SOLUTION: A long-distance runner started on a course at an average speed of 7 mph. Half an hour later, a second runner began the same course at an average speed of 9 mph. How long after the

        Quick and short Physics solution

In half an hour, 1st runner is   = 3.5 miles ahead the 2nd runner.


Moving faster than 1st runner, the 2nd runner has approaching speed of 9-7 = 2 miles per hour.


Therefore, it is clear that the 2nd runner will overtake the 1st runner in 


     = 1.75 hours = 1 hour and 45 minutes.    ANSWER

        A slow Algebra solution

Let t be the time after the 2nd runner started.

So, to the catching moment, the 2nd runner moves t hours.


The 1st runner started half an hour earlier, 
so the 1st runner moved (t+1/2) hours till the catching up moment.


They cover the same distance, so we write this distance equation

    9*t = 7*(t+1/2)


Left  side is the distance covered by the 1st runner; 
right side is the distance covered by the 2nd runner.


To solve, multiply both sides of the equation by 2. You will get

    18t = 7*(2t+1)


Simplify and find t

    18t = 14t + 7

    18t - 14t = 7

        4t    = 7

          t     = 7/4 = 1 3/4  hour = 1 hour and 45 minutes.


So, the 2nd runner overtakes the 1st runner in 1 hour and 45 minutes after the 2nd runner start.


You get the same answer.