A,B and C do certain investments for time periods in the ratio of `2:1:8`. At the end of the business term, they received the profits in the ratio of `3:4: 2`. Find the ratio of investments of A,B and C.

Options

A. `6:16:1`
B. `2:5:1`
C. `6:17:1`
D. `6:19:3`

Answer

To find the ratio of investments of A, B, and C based on the given profit ratios and time periods, we can follow these steps:

### Step 1: Understand the given ratios
We are given:
- Investment time periods in the ratio of A:B:C = 2:1:8
- Profit received in the ratio of A:B:C = 3:4:2

### Step 2: Set up the equations
Let the investments of A, B, and C be represented as \( I_A, I_B, I_C \) respectively. The profit received can be expressed in terms of their investments and time periods.

The formula for profit is:
\[ \text{Profit} = \text{Investment} \times \text{Time} \]

### Step 3: Express profits in terms of investments
Using the ratios:
- For A: Profit_A = \( I_A \times 2 \)
- For B: Profit_B = \( I_B \times 1 \)
- For C: Profit_C = \( I_C \times 8 \)

### Step 4: Write the profit ratios
From the profit ratios given, we can express:
\[ \frac{Profit_A}{Profit_B} = \frac{3}{4} \quad \text{and} \quad \frac{Profit_B}{Profit_C} = \frac{4}{2} \]
This leads us to:
\[ \frac{I_A \times 2}{I_B \times 1} = \frac{3}{4} \quad \text{(1)} \]
\[ \frac{I_B \times 1}{I_C \times 8} = \frac{4}{2} \quad \text{(2)} \]

### Step 5: Solve equation (1)
From equation (1):
\[ \frac{2I_A}{I_B} = \frac{3}{4} \]
Cross-multiplying gives:
\[ 8I_A = 3I_B \quad \Rightarrow \quad I_A = \frac{3}{8} I_B \quad \text{(3)} \]

### Step 6: Solve equation (2)
From equation (2):
\[ \frac{I_B}{8I_C} = \frac{4}{2} \]
Cross-multiplying gives:
\[ 2I_B = 32I_C \quad \Rightarrow \quad I_B = 16I_C \quad \text{(4)} \]

### Step 7: Substitute equation (4) into equation (3)
Substituting \( I_B \) from equation (4) into equation (3):
\[ I_A = \frac{3}{8} \times 16I_C = 6I_C \]

### Step 8: Write the ratios of investments
Now we have:
- \( I_A = 6I_C \)
- \( I_B = 16I_C \)
- \( I_C = I_C \)

Thus, the ratio of investments \( I_A : I_B : I_C \) is:
\[ 6I_C : 16I_C : 1I_C = 6 : 16 : 1 \]

### Final Ratio
The ratio of investments of A, B, and C is \( 6 : 16 : 1 \).

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