To find out how many minutes Charlotte can earn to play video games tonight, we first need to determine how many math homework problems she can complete.
Charlotte earns \( 2 \frac{1}{2} \) minutes for every 5 math homework problems completed. First, we convert \( 2 \frac{1}{2} \) minutes to an improper fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \text{ minutes} \]
Next, we calculate how many sets of 5 problems are in the total of 24 problems:
\[ \text{Number of sets} = \frac{24}{5} = 4.8 \]
Since Charlotte can only complete whole sets of 5 problems, she can complete 4 full sets. This means she can complete:
\[ 4 \times 5 = 20 \text{ problems} \]
Now we calculate the amount of time she earns for these 20 problems. Each set of 5 problems earns her \( \frac{5}{2} \) minutes. Therefore, for 4 sets, she earns:
\[ 4 \times \frac{5}{2} = \frac{20}{2} = 10 \text{ minutes} \]
Now, we need to check if she can earn time for the remaining problems:
She completes 20 problems out of 24, leaving her with:
\[ 24 - 20 = 4 \text{ problems} \]
Charlotte earns \( \frac{5}{2} \) minutes for every 5 problems, so for 4 problems, she will not earn any additional time, as she has not completed another full set of 5 problems.
Thus, the most time Charlotte can earn to play video games tonight is:
\[ \boxed{10} \text{ minutes} \]
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