A, B and C did certain investments and the ratio of their time periods is $3:2:7$ respectively. The ratio of the profits of A, B and C is $4:3:14$ respectively. What is the ratio of the investments of A, B and C?

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**Solution:**

1. Given that profit is directly proportional to the product of investment and time period, we have:
   $$P \propto I 	imes T$$

2. Let the investments of A, B, and C be $I_A, I_B,$ and $I_C$ respectively.

3. Given:
   - Time periods ratio: $T_A : T_B : T_C = 3 : 2 : 7$
   - Profits ratio: $P_A : P_B : P_C = 4 : 3 : 14$

4. Using the proportionality:
   $$\frac{P_A}{P_B} = \frac{I_A 	imes T_A}{I_B 	imes T_B} \implies \frac{I_A}{I_B} = \frac{P_A}{P_B} 	imes \frac{T_B}{T_A}$$

Similarly for other pairs.

5. Calculate the investment ratios:
   - For A and B:
     $$\frac{I_A}{I_B} = \frac{4}{3} 	imes \frac{2}{3} = \frac{8}{9}$$

- For B and C:
     $$\frac{I_B}{I_C} = \frac{3}{14} 	imes \frac{7}{2} = \frac{21}{28} = \frac{3}{4}$$

6. From above:
   - $I_A : I_B = 8 : 9$
   - $I_B : I_C = 3 : 4$

7. To combine these ratios, express $I_B$ with a common denominator:
   - $I_B$ is 9 in first ratio and 3 in second ratio.
   - Multiply second ratio by 3 to get $I_B = 9$:
     $$I_B : I_C = 9 : 12$$

8. Therefore, the combined ratio is:
   $$I_A : I_B : I_C = 8 : 9 : 12$$

**Answer:** $8 : 9 : 12$

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