What is the length of the race in miles?

Question

Sarah races at 10 mph and Kim races at 9 mph. When they both ran the same long distance race last weekend, Sarah finished eight minutes ahead of Kim. 

Answer 1

The slow runner needs more time to finish the race.  Kim is the slow runner at 9 mph.  Sarah is the fast runner at 10 mph.  The distance run by both is the same.  Sarah's time was 8 minutes less than Kim's time.  

 

Let t = Kim's time = the longer time

Then Sarah's time = (t - 8/60) hours = (t - 2/15) hours

 

Sarah's distance:  10(t - 2/15) (in miles with t in hours)

Kim's distance:  9t (in miles with t in hours)

The 2 distances are the same.

 

9t = 10(t - 2/15)

 

t = 20/15 hrs = 1 hour + 20 min

 

Kim's time is 1 hour + 20 min.

 

t - 2/15 = 20/15 - 2/15 = 18/15 = 1 hour + 12 min.

 

Sarah's time is 1 hour + 12 min.

 

Test the answers.  Take the difference of the times and see that Sarah finishes 8 minutes  before Kim.  Multiply the times by their respective rates and see that the distances are the same.

 

d = 9t = (9 mph)(1 hr + 20 min) = 9(4/3) miles = 12 miles

 

Answer 2

d = r · t

9(t + 2/15) = 10(t)   eight minutes is 2/15 of an hour

 

Can you solve for t and answer?