Alex wants to take his two sons to visit their grandmother,
living 33 kilometers away.
His motorcycle can run 25 kilometers per hour if he rides alone, but the speed drops to 20 kph if he carries one passenger and he cannot carry two.
The walking speed of each of the boys is 5 kph.

How can the three of them reach grandmother’s house in 3 hours?

Source : Russian Mathematical Olympiad (1999)

While Alex takes one son partway to grandmother's, the other starts walking.  Alex drops off the first son before reaching the destination.  That son walks the rest of the way while Alex heads back to get the other son.  They all arrive together.

A solution with a time vs distance graph can help visualize this.  Say Alex took his first son for a hours.  They reach the point (a,20a).  If Alex heads back and takes another b hours he reaches the other son at a point that can be thought of in two ways:
From the son's perspective walking continuously (a+b,5a+5b)
From Alex's perspective having turned back riding alone (a+b,20a-25b).
So 5a+5b=20a-25b
Solving for b gives b=a/2
So this point can be rewritten as (1.5a,7.5a)
The final leg for Alex mirrors the first: a hours and 20a km
Adding this gives the destination point as (2.5a,27.5a)
The known distance is 33km so
27.5a=33
a=1.2 hours
And the total time 2.5a=3 hours.

The drawing makes a nice parallelogram with the short diagonal for the return journey.
Other facts about the trip:
Each kid had to walk 9km.
Alex had to travel a total of 63km.

Posted by Jer on 2015-09-15 13:20:13