\)
- Bhavya = \(xk\)
- Ashu = \(\frac{4xk}{3}\)

The total investment ratio can be expressed as:
\[ 4k : xk : \frac{4xk}{3} \]

To eliminate \(k\), we can multiply through by 3:
\[ 12 : 3x : 4x \]

### Step 4: Determine the Profit Sharing Ratio
Since the time period for all partners is the same (2 years), the profit-sharing ratio will be the same as the investment ratio:
\[ \text{Profit Ratio} = 12 : 3x : 4x \]

### Step 5: Calculate Total Profit and Ashu's Profit
The total profit is given as Rs 3700, and Ashu's profit is Rs 1850.

### Step 6: Set Up the Equation
The total parts in the profit ratio can be calculated as:
\[ 12 + 3x + 4x = 12 + 7x \]

Ashu's share of the profit can be expressed as:
\[ \frac{4x}{12 + 7x} \times 3700 = 1850 \]

### Step 7: Solve the Equation
Now we can set up the equation:
\[ \frac{4x}{12 + 7x} = \frac{1850}{3700} \]
\[ \frac{4x}{12 + 7x} = \frac{1}{2} \]

Cross-multiplying gives:
\[ 2 \times 4x = 12 + 7x \]
\[ 8x = 12 + 7x \]

Now, isolate \(x\):
\[ 8x - 7x = 12 \]
\[ x = 12 \]

### Conclusion
The value of \(x\) is \(12\).