Can someone please help me understand ?

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Two friends enjoy competing with each other to see who has the best time in running a mile. Initially (before they ever raced each other), the first friend runs a mile in 7 minutes, and for each race that they run, his time decreases by 11 seconds. Initially, the second friend runs a mile in 7 minutes and 17 seconds, and for each race that they run, his time decreases by 15 seconds. Which will be the first race in which the second friend beats the first?

race number   
Oct 3rd, 2015

Thank you for the opportunity to help you with your question!

First, recall that 1 minute = 60 seconds, so 7 minutes is actually 7 * 60 = 420 seconds.

Next, represent the time for each runner on the nth race. Let T(1,n) be the time for the 1st friend on the nth race.
Let T(2,n) be the time for the 2nd friend on the nth race. We want to find the smallest value of n for which 
T(2,n) > T(1,2).

According to the problem statement T(1,n) = 420 - 11*n and T(2,n) = 420 + 17 - 15*n.

If T(2,n) > T(1,n) then 437- 15n > 420 - 11n. Now subtract 420 from both sides of the inequality, getting
17 -15n > -11n. Next, add 15n to both sides, getting 17 > 4n, Next, divide both sides by 4, getting n < 17/4.

Now 17/4 < 16/4 = 4, so n must be equal to 4.

Check T(1,4) = 420 - 44 =  376 and T(2,4) = 437 - 60 = 377 so (as expected) T(1,4) < T(2,4).

Please let me know if you need any clarification. I'm always happy to answer your questions.
Sep 21st, 2015

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