Consider Fred and Frank as posed in Mason: Fred and Frank are two fitness fanati

Question

Consider Fred and Frank as posed in Mason: Fred and Frank are two fitness fanatics on a run from A to B. Fred runs half the way and walks the other half. Frank runs for half the time and walks for the other half. They both run and walk at the same speeds. Who finishes first? In contrast to the problem in Homework 2, there is no indication of actual distances and speeds. Use your experience solving the homework question to set up a general algebraic model. Does the same person always win or is the answer dependent on the relative speeds of running and walking? You need to justify your answers to these questions. Make sure that you can present your work in full ordinary language sentences. Justify the following statement. Whoever runs over a longer portion of the course will win the race. Use the statement to obtain a solution to the problem which does not require algebra. Francis joins Fred and Frank and teaches them to jog. Fred now runs one-third of the way, jogs one-third of the way and walks the rest, while Frank jogs for one-third of the time, runs for one-third and walks the rest. Who finishes first? Solve two ways by extending each of the methods used in 1. and 2.

Explanation / Answer

Here let both run at a speed of x miles per hour and both walk at a speed of y miles per hour and also let total distance between A and B is z miles.

Now as time = distance/speed

so total time in run by Fred = (z/2)/x + (z/2)/y = (z/2)(1/x + 1/y) ..........(i)

Again let time taken by Frank in completing z distance is t hours.

and it covers d miles in half time running, that means

t/2=d/x or d= xt/2

and then he completes remaining (z-d) miles in remaining t/2 hours with the speed of y miles per hour.

or t/2 = (z-d)/y

or t/2= (z-xt/2)/y

or t/2= (2z-xt)/2y

or ty = 2z-xt

or ty+xt= 2z

or t(x+y) = 2z

or t=2z/(x+y)

so total time taken by frank in completing the distance AB is 2z/(x+y).

Now clearly for any positive values of x,y and z,

2z/(x+y) > (z/2)(1/x+1/y)

that means FRed took less time and that is why FRED came FIRST.

And winner will be depended on the relative speeds of their running and walking. It is never necessary that the same person will win always.

This is the answer of first question.

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