When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?
1. Let \( x \) be the number of hours that have already passed in a 24-hour day.
2. According to the problem, a quarter of the time that has passed plus half of the remaining time equals the current time. This can be written as:
   \[
   \frac{x}{4} + \frac{24 - x}{2} = x
   \]
3. To solve this equation, first find a common denominator for the fractions. The common denominator for 4 and 2 is 4. Rewrite the equation:
   \[
   \frac{x}{4} + \frac{2(24 - x)}{4} = x
   \]
4. Combine the fractions:
   \[
   \frac{x + 2(24 - x)}{4} = x
   \]
5. Simplify the numerator:
   \[
   \frac{x + 48 - 2x}{4} = x
   \]
6. Combine like terms:
   \[
   \frac{48 - x}{4} = x
   \]
7. Multiply both sides by 4 to clear the fraction:
   \[
   48 - x = 4x
   \]
8. Add \( x \) to both sides to isolate \( x \):
   \[
   48 = 5x
   \]
9. Divide both sides by 5 to solve for \( x \):
   \[
   x = \frac{48}{5} = 9.6
   \]
10. Convert 9.6 hours to hours and minutes. Since 0.6 hours is 36 minutes (0.6 * 60 = 36), the time is:
    \[
    9 \text{ hours and } 36 \text{ minutes}
    \]

The final answer is \(\boxed{9:36}\).