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# Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than

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Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?

May 5, 2022

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Call Ian's rate  =  R

And call the time he drove in hours  ,T

So

Han drove   at  (R + 5)  for a time of (T + 1)

And

Jan drove  at (R + 10)  for a time of (T + 2)

We have  this  equation

(R + 5) (T + 1)  -  RT  =  70

RT + 5T + R  + 5  - RT  =70

5T + R  =  65

R =  65 - 5T

So  the distance that  Ian drove =  RT  =   (65 - 5T)(T)   =  65T - 5T^2      (1)

And the distance  that Jan drove  =  (R + 10) (T + 2)  =   (65 - 5T + 10) (T + 2)  =  (75 - 5T)(T + 2)  =

75T - 5T^2 + 150 - 10T  =   65T - 5T^2  + 150    (2)

Subtract  (1)  from (2)

65T - 5T^2 + 150  = 65T - 5T^2  =   150

So  Jan drove 150 miles farther than Ian

May 6, 2022