Comparing Running Rates

Date: 01/20/2005 at 18:50:05
From: Michael
Subject: The race...

Jack and Jane ran the 100 yard dash.  When Jane won, Jack was 10 yards
behind her.

In the next race, Jane started 10 yards behind the starting line, and
both Jane and Jack ran at the same rate as before.  Who won?

Reasoning tells me the second race should be a tie, but the book says
it isn't.  Why?  What would it take to run a tie race?

Date: 01/20/2005 at 19:59:36
From: Doctor Jeremiah
Subject: Re: The race...

Hi Michael,

While Jane runs 100 yards, Jack only runs 90.  So Jack can only run 9 
yards for every 10 that Jane runs.  If Jane is made to run 10 more 
yards (total of 110 yards) then Jack will run an extra 9 yards (total
of 99 yards), so when Jane finishes the race the second time Jack will
have only gone 99 yards.  He only loses by a yard instead of 10 yards
this time, but he still loses.

In general, Jack only covers 90% of the distance Jane does.  So in the
second race Jane runs 110 yards and Jack covers 90% of that which is
99 yards.  For it to be a tie, 90% of Jane's distance must equal 100
yards.

Let's represent Jane's distance as "d" and do some calculations:

    90% of d = 100
  90/100 * d = 100
           d = 100 divided by 90/100
           d = 100 * (100/90) = 10000/90 = 111 and 10/90 yards.

So if Jane is made to run 111 and 1/9 yards while Jack runs 100, it
will be a tie.  If she has to run more than 111 and 1/9 while he runs
100, she will lose.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/