APPROACH #1: Use shrinking gap properties
Let d = the distance between X and Y

A can travel the entire distance in 4 hours, which means A's RATE = distance/time = d/4
B can travel the entire distance in 6 hours, which means B's RATE = distance/time = d/6
So, the rate at which the gap between A and B shrinks = d/4 + d/6 = 3d/12 + 2d/12 = 5d/12

At 9am, the distance between A and B is d.
Over the next hour, the distance person A travels = (rate)(time) = (d/4)(1) = d/4
So, at 10am, the distance between A and B = d - d/4 = 3d/4

Time for gap to shrink to zero = distance/(shrink rate) = (3d/4)/(5d/12) = (3d/4)(12/5d) = 36d/20d = 36/20 = 9/5 = 1 4/5 hours = 1 hour and 48 minutes

So the time at which they met = 10:00 am + 1 hour and 48 minutes = 11:48 am

Answer: C


APPROACH #2: Start with the word equation
Let d = the distance between X and Y

A can travel the entire distance in 4 hours, which means A's RATE = distance/time = d/4
B can travel the entire distance in 6 hours, which means B's RATE = distance/time = d/6

Since A starts traveling 1 hour before B starts traveling, we know that B's travel time is 1 hour less than A's travel time
So, if we let t = A's travel time, it must be the case that t - 1 = B's travel time.

When A and B meet, the SUM of their distances traveled must be d.
So, we can write the following word equation: (A's distance traveled) + (B's distance traveled) = d

Distance = (rate)(time), which means we can substitute values to get: (d/4)(t) + (d/6)(t - 1) = d
Eliminate the fractions by multiplying both sides of the equation by 12 to get: (3d)(t) + (2d)(t - 1) = 12d
Divide both sides of the equation by d to get: 3t + 2(t - 1) = 12
Expand and simplify the left side: 5t - 2 = 12
Solve: t = 14/5 = 2 4/5 hours = 2 hours and 48 minutes

Since t = A's travel time, we know that the two travelers meet 2 hours and 48 minutes after A started traveling.
9:00 am + 2 hours and 48 minutes = 11:48 am

Answer: C