Two runners, starting at the same time, run towards each other from opposite ends of an 8 mile trail. One runner is running at a rate of 5 mph, and the other is running at a rate of 7 mph. How long after they start will the two runners meet?

r_1 = 5 mph

r_2 = 7 mph

d_1 = distance first runner has run

d_2 = distance second runner has run

d_1 + d_2 = 8 miles

t_1 = time first runner has been running

t_2 = time second runner has been running

t_1 = t_2 = t = time they will meet

d_1 = r_1 * t = 5 t

d_2 = r_2 * t = 7 t

d_1 + d_2 = 8

5 t + 7 t = 8

t (5 + 7) = 8

t = 8/12 = 2/3 hours = 40 minutes