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. `3:4:5`
C. `3:4:5`
D. `4:3:5`

Answer

To solve the problem, let's denote the total investment in the business as \( T \).

1. **Calculate Raman's Investment**:
- Raman invests \( \frac{1}{4} \) of the total investment.
- Therefore, Raman's investment = \( \frac{1}{4} T \).

2. **Calculate Rohit's Investment**:
- Rohit invests \( \frac{1}{3} \) of the total investment.
- Therefore, Rohit's investment = \( \frac{1}{3} T \).

3. **Calculate Raja's Investment**:
- To find Raja's investment, we first need to calculate the total investment made by Raman and Rohit.
- Total investment by Raman and Rohit = \( \frac{1}{4} T + \frac{1}{3} T \).
- To add these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12.
- Converting the fractions:
- \( \frac{1}{4} T = \frac{3}{12} T \)
- \( \frac{1}{3} T = \frac{4}{12} T \)
- Now, adding these:
\[ \frac{3}{12} T + \frac{4}{12} T = \frac{7}{12} T \]
- Thus, the total investment by Raman and Rohit is \( \frac{7}{12} T \).
- Therefore, Raja's investment = Total investment - (Raman's investment + Rohit's investment)
\[ Raja's investment = T - \frac{7}{12} T = \frac{5}{12} T \]

4. **Calculate the Ratio of Investments**:
- Now we have:
- Raman's investment = \( \frac{1}{4} T = \frac{3}{12} T \)
- Rohit's investment = \( \frac{1}{3} T = \frac{4}{12} T \)
- Raja's investment = \( \frac{5}{12} T \)
- The ratio of their investments is:
\[ \text{Raman : Rohit : Raja} = \frac{3}{12} : \frac{4}{12} : \frac{5}{12} \]
- Removing the common denominator (12):
\[ \text{Raman : Rohit : Raja} = 3 : 4 : 5 \]

5. **Conclusion**:
- The ratio of profit of Raman, Rohit, and Raja respectively is \( 3 : 4 : 5 \).