A,B and C invested capitals in the ratio of `2:3: 5` .At the end of the business terms, they received the profit in the ratio of `5:3:12`. Find the ratio of time for which they contributed their capitals.

Answer

To find the ratio of 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:

\[ \text{Profit Ratio} = \frac{\text{Capital} \times \text{Time}}{\text{Capital} \times \text{Time}} \]

Given:
- Capital ratio of A, B, and C = \(2:3:5\)
- Profit ratio of A, B, and C = \(5:3:12\)

Let’s denote:
- Capital of A = \(2x\)
- Capital of B = \(3x\)
- Capital of C = \(5x\)

Let’s denote the time for which A, B, and C contributed their capitals as \(t_A\), \(t_B\), and \(t_C\) respectively.

### Step 1: Set up the equation using the profit and capital ratios

Using the formula, we can express the time in terms of the profit and capital:

\[ \frac{P_A}{P_B} = \frac{C_A \times t_A}{C_B \times t_B} \]

Substituting the values we have:

\[ \frac{5}{3} = \frac{2x \times t_A}{3x \times t_B} \]

### Step 2: Simplify the equation

The \(x\) cancels out:

\[ \frac{5}{3} = \frac{2t_A}{3t_B} \]

Cross-multiplying gives:

\[ 5 \cdot 3t_B = 2t_A \cdot 3 \]

This simplifies to:

\[ 15t_B = 2t_A \]

### Step 3: Rearranging the equation

Rearranging gives:

\[ t_A = \frac{15}{2} t_B \]

### Step 4: Set up the second equation for B and C

Now we will set up a similar equation for B and C:

\[ \frac{P_B}{P_C} = \frac{C_B \times t_B}{C_C \times t_C} \]

Substituting the values:

\[ \frac{3}{12} = \frac{3x \times t_B}{5x \times t_C} \]

Cancelling \(x\):

\[ \frac{3}{12} = \frac{3t_B}{5t_C} \]

Cross-multiplying gives:

\[ 3 \cdot 5t_C = 12 \cdot 3t_B \]

This simplifies to:

\[ 15t_C = 36t_B \]

### Step 5: Rearranging the equation

Rearranging gives:

\[ t_C = \frac{15}{36} t_B = \frac{5}{12} t_B \]

### Step 6: Now we have expressions for \(t_A\) and \(t_C\) in terms of \(t_B\)

From the previous steps, we have:

\[ t_A = \frac{15}{2} t_B \]
\[ t_C = \frac{5}{12} t_B \]

### Step 7: Express the ratios of \(t_A\), \(t_B\), and \(t_C\)

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

\[ t_A : t_B : t_C = \frac{15}{2}t_B : t_B : \frac{5}{12}t_B \]

### Step 8: Eliminate \(t_B\) and find a common denominator

To eliminate \(t_B\) and find a common ratio, we can express all terms with a common denominator. The common denominator for \(2\) and \(12\) is \(12\):

\[ t_A = \frac{15}{2}t_B = \frac{15 \times 6}{2 \times 6} = \frac{90}{12}t_B \]
\[ t_B = \frac{12}{12}t_B \]
\[ t_C = \frac{5}{12}t_B \]

Thus, the ratios become:

\[ 90 : 12 : 5 \]

### Step 9: Simplify the ratio

Now we simplify the ratio \(90 : 12 : 5\):

To simplify, we can divide each term by the greatest common divisor (GCD). The GCD of \(90\), \(12\), and \(5\) is \(1\), so the ratio remains:

\[ 90 : 12 : 5 \]

### Final Answer

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

\[ \boxed{90 : 12 : 5} \]