Raman, Manan and Kamal are partners and invest in a business such that Raman Invests `2/5` th of total and Manan Invests th of `3/8` the total. What is the ratio of prof Its of Raman, Manan and Kamal respectively?

Options

A. `16:15:9`
B. `16:15:31`
C. `2:3:5`
D. `15:16:9`

Answer

To solve the problem of finding the ratio of profits of Raman, Manan, and Kamal based on their investments, we can follow these steps:

### Step 1: Determine the investments of Raman and Manan
- Raman invests \( \frac{2}{5} \) of the total capital.
- Manan invests \( \frac{3}{8} \) of the total capital.

### Step 2: Calculate Kamal's investment
To find Kamal's investment, we need to subtract the investments of Raman and Manan from the total capital.

Let the total capital be represented as 1 (or 100%).
- Total investment by Raman and Manan:
\[ \text{Total investment} = \frac{2}{5} + \frac{3}{8} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 8 is 40.

Converting the fractions:
- For Raman:
\[ \frac{2}{5} = \frac{2 \times 8}{5 \times 8} = \frac{16}{40} \]
- For Manan:
\[ \frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40} \]

Now, adding these fractions:
\[ \frac{16}{40} + \frac{15}{40} = \frac{31}{40} \]

So, the investment by Kamal is:
\[ \text{Kamal's investment} = 1 - \frac{31}{40} = \frac{9}{40} \]

### Step 3: Write down the investments
Now we have:
- Raman's investment = \( \frac{16}{40} \)
- Manan's investment = \( \frac{15}{40} \)
- Kamal's investment = \( \frac{9}{40} \)

### Step 4: Determine the ratio of their investments
The ratio of their investments is:
\[ \text{Raman : Manan : Kamal} = \frac{16}{40} : \frac{15}{40} : \frac{9}{40} \]

Since the denominators are the same, we can simplify this to:
\[ \text{Raman : Manan : Kamal} = 16 : 15 : 9 \]

### Step 5: Conclusion
Thus, the ratio of profits of Raman, Manan, and Kamal is:
\[ \text{Profit Ratio} = 16 : 15 : 9 \]