There is no unique solution to your problem.
Let the time spent driving by man1 be 1 hours
then the time for man2 when he overtook man1 is t-1 hours.
let the speed of man2 be x mph
distance driven by man1 = 50t miles
distance driven by man2 = x(t-1) miles
but at the overtake, the both went the same distance, so
x(t-1) = 50t
xt - x = 50t
xt-50t = x
t(x-50) = x
t = x/(x-5) , clearly x > 50
let x = 60, then
t = 60/10 = 6 hours
check: when t = 6 hours,
distance of man 1 = 300 miles
time for second man = 5 hours, speed of second man = 60 mph, distance = 300
try x=70 , so t = 3.5
distance of man1 = 3.5x50 = 175 miles
time for second man = 2.5, speed of second man = 70
distance of second man = 2.5x70 = 175
notice both combination of answers work, as will as many others as you want.
We know man 1 drives 50 miles per hour
man 2 goes 55 mph
The answer differs from yours.
Please help! I'm still waiting!
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