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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?
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So, in this problem, Dan starts first and runs with the speed of a mile per minute.
Mike starts one minute after Dan and runs with the speed of a mile per minute.
Let t be the time after Dan starts.
The distance formula for Dan is D(t) = miles.
The distance formula for Mike is M(t) = miles.
Mike passes Dan when the distances are the same
= ,
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 = miles = miles from the start. <<<---=== ANSWER to (a)
Solved.