A,B and C are partners and invest in a business. A invests `1/5` th of total and B invests `1/4` th of the total. What is the ratio of profit of A, B and C respectively?

Options

A. `5 : 4 : 9`
B. `5 : 4 : 1`
C. ` 4 : 5 : 1 `
D. ` 4 : 5 : 11`

Answer

To find the ratio of profits of A, B, and C, we can follow these steps:

### Step 1: Define the total investment
Let the total investment be represented as \( x \).

### Step 2: Calculate A's investment
A invests \( \frac{1}{5} \) of the total investment:
\[ \text{A's investment} = \frac{1}{5}x \]

### Step 3: Calculate B's investment
B invests \( \frac{1}{4} \) of the total investment:
\[ \text{B's investment} = \frac{1}{4}x \]

### Step 4: Calculate C's investment
To find C's investment, we subtract A's and B's investments from the total investment:
\[ \text{C's investment} = x - \left(\frac{1}{5}x + \frac{1}{4}x\right) \]

### Step 5: Find a common denominator for A's and B's investments
The common denominator for 5 and 4 is 20. Therefore, we convert A's and B's investments:
\[ \text{A's investment} = \frac{4}{20}x \]
\[ \text{B's investment} = \frac{5}{20}x \]

### Step 6: Combine A's and B's investments
Now we can add A's and B's investments:
\[ \frac{1}{5}x + \frac{1}{4}x = \frac{4}{20}x + \frac{5}{20}x = \frac{9}{20}x \]

### Step 7: Calculate C's investment
Now substituting back to find C's investment:
\[ \text{C's investment} = x - \frac{9}{20}x = \frac{20}{20}x - \frac{9}{20}x = \frac{11}{20}x \]

### Step 8: Write down the investments of A, B, and C
Now we have:
- A's investment: \( \frac{4}{20}x \)
- B's investment: \( \frac{5}{20}x \)
- C's investment: \( \frac{11}{20}x \)

### Step 9: Find the ratio of their investments
The ratio of A's, B's, and C's investments can be expressed as:
\[ \text{Ratio} = \frac{4}{20} : \frac{5}{20} : \frac{11}{20} \]
This simplifies to:
\[ 4 : 5 : 11 \]

### Conclusion
Thus, the ratio of profits of A, B, and C is:
\[ \boxed{4 : 5 : 11} \]