Question 1208987
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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  {{{1/9}}}  of a mile per minute.
                     
                     Mike starts one minute after Dan and runs with the speed  {{{1/6}}}  of a mile per minute.


Let t be the time after Dan starts.


The distance formula for Dan  is  D(t) = {{{t/9}}}  miles.

The distance formula for Mike is  M(t) = {{{(t-1)/6}}}  miles.


Mike passes Dan when the distances are the same

    {{{t/9}}} = {{{(t-1)/6}}},


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.    <<<---=== <U>ANSWER</U> to (b)


Mike will pass Dan at the distance  {{{(t-1)/6}}} = {{{2/6}}} miles = {{{1/3}}} miles from the start.   <<<---=== <U>ANSWER</U> to (a)
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Solved.