Please help answer with steps thanks so much.

Question

After robbing a bank in Dodge City, a robber gallops off at 13 mi/h. 30 minutes later, the marshall leaves to pursue the robber at 16 mi/h. How long (in hours) does it take the marshall to catch up to the robber?

Answer 1

This is a classic D-I-R-T (Distance-Is-Rate-times-Time) problem. You will see them often. With such problems, there is usually one variable that is the same or related between two equations so that we may solve the problem.

For this problem, notice that “30 minutes later” is a relationship between the times of the two people. Also, “catch up” means that the two people have traveled the same distance. That’s how we can solve this problem.

Now, you must establish elapsed time (in hours). You may choose either the time the “robber gallops off” (which makes the marshal’s elapsed time t-0.5 hours) or the time the “marshal leaves” (which makes the robber’s elapsed time t+0.5 hours). Since the problem asks, “How long does it take the marshal to catch up,” let’s count t as the elapsed time for the marshal.

Set the two distances equal: (D=RT)
                 Robber                 =                  Marshal
      (13 mph)(0.5 hr + t hr)    =          (16 mph)(t hr)
         13 mi + 26t mi             =        32t mi                   [multiply all by 2]
         13 mi                          =          6t mi                    [subtract 26t from both sides]
                            13/6 = t

At 2 hours 10 minutes [that is 2 1/6 hours], the marshal catches up with the robber.

Check:
             Robber        =          Marshal

      Is (13)(1/2 + 13/6) = 16(13/6) ?

           (13)((3+13)/6)  = 16(13/6)   ?

           13(16/6) = 16(13/6) ?

            208/6 = 208/6 ?yes