Raman, Rohit and Raja are partners and Invest in a business. Raman invests `1/4` th of total and Rohit invests `1/3` rd of the total.
What is the ratio of profit of Raman, Rohit and Raja respectively?

Options

A. `4 : 3 : 1`
B. `3 : 4 : 4`
C. ` 3 : 4 : 5 `
D. `4 : 3 : 5`

Answer

To find the ratio of profits of Raman, Rohit, and Raja based on their investments, we can follow these steps:

### Step 1: Define the total investment
Let the total investment in the business be denoted as \( T \).

### Step 2: Calculate Raman's investment
Raman invests \( \frac{1}{4} \) of the total investment. Therefore, his investment can be calculated as:
\[ \text{Raman's Investment} = \frac{1}{4} T \]

### Step 3: Calculate Rohit's investment
Rohit invests \( \frac{1}{3} \) of the total investment. Therefore, his investment can be calculated as:
\[ \text{Rohit's Investment} = \frac{1}{3} T \]

### Step 4: Find a common denominator
To combine these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12. Therefore, we can express the investments as:
\[ \text{Raman's Investment} = \frac{1}{4} T = \frac{3}{12} T \]
\[ \text{Rohit's Investment} = \frac{1}{3} T = \frac{4}{12} T \]

### Step 5: Calculate Raja's investment
Now, we can find Raja's investment. Since the total investment is \( T \), we can calculate Raja's investment as follows:
\[ \text{Raja's Investment} = T - \left( \text{Raman's Investment} + \text{Rohit's Investment} \right) \]
Substituting the values we have:
\[ \text{Raja's Investment} = T - \left( \frac{3}{12} T + \frac{4}{12} T \right) = T - \frac{7}{12} T = \frac{5}{12} T \]

### Step 6: Write the ratio of investments
Now, we can express the investments of Raman, Rohit, and Raja in terms of a common unit:
- Raman's Investment: \( \frac{3}{12} T \)
- Rohit's Investment: \( \frac{4}{12} T \)
- Raja's Investment: \( \frac{5}{12} T \)

The ratio of their investments is:
\[ \text{Raman : Rohit : Raja} = 3 : 4 : 5 \]

### Step 7: Conclusion
Since the profit is distributed in the same ratio as the investments (assuming the time of investment is the same), the ratio of profits for Raman, Rohit, and Raja will also be:
\[ \text{Profit Ratio} = 3 : 4 : 5 \]