Equating the Distances Traveled

Many distance problems have the scenario that one person catches up with another or one train or plane leaves later and finally passes the first one. The common thread or theme for these problems is that the distance traveled by the two participants is the same. You equate the distances. Some problems have you solve for time — how long it took to catch up. Or you may solve for how fast one or the other is traveling. And the question may even be about the distance — how far they traveled before arriving at the same place. In each case, though, the equation setup is the same. 

The formula is d = rt or distance = rate x time. And when two distances are equated, the individual rates and times are multiplied together first and then set equal to one another  

When two people leave the same place at different times, one has to travel more quickly than the other to catch up — assuming that they re both using the same route. 

The Problem Henry left for work at 7 a.m. and drove at an average speed of 45 mph. Unfortunately, he forgot to put some important papers in his briefcase. At 7 30, Betty found the papers, jumped in her car, and chased after Henry. She averaged 55 mph. How long did it take for Betty to catch up to 

The distance that Henry drives and the distance that Betty drives is the same — at the moment Betty catches up to Henry. So multiply the rate that Henry drives times how long he drives and equate that to the rate that Betty drives times her amount of time. Let the amount of time that Henry drives be represented by t hours. Then Betty drives for t— 0.5 hours. Multiplying rate times time, the equation and solution are  

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