word problem

Question

Mark left the movie theater and drove
toward the ferry office at an average speed
of 32 km/h. Julia left one hour later and
drove in the same direction but with an
average speed of 40 km/h. How long did
Mark drive before Julia caught up?

Answer 1

d=rt

d=32t

d=40(t-1)

d=40t-40

40t-40=32t

40t-32t=40

8t=40

t=40/8

t=5 hours

Answer 2

1. Distance (D) = Rate (R) * Time (T),  where rate, in this case, is the average speed
2. The problem is essentially asking when will the distance traveled be the same.  In other words, when they have traveled the same distance at the same point in time, Julia has caught up with Mark
3. So, at this point, we don't know the distance traveled nor the time it takes for Julia to catch up.  But, we know that the distance must be the same for Mark and Julia.  So:
1. D_M = R_M * T_M    where the M subscript refers to Mark
2. D_M = 32 * T_M
3. D_J = R_J * T_J   where the J subscript refers to Julia
4. D_J = 40 * T_J
5. Now set D_M equal to D_J.   
6. 32*T_M = 40 * T_J    {Eqn 1}
7. Eqn 1 has two unknowns.  So it can't be solved uniquely.  However, we know that Julia left 1 hour after Mark.  So we have a new relationship :  T_J = T_M - 1 hour.  {Eqn 2}  In other words, however long it took Julia to get to the point where she has caught up with Mark, she is able to do it in one less hour.
8. Now substitute Eqn 2 into Eqn 1
9. 32*T_M = 40 *(T_M -1)
10. 32T_M = 40T_M - 40
11. -8T_M = -40
12. T_M = 5 hours
13. Mark drove for 5 hours before Julia caught up with him.  Nice thing about this problem is you can easily check to make sure this is the correct answer.  If Mark traveled 5 hours @ 32km/hr, then he traveled 160 km.  Since Julia left an hour later, she must only have driven 4 hours.  So Julia traveled 4 hours @ 40kms/hr =  160km.  So, it all checks out - when they travel 160km, Julia will have caught up to Mark.