To determine how long it will take Chester to drive all the way home, we first need to understand the proportion of the journey he has completed and the time taken for that portion.
Chester drove \( \frac{2}{3} \) of the way home in \( \frac{1}{4} \) of an hour. We can denote the total time to drive all the way home as \( T \).
Since \( \frac{2}{3} \) of the trip takes \( \frac{1}{4} \) of an hour, we can set up the following proportion to find out how long the entire journey will take:
\[ \frac{2}{3}T = \frac{1}{4} \]
To solve for \( T \), we can first isolate \( T \) by multiplying both sides of the equation by \( \frac{3}{2} \):
\[ T = \frac{1}{4} \times \frac{3}{2} \]
Now we compute the right-hand side:
\[ T = \frac{1 \times 3}{4 \times 2} = \frac{3}{8} \]
Thus, the total time \( T \) it takes Chester to drive all the way home is \( \frac{3}{8} \) of an hour.
To recap, it took Chester \( \frac{1}{4} \) hour to drive \( \frac{2}{3} \) of the way home, which means it will take him \( \frac{3}{8} \) hour to drive the entire distance home.
Therefore, the final answer is:
\[ \frac{3}{8} \text{ hours} \]