SOLUTION: Mike can run the mile in 6 minutes, and Dan can run the mile in 9 minutes. If Mike gives Dan a head start of 1 minute, how far from the start will Mike pass Dan? How long does

Algebra ->  Customizable Word Problem Solvers  -> Travel -> SOLUTION: Mike can run the mile in 6 minutes, and Dan can run the mile in 9 minutes. If Mike gives Dan a head start of 1 minute, how far from the start will Mike pass Dan? How long does      Log On

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Question 1208987:
Mike can run the mile in 6 minutes, and Dan can run the mile in 9 minutes. If Mike gives Dan a head start of 1 minute, how far from the start will Mike pass Dan? How long does it take?

Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
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Mike can run the mile in 6 minutes, and Dan can run the mile in 9 minutes.
If Mike gives Dan a head start of 1 minute,
(a) how far from the start will Mike pass Dan?
(b) How long does it take?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

So, in this problem, Dan  starts first and runs with the speed  1%2F9  of a mile per minute.
                     
                     Mike starts one minute after Dan and runs with the speed  1%2F6  of a mile per minute.


Let t be the time after Dan starts.


The distance formula for Dan  is  D(t) = t%2F9  miles.

The distance formula for Mike is  M(t) = %28t-1%29%2F6  miles.


Mike passes Dan when the distances are the same

    t%2F9 = %28t-1%29%2F6,


or, cross-multiplying

    6t = 9(t-1)  --->  6t = 9t - 9  --->  9 = 9t - 6t  --->  9 = 3t  --->  t = 9/3 = 3 minutes.


So, Mike passes Dan in 3 minutes after Dan starts, or in 2 minutes after Mike starts.    <<<---=== ANSWER to (b)


Mike will pass Dan at the distance  %28t-1%29%2F6 = 2%2F6 miles = 1%2F3 miles from the start.   <<<---=== ANSWER to (a)

Solved.