Question

An officer on horseback starts at the back of a column of marching soldiers and rides to the front of the column, then turns around and rides back to the rear of the column. If the rider travels three times as fast as the column moves, and the column is 100 metres long, how far does the column move by the time the officer arrives back at the rear of the column?

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Different ways to solve this problem would be great :)
Please explain the steps you took to get to each answer

Answer 1

This is tricky because everyone is moving! As usual, a picture will help, and then some guessing and checking (or algebra).
Let's make a specific assumption -- and suppose that the column is moving forward at a rate of 50 meters per minute. Then the officer moves ahead at a rate of 150 meters per minute. Here's the situation at the beginning of the problem, just as the officer begins his ride:
In the first minute, the front of the column moves ahead 50 meters, and the officer traverses a total of 150 meters. The first 100 meters gets him to where the front originally started, and the extra 50 meters that the front of the column has moved forward
So at the end of one minute, the officer is in front of the column. Now let's think about what happens in the next half minute.
The back of the column will move forward 25 meters, and the officer will ride 75 meters. Thus, they'll meet up at that time and that point. Altogether, the column has moved forward 75 meters (in a minute and a half).
so Is it always 75 meters