A,B and C invested capital in the ratio `5:9:8` .At the end of the business term, they received the profits in the ratio `5:3:4`. Find the ratio of time for which they contributed their capital ?

Answer

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

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

Given:
- The ratio of capital invested by A, B, and C is \(5:9:8\).
- The ratio of profits received by A, B, and C is \(5:3:4\).

### Step 1: Write down the ratios
- Capital ratio: \(C_A : C_B : C_C = 5 : 9 : 8\)
- Profit ratio: \(P_A : P_B : P_C = 5 : 3 : 4\)

### Step 2: Set up the ratio of time
Using the formula for time, we have:

\[ T_A : T_B : T_C = \frac{C_A}{P_A} : \frac{C_B}{P_B} : \frac{C_C}{P_C} \]

### Step 3: Substitute the values
Substituting the values of capital and profit into the equation:

\[ T_A : T_B : T_C = \frac{5}{5} : \frac{9}{3} : \frac{8}{4} \]

### Step 4: Simplify the ratios
Now we simplify each term:

\[ T_A : T_B : T_C = 1 : 3 : 2 \]

### Final Answer
Thus, the ratio of time for which A, B, and C contributed their capital is:

\[ \boxed{1 : 3 : 2} \]