"at a cross-country track meet, alicia ran 8 mph for the first part of the race, then increased her speed to 12 mph for the second part. if the race was 21 miles long and alicia finished in 2 hours, how far did she run at the faster pace? "

15 miles

Explanation:

Alicia's speed in the first part of the race is:

where is the length of the first part and the time taken to cover this part.

Alicia's speed in the second part of the race is:

where is the length of the second part and the time taken to cover this part.

The total length of the race is

And the total time taken by Alicia to complete the race is

We have in total 4 equations with 4 unknown quantities. From the first 2 equations we find:

And substituting into the second group of equations:

Calling A the first equation and B the second equation, if we compute (A-8B) we get:

And so

So Alicia runs for 1.25 hours at 12 mph (second part of the race). Therefore, the distance travelled during this part is:

Therefore the answer is 15 miles.