D01-24

Question

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

Show Answer

Official Answer

E

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

Bunuel · Accepted Answer

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

Show Answer

Official Answer

E

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

sanket1991 · Answer

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

kritiu · Answer

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.

qw1981 · Answer

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

gravito123 · 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!!

nikhp · Answer

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

Krishna7568 · Answer

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

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

bb · Answer

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

BottomJee · Answer

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

kop18 · Answer

Hi  Bunuel

I have a doubt in this question.....why is d/4 (the extra distance covered by the bus that left early) not equal to 24 Miles? Both the buses would ideally travel d/2 in 2 hrs, since this bus had a head start of an hour it covered   some more miles = x    than the other bus that started late. Why is x not equal to 24? Shouldn't that extra 1 hour account for the extra 24 miles travelled by one of the buses?

In this case the distance would be d/4= 24mile =>> d =96 miles?

Bunuel · Answer

The early bus, in its one-hour head start, covers an extra distance which we can call "x". This "x" is a quarter of the total distance between the two cities.

When both buses eventually meet, they're 24 miles away from point P. This 24 miles doesn't represent the full extra distance "x" traveled by the early bus. Instead, it's only half of "x", because both buses are moving towards each other, effectively halving the advantage the early bus had.

So, while the early bus travels an extra distance equivalent to a quarter of the total distance in its head start, the point where they meet being 24 miles from point P shows that the extra advantage it maintains when they meet is only an eighth of the total distance, which is 24 miles.

Hope it's clear.

varad98 · Answer

I was almost there with this method, but made mistake of assuming speed of bus at 24 miles & I marked 96 miles as answer. Thanks for providing additional insight into this.

ankitk241997 · Answer

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

Gemmie · Answer

D1:  
 + Two buses depart simultaneously from cities M and N and travel towards each other at the same constant speed (s miles/hour), 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 distance covered by each bus: 2s
=> Distance between M&N: 2 * 2s = 4s

D2: 
 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.

=> Difference between the start time: 24 + 36 = 60 min = 1h
=> One bus travel 1h earlier than the other

=> The distance the bus complete together (excluding the 1hour headstart): 4s - s = 3s
=> The distance each bus covers (excluding the 1hour headstart):  \frac{3s}{2 }

They meet 24 miles from point P  =>  24 = 2s - \frac{3s}{2} = \frac{s}{2} 
=> s = 48
=> Distance between M&N: 4s = 4 * 48 = 192

Rohanx9 · Answer

I think i think a bit mechanically about this....

Let the speed of both buses be  v  and the distance between cities M and N be  D .

Day 1:  
 The buses meet at point P after traveling for 2 hours, so:
D = 4v (Equation 1)

Day 2:  
 One bus departs 36 minutes early (0.6 hours), and the other departs 24 minutes late (0.4 hours). This creates a 1-hour head start. The buses meet 24 miles from point P. 
 
  The early-departing bus travels for (T + 1) hours and covers half the distance (D/2) plus 24 miles:
 v(T + 1) = D/2 + 24 (Equation 2)   
  The late-departing bus travels for T hours and covers half the distance minus 24 miles:
 vT = D/2 - 24 (Equation 3) 
   
   Solve for  v :   
 Subtract Equation 3 from Equation 2: 
 
v(T + 1) - vT = (D/2 + 24) - (D/2 - 24) 
   v = 48

Solve for  D :

Substitute v = 48 into Equation 1:
D = 4v = 4(48) = 192

phmahi1997 · Answer

I like the solution - it’s helpful.

kouassidrissa047 · Answer

I like the solution - it’s helpful.

AswinSundar · Answer

I like the solution - it’s helpful.

D01-24

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards