Word problem

Question

You ride your bike to campus a distance of 8 miles and return home on the same route. Going to campus, you ride mostly downhill and average 3 mph faster then on your return trip home. If the round trip takes 2 hours and 36 minutes- that is,13/5 hours- what is your average rate on the return trip?

Answer 1

36 min = 36/60 hours = 3/5 hours 

so 2 hours and 36 min = 2 and 3/5 hours = 13/5 hours

 

T is the time going uphill, the longer amount of time

 

 Downhill:    8 = R* (13/5 - T) = (13/5)R - RT

 

 Uphill:  8 = (R - 3 ) *T  = RT - 3T <--- keep this second equation in mind!!!!

 

 

16 = (13/5)R - 3T <--- adds the equations together; RT cancels

 

16 + 3T = (13/5)R <---  adds 3T to both sides

 

 (5/13) ( 16 + 3T) = R <--- multiplies both sides by 5/13 to solve for R

 

 

  (5/13)(16 + 3T) T = RT <--- multiplies both sides by T

 

 (5/13)(16 + 3T) T - 3T = RT - 3T <--- subtracts 3T from both sides

 

   notice the right side is the same as the second equation in bold above, which equals 8

 

(5/13)(16 + 3T)T - 3T = 8 <--- substitution/transitive

 

 (5/13)(16 + 3T)T - 3T - 8 = 0

 

 ((80/13) + (15/16)T )T - 3T - 8 = 0  <--- distributes 5/13

 

  (15/16)T^2 + (80/13)T - 3T - 8 = 0 <--- distributes T

 

(15/16)T^2 + (41/13)T - 8 = 0  <--- combines like terms; 3 = 39/13, so 80/13 - 39/13 = 41/13

 

  common denominator is 16*13 = 208

 

  195 T^2  + 615 T -   1264 = 0

 

  quadratic formula:

 

  [-615 +or- sqrt ( 615^2 + 4*195*1264)]/390

 

  the negative branch results in negative times

 

  t = 1.417862 hours or approximately 1 hour and 25 minutes

 

                                               25/60 = 5/12 so

                                           1 and 5/12 hours = 17/12 hours

 

Downhill time:  13/5 - 17/12 = (156 - 85)/60 = 81/60 = 27/20 hours

 

 8 miles in 27/20 hours = 8 * 20/27 = 160/27 = 5.925925925... mph

 

 

Uphill time:

8 miles in 17/12 hours = 8(12)/17 = 5.64705 mph