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?
+130085 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
