Question

Two runners start running at the same time, from the same starting position. George runs a lap in 50 seconds; Sue runs a lap in 30 seconds.When will the runners next be side by side?
I know the answer is 75 seconds but i dont know how to prove it

Answer 1

Since one is running faster than the other, the fast one will overlap the slower one. So lets get some formulas down. Let a be the slow guy and b be the fast guy:
\[d_{a} = \frac{1}{50}t\]
\[d_{b} = \frac{1}{30}t\]

Answer 2

Those are the distances, we want the fast guy to be one lap ahead of the slow guy so:
\[d_{a}+1 = d_{b} \Leftrightarrow \frac{1}{50}t+1=\frac{1}{30}t\]

Answer 3

Solving for t gives us:
\[(\frac{1}{30}-\frac{1}{50})t = 1 \Leftrightarrow t = \frac{1}{\frac{1}{30}-\frac{1}{50}}\]

Answer 4

Which is 75 secs :)

Answer 5

If you need anymore explanations just ask :)

Answer 6

ok i get it!! howd you think of the fast guy being one lap ahead though?

Answer 7

Well they started at the same place at the same time. If they are going different speeds (constant speeds), there is no way they would ever meet again...unless the track loops and the faster guy catches up to the slow one.

Answer 8

that makes sense! do you mind if I ask you another question? it sounds like a trick question but im not sure if it is:/

Answer 9

Sure thing, fire away :)

Answer 10

K!
Player A has a higher batting average than player B for the first half of the baseball season. Player A also has a higher batting average than player B for the second half of the season. Is it necessarily true that player A has a higher batting average than player B for the entire season?

Answer 11

im racking my brain to come up with a counter example (to see if its false), but im not getting one. I believe its true.

Answer 12

if x1 and x2 are player A's batting averages for the 1st and 2nd half, and y1 and y2 are the same for player B, then we have:\[x_{1} > y_{1}, x_{2} > y_{2} \Rightarrow x_{1}+x_{2} > y_{1}+y_{2}\]
Now those arent averages, but you will divide both sides by the same number to get the average. So that should be true no matter how many times they swing.

Answer 13

That's what I was thinking but now that reassures me! Did u mind if I asked two more??