A Strange Race

For every 100 metres John runs, David will run 99 metres. So, at the end they both will have 1 metre more to run and since John is faster, he will win!

By the first paragraph, we can conclude that if John runs a meter, David will not run as much as one meter. Thus, when John reaches 100 meter and runs a meter to reach 101 meter , David who still at 99 meter can't reach a meter needed to finish the 100 m race. John wins.

For simplicity, let's say that John finished the race in 1 minute, and let $x$ represent minutes. Then the expression for David's running is $99x$ , whereas the expression for John's is $100x-1$ (we subtract 1 for the one meter difference). When $99x = 100x-1$ , that will represent the amount of time it will take for John to catch up to David. Solving the equation... $x = 1$ , meaning it would take one minute for John to catch up to David. Now, if John takes longer than one minute to reach the finish line, then he will beat David, and if he doesn't, he will not, since he couldn't catch up to David in time. So $100x-1=100$ . Solving the equation... $x=1.01$ , in other words, he has enough time to reach (and pass) David, and thus, $\boxed{John}$ reaches the finish line first.

David's speed is 99% of John's. So in the 2nd race when John runs 101m, David will run 99% of 101 or 99.99m. Hence John wins again.

Suppose that John finished the race for 1 hour. So John's speed is 100m/h, while David's is 99 m/h. In the 2nd race, the duration for John to finish the race is: = 1/100 h/m x 101 m = 1,01 h

While for David it takes: = 1/99 h/m x 100 = 1,0101 h

So, it take a little bit longer for David to finish the race.