First, we need to determine how many problems Yan has left to complete. Since he has finished 27 out of 45 problems, he has:
\[ 45 - 27 = 18 \text{ problems left} \]
Next, we know that each problem takes him \( \frac{4}{5} \) of a minute to complete. To find out how much time it will take him to complete the remaining 18 problems, we calculate the total time for the remaining problems:
\[ \text{Time for remaining problems} = \text{Number of problems} \times \text{Time per problem} \]
Substituting in the values:
\[ \text{Time for remaining problems} = 18 \times \frac{4}{5} \]
To carry out the multiplication:
\[ 18 \times \frac{4}{5} = \frac{18 \times 4}{5} = \frac{72}{5} = 14.4 \text{ minutes} \]
Now we round 14.4 minutes to the nearest minute, which gives us:
\[ 14.4 \approx 14 \text{ minutes} \]
Since the question asks how much more time to the nearest minute the rest of the problems will take to finish, we conclude that the nearest minute is:
\[ 14 \text{ minutes} \]
However, this option does not appear listed in your choices (18 minutes, 27 minutes, 32 minutes, 43 minutes). Thus, none of the provided answers match the calculated time. Please double-check the options or any assumptions.