To find the ratio of investments of A, B, and C based on the given profit ratios and investment ratios, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Ratios**:
- Profit ratio of A, B, and C is given as \(5:3:12\).
- Investment ratio of A, B, and C is given as \(5:6:8\).
2. **Set Up the Equations**:
- Let the investments of A, B, and C be represented as \(I_A\), \(I_B\), and \(I_C\) respectively.
- Let the profits of A, B, and C be represented as \(P_A\), \(P_B\), and \(P_C\) respectively.
3. **Use the Formula for Profit**:
- The profit earned by each partner is proportional to their investment and the time for which the investment is made.
- The formula can be expressed as:
\[
P_A : P_B : P_C = I_A \cdot T_A : I_B \cdot T_B : I_C \cdot T_C
\]
- Given the time ratios are in the same ratio as the investment ratios, we can simplify this to:
\[
P_A : P_B : P_C = I_A : I_B : I_C
\]
4. **Set Up the Proportionality**:
- From the profit ratio:
\[
\frac{P_A}{P_B} = \frac{5}{3}, \quad \frac{P_B}{P_C} = \frac{3}{12}
\]
- From the investment ratio:
\[
\frac{I_A}{I_B} = \frac{5}{6}, \quad \frac{I_B}{I_C} = \frac{6}{8}
\]
5. **Calculate the Investment Ratios**:
- Using the profit ratios:
\[
\frac{I_A}{I_B} = \frac{5}{3} \cdot \frac{I_B}{I_C}
\]
- Substitute \(I_B\) and \(I_C\) from the investment ratio:
\[
I_A : I_B : I_C = 5x : 6x : 8x
\]
6. **Combine the Ratios**:
- To find the combined ratio of investments, we can express everything in terms of a common variable:
\[
I_A : I_B : I_C = 5k : 6k : 8k
\]
- This gives us the ratio of investments as \(5:6:8\).
7. **Final Ratio**:
- The final ratio of investments of A, B, and C is \(5:6:8\).
