Rational numbers

Question

Two motorcycles made the same 315 mile drive. One motorcycle traveled 10 mph faster than the other and arrived 2 hours earlier. Find the speed of each motorcycle

Answer 1

Motorcycle 1 moves a distance of 315 at speed v and time t.  Motorcycle 2 moves a distance 315 at speed v' = v+10 at time t' = t-2.  This gives us 2 equations in 2 unknowns:

 

vt = 315

(v+10)(t-2) = 315

 

Get t from the top equation and substitute into the bottom equation:

 

t = 315/v

(v+10)(315/v - 2) = 315

 

Solve for v:

 

(v+10)(315-2v) = 315v

315v - 2v² +3150 -20v = 315v

-2v² - 20v + 3150 = 0

v² +10v - 1575 = 0

v = (1/2){-10±sqrt[10²-4(-1575)]} = (1/2)(-10±80)

 

Note only the positive square root is valid (need positive time!)

 

v = (1/2)(-10+80) = 35 mph

 

That is speed of motorcycle 1.  Then speed of motorcycle 2 is v+10 = 35+10 = 45 mph

 

Note we can also get the duration of the trips:

 

t = 315/v = 315/35 = 9 hrs for motorcycle 1

t-2 = 7 hours for motorcycle 2