Math Word Problems: Solving Motion Problems Part 3

This is the third part of the Motion Word Problem Series, a subseries of the Mathematics Word Problem Solving Series. The first part discusses about the basic concepts of rate, time, and distance. The second part discusses objects traveling in opposite direction and objects traveling toward each other.

In this post, we discuss about objects traveling on the same direction.

Problem 3

Fred leaves Manila in a car heading to Batangas traveling at an average speed of 60 kilometers per hour. After 2 hours, Ben decided to follow. If Ben’s car is traveling at an average speed of 80 kilometers per hour, how long will Ben’s car overtake Fred’s?

Solution

First, we note that Ben’s car travels 2 hours less than Fred’s. So, if the travel time of  Fred is $t$, then Ben’s travel time is $t - 2$. Second, Ben overtaking Fred means that they will have traveled the same distance.  Therefore, we must equate the distances in the table below.

Equating the distance, we have

$60t = 80t - 160 \Rightarrow -20t = -160 \Rightarrow t = 8$

In $8$ hours, Ben’s car will overtake Fred’s car.

Problem 4

Dan went biking. After 20 minutes, his brother Jim, decided to follow him on the same path. If Dan is traveling at an average speed of 35 kilometers per hour, and Jim’s average speed is 42 kilometers per hour, how long before Jim overtakes Dan?

Solution

This problem is very similar to Problem 3. The only difference is instead of hours, Jim followed after 20 minutes. Although I have not explicitly stated that $t$ is in hours, it should be clear from the previous examples. Since 20 minutes is $\frac{1}{3}$ of an hour, Jim’s travel time is $\frac{1}{3}$ of an hour less that of Dan’s.

Again, the instance that Jim overtakes Dan, the distance they have traveled are equal, so we equate the distances they traveled.

$35 t = 42t - 14 \Rightarrow -7t = -14 \Rightarrow t = 2$

That is, in $2$ hours, Jim will overtake Dan.

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Photo Credit (Creative Commons):  Ed Callow via Flickr