A,B and C invested capital in the ratio of `4:6:9` . At the end of the business term, they received the profit in the ratio of `2:3: 5`. Find the ratio of their time for which they contributed their capitals.

Options

A. `1:1:9`
B. `2:2:9`
C. `10:10:9`
D. `9:9:10`

Answer

To find the ratio of the time for which A, B, and C contributed their capitals, we can use the relationship between capital, profit, and time. The formula we will use is:

\[ \text{Profit} \propto \text{Capital} \times \text{Time} \]

Given:
- The capital ratio of A, B, and C is \(4:6:9\).
- The profit ratio of A, B, and C is \(2:3:5\).

Let the time periods for A, B, and C be \(t_A\), \(t_B\), and \(t_C\) respectively.

### Step 1: Set up the equations based on the ratios

From the capital and profit ratios, we can express the relationship as:

\[ \frac{\text{Profit}_A}{\text{Profit}_B} = \frac{\text{Capital}_A \times t_A}{\text{Capital}_B \times t_B} \]

Substituting the values we have:

\[ \frac{2}{3} = \frac{4 \times t_A}{6 \times t_B} \]

### Step 2: Simplify the equation

Cross-multiplying gives us:

\[ 2 \times 6 \times t_B = 3 \times 4 \times t_A \]

This simplifies to:

\[ 12 t_B = 12 t_A \]

Dividing both sides by 12:

\[ t_B = t_A \quad \text{(1)} \]

### Step 3: Set up the second equation

Now, we can set up the second equation using the profit ratio of A and C:

\[ \frac{\text{Profit}_A}{\text{Profit}_C} = \frac{\text{Capital}_A \times t_A}{\text{Capital}_C \times t_C} \]

Substituting the values we have:

\[ \frac{2}{5} = \frac{4 \times t_A}{9 \times t_C} \]

### Step 4: Simplify this equation

Cross-multiplying gives us:

\[ 2 \times 9 \times t_C = 5 \times 4 \times t_A \]

This simplifies to:

\[ 18 t_C = 20 t_A \]

Dividing both sides by 2:

\[ 9 t_C = 10 t_A \]

Rearranging gives:

\[ t_C = \frac{10}{9} t_A \quad \text{(2)} \]

### Step 5: Express all times in terms of \(t_A\)

From equation (1), we have \(t_B = t_A\).

From equation (2), we have \(t_C = \frac{10}{9} t_A\).

### Step 6: Find the ratio of times

Now we can express the ratio of \(t_A\), \(t_B\), and \(t_C\):

\[ t_A : t_B : t_C = t_A : t_A : \frac{10}{9} t_A \]

This simplifies to:

\[ 1 : 1 : \frac{10}{9} \]

To eliminate the fraction, we can multiply the entire ratio by 9:

\[ 9 : 9 : 10 \]

### Final Answer

Thus, the ratio of the time for which A, B, and C contributed their capitals is:

\[ \boxed{9 : 9 : 10} \]