Rate Problem

A bus from city M is traveling to city N at a constant speed while another bus is making the same journey in the opposite direction at the same constant speed. They meet in point P after driving for 2 hours. The following day the buses do the same trip again at the same constant speed. One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?

A. 48
B. 72
C. 96
D. 120
E. 192

Let:
P1 = the distance traveled by EACH bus on Day 1.
P2 = the distance traveled by the LATER bus on Day 2.
d = the total distance between M and N.

The two buses travel at the SAME SPEED.
Thus, when they travel toward each other, each bus travels HALF the distance between them.

Day 1:
When the two buses travel toward each other, each travels (1/2)d.
Thus:
P1 = (1/2)d.
Since each bus travels for 2 hours, the total time required to travel the d miles between M and N = 2+2 = 4 hours.

Day 2:
The earlier bus leaves 36 minutes AHEAD of schedule, while the later bus leaves 24 minutes BEHIND schedule.
Since 36+24 = 60, the earlier bus travels for 60 minutes -- in other words, for 1 hour -- by itself.
Since it takes 4 hours to travel the d miles from M to N, in 1 hour -- 1/4 the time -- the earlier bus travels (1/4)d miles.
Then, when the two buses travel toward each other, each travels 1/2 of the remaining (3/4)d miles.
Thus:
P2 = (1/2)(3/4)d = (3/8)d.

Result:
P1 - P2 = (1/2)d - (3/8)d = (1/8)d.
Since the distance between P1 and P2 is 24 miles, we get:
(1/8)d = 24
d = 192.

The correct answer is E.

Uva@90 wrote:A bus from city M is travelling to city N at a constant speed while another bus is making the same journey in the opposite direction at the same constant speed. They meet in point P after driving for 2 hours. The following day the buses do the return trip at the same constant speed. One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?
A. 48
B. 72
C. 96
D. 120
E. 192

Ans E

Alternate approach is to PLUG IN THE ANSWERS, which represent the total distance.
Let E = the earlier bus on the 2nd day and L = the later bus on the 2nd day.

1st day:
E and L start at the same time and travel at the same speed, so each covers 1/2 the total distance between them.

2nd day:
E leaves 36 minutes early, while L leaves 24 minutes late -- a difference of ONE HOUR.
Implication:
E travels for 1 hour ON ITS OWN, then E and L WORK TOGETHER to cover the remaining distance between them.
Since E and L each travel at the same speed, each will travel 1/2 the remaining distance between them.

Required result:
The meeting point on the 2nd day must be 24 miles from the meeting point on the 1st day.
Thus, when the correct distance is plugged in, E must travel 24 miles MORE on the 2nd day than on the 1st day.

Answer choice C: 96 miles
1st day:
Since E and L each travel 1/2 the distance, the distance traveled by E = 96/2 = 48 miles.
Since each bus travels 48 miles in 2 hours, the rate for each bus = d/t = 48/2 = 24 miles per hour.

2nd day:
In the first hour, the distance traveled by E = r*t = 24*1 = 24 miles.
Since E travels 1/2 the remaining distance, the remaining distance traveled by E = (96-24)/2 = 36.
Total distance traveled by E on the 2nd day = 24+36 = 60 miles.

Result:
(E's 2nd-day distance) - (E's 1st day distanc)e = 60-48 = 12 miles.
Since the required difference is 24 miles -- TWICE the result here -- the correct answer choice must be TWICE as great as answer choice C.

The correct answer is E.

Thanks a ton MITCH

Regards,
Uva.