D01-24

Official Solution:

Two buses depart simultaneously from cities M and N and travel towards each other at the same constant speed, heading towards cities N and M, respectively. After two hours of travel, they meet at point P and pass each other to continue towards their destination cities. The next day, both buses are scheduled to simultaneously depart back to their original cities at the same constant speed as the day before. However, one of the buses is delayed by 24 minutes while the other departs 36 minutes earlier than originally scheduled. If they meet 24 miles from point P, what is the distance between cities M and N?

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


Let the distance between cities M and N be \(d\) miles.

Since both buses travel at the same constant speed and leave their respective cities at the same time, they will meet each other at the halfway point. Therefore, the first meeting point P is located at a distance of \(\frac{d}{2}\) miles from both cities M and N.

Next, since the buses meet each other after 2 hours of travel, the time taken by each bus to cover the entire distance is 4 hours.

On the second day, one of the buses departs 36 minutes earlier than scheduled, while the other bus is delayed by 24 minutes. As a result, the first bus travels alone for 1 hour and covers a distance of \(\frac{d}{4}\) miles. Therefore, when the second bus begins to move, the distance between the two buses is \(d - \frac{d}{4} = \frac{3}{4}*d\) miles.

Since both buses travel at the same constant speed, they will meet each other at the halfway point of the remaining distance of \(\frac{3}{4}*d\) miles, hence \(\frac{3}{8}*d\) miles from the city that the delayed bus departed from.

Given that this meeting point is 24 miles from point P, we have the equation \(\frac{d}{2} - 24 = \frac{3}{8}*d\).

Solving this equation gives us \(d = 192\) miles.


Answer: E
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M------------------------------N
Bus-A Bus-B
Both at constant equal speed meet at point P which is 2 hrs away so total time is 4 hours for each bus
Now on Second day Bus-A departs 36 mins early and Bus-B departs 24 minutes late
that means when A traveled for 1 hour, B just left N
1hr.
M------|-----------------N
A B

Now time left is 3 hrs. Both travel at same constant speed so they will meet after 1.5 hrs
In this time A has traveled for 2.5 hours (.5 hour more than yesterday which represents 24miles.)
So 4 hours = .5 * 8 is equal to distance of 24*8 = 192miles

The buses are travelling at the same speed. One bus is at the point 0.75d when the other bus starts. So, the distance between them is 0.75d and they are travelling at the same speed. Therefore, they will meet halfway through the journey. Hope that helps.

its given that 1 bus starts 36 minutes late and the other 24 minutes early which means effectively 1 bus had head start of 1 hr. Since they are traveling at same speed and they meet 24 miles from where they met yesterday then the bus which had head start of 1 hr traveled 24 miles in 1 hr. Thus its speed is 24 miles. The speed of 2nd bus is also 24 (since both bus travels at constant speed)..
Thus both bus travels for 4 hrs hence distance traveled by 1 bus =24*4

Total distance= 2*24*4= 192.. Answer

Really good question. Here's my approach:

Suppose both bus travels with V mph both days (as same constant speed).

Day1:
Bus A: Distance covered in 2 hours = 2v
Bus B: Distance covered in 2 hours = 2v
Total distance between city M and N = total distance covered by Bus A + Bus B = 2v + 2v = 4v.

Day2:
We need to account for delay to get proper time here. Bus A is delayed by 36 minutes, so it travels 36 minutes less. Bus B leaves 24 minutes early, so it travels 24 minutes extra. So we can say, bus B travels for 36 (delay of bus A) + 24 (early of bus B) = 60 minutes = 1 hour extra. So let t = Bus A travel time, t+1 = Bus B travel time.

Now, distance travelled by Bus A: vt
distance travelled by Bus B: v(t+1)

Total distance between City N and M = combined distance travelled of both bus A and bus B = vt + v(t+1).

Since total distance remains same on both days, we can say vt + v(t+1) = 4v. Solving this, we get t = 3/2 = 1.5 hours.

Now, we are given that both bus meet 24 miles from where they met yesterday (point P). Think logically, the bus who travels more (has more t+1 as V constant) will naturally travel 24 miles extra. And the bus who travels less, will travel 24 miles less. So Bus A travels 24 miles less than what it travelled on Day 1 (2v), and Bus B travels 24 miles extra than what it travelled on Day 2 (2v). Take any:

2v - 24 = v*3/2. Solving for V = 48.
2v + 24 = v*(3/2 + 1). Solving for V = 48.

So we got V = 48. Total distance = 4V, or V*3/2 + V(3/2 + 1). Put V = 48 in any. We get distance = 192 miles.

Your Kudos keeps us motivated to post more answers. Thanks!!

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

I have edited the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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I had a different approach to the solution that is a bit more visual/logical but I don't think it is better or more reliable when you are under pressure. I guess it may work better for some.

1. We know that on day 2, the buses have started moving with a 1-hour difference.

2. This difference results into the meeting point moving by 24 miles. This point intrigued me and I immediately wanted to start approaching the question from this perspective to find out what speed they were moving at, and at what distance the cities are from each other.

3. If the distance is 24 miles away from the original point and one bus was driving an extra hour, does it mean they are moving 24 miles an hour? I felt that was a bit slow and unrealistic so likely it has to be more... and I think it is because we have 2 buses moving towards each other, so they are actually covering 2x the distance when moving towards each other, which means that if in 1 hour they covered 24 miles, they were actually driving 48 miles per hour (24x2). Honestly if the speed was more realistic, I may not have thought about the double speed effect.

4. We know they drove for 4 hours. 4x48 = 192

Curious if anyone else solved this way.
-B
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