maria runs through the countryside at a rate of 10mph.

Question

she returns along the same route at 6 mph. if the total trip took 1 hour and 36 minutes, how far did she run total

Answer 1

 

distance=rate*time

d=r*t

d=10*t

d=6*(1.6-t)   (36 minutes is 36/60=0.6 hours)

because they both equal d set the equations equal to each other

10*t=6*(1.6-t)

10t=9.6-6t

10t+6t=9.6

16t=9.6

t=9.6/16

t=96/160

t=0.6 hours

d=10*0.6=6 miles

d=6*(1.6-0.6)

d=6*1=6 miles

total distance run is 6+6=12 miles

the average rate is found by using the formula total distance=rate*total time

12=r*1.6

r=12/1.6

r=120/16

r=7.5 mph

you can't find the average rate by adding 6 and 10 to get 16 and dividing by 2 to get 8 mph

if you are familiar with the harmonic mean you can find the average rate very easily:

2*6*10/(6+10)=120/16=15/2=7.5 mph

now d=r*t

d=7.5*1.6

d=12 miles

 

Answer 2

Answer: 12.8 miles  Formula for distance is D = R times T (where D is distance, R is the rate in mph, and T is time in hours and minutes).  Since for this problem 1/2 the distance is at 10 mph and 1/2 the distance is at 6 mph we can easily compute that the average distance for the entire run was 8mph (10+6)/2.  The time of one hour and 36 minutes can be converted to 1 and 3/5 hours or better yet the improper fraction 8/5.  Now plug it into the formula to solve for D:

 

D = 8mph times 8/5    This gives you 64/5 which simplifies to 12 4/5 or 12.8 miles.