Question 1088421
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The Alan's rate is {{{D/40}}} of the distance D to the library per minute.

The Bob's rate is {{{D/20}}} of the distance D to the library per minute.


During t minutes Alan will cover the distance  {{{t*(D/40)}}}, and will be at the distance {{{D-t*(D/40)}}} from the library.

          while  Bob  will cover the distance  {{{t*(D/20)}}}, and will be at the distance {{{D-t*(D/20)}}} from the library.


They ask you: find t in the way that

{{{D-t*(D/20)}}} = {{{(2/5)*(D-t*(D/40))}}}.


Cancel D in both sides and multiply both sides by 5. You will get

{{{5*(1-t/20)}}} = {{{2*(1-t/40)}}}.


Now multiply by 40 both sides. You will get

5*(40-2t) = 2*(40-t)  ====>  200 - 10t = 80 - 2t  ====>  200 - 80 = 10t - 2t  ====>  8t = 120  ====>  t = 15 minutes.


It is your 

<U>Answer</U>.  The time under the question is 15 minutes.
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Solved.