M10-08

Question

To arrive at its destination on time, a bus should have maintained a speed of \(v\) kilometers per hour throughout the journey. However, after traveling the first third of the distance at \(v\) kilometers per hour, the bus increased its speed and covered the rest of the distance at \(1.2v\) kilometers per hour. As a result, the bus arrived at its destination \(x\) minutes earlier than planned. What was the actual duration of the trip?

(1) \(v = 60\)

(2) \(x = 20\)

(1)  v = 60 
 
(2)  x = 20

Bunuel · Accepted Answer

Official Solution:

To arrive at its destination on time, a bus should have maintained a speed of  v  kilometers per hour throughout the journey. However, after traveling the first third of the distance at  v  kilometers per hour, the bus increased its speed and covered the rest of the distance at  1.2v  kilometers per hour. As a result, the bus arrived at its destination  x  minutes earlier than planned. What was the actual duration of the trip? 
 
The bus covered  \frac{1}{3}  of the distance at  v  kilometers per hour and the remaining  \frac{2}{3}  of the distance at  1.2v  kilometers per hour.
 
Let the actual duration of the trip be  t  hours and the total distance be  d  kilometers. Then we have:
 
 t = \frac{(\frac{d}{3})}{v}+\frac{(\frac{d2}{3})}{1.2v} , which simplifies to  t=\frac{d}{v}*(\frac{1}{3}+\frac{2}{3.6}) , and finally to  t=\frac{d}{v}*\frac{8}{9} 
 
We also know that if the speed throughout the journey had been  v  kilometers per hour, the bus would have needed  \frac{x}{60}  hours more time to cover the same distance:  t + \frac{x}{60} = \frac{d}{v} .
 
Substituting  \frac{d}{v}  into the first equation, we get:  t = (t + \frac{x}{60})*\frac{8}{9} . Therefore, to find the value of  t , we need to know the value of  x .
 
(1)  v = 60 . Not sufficient.
 
(2)  x = 20 . Sufficient.

Answer: B

banerjeeankush99 · Answer

I did not understand this part:

Also we know that if the speed throughout the journey had been v kilometers per hour the bus would need x/60 hours more time to cover the same distance: t+x/60=d/v;

Please explain

Bunuel · Answer

Usual time to cover d kilometres at v kilometers per hour is (time) = (distance)/(rate) = d/v. We know that as a result of an acceleration, the bus arrived at its destination x minutes (which is x/60 hours) earlier than planned. Therefore  t+\frac{x}{60}=\frac{d}{v} .

Does this make sense?

bedarkaryashas · Answer

I think a faster way to arrive at the correct option is as follows:
V= Speed; D= Distance' T= Original time required.

Given that: (1.2V)= (2D/3)/(2T/3 - x)
We have to find: T
Since we already know that: V= D/T, we substitute this in the LHS of the above eqn
Hence: 1.2*(D/T)= (2D/3)/ (2T/3 - x) i.e. (1.2/T)= (2/3) / (2T/3 - x)
Now from the above eqn, knowing the value of x will give us the value to T

Answer: Only (B)

carouselambra · Answer

This is such a beautiful question with an even more beautiful 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.

Mdabhi1 · Answer

Hi Bunuel,

Questions say that the bus travelled first 1/3 of the distance at V km/h and statement 1: v= 60 km/h so we can infer that 1/3 distance = 60 km and total distance = 180 km.

Now the bus travelled the remaining distance at 1.2v km/h = 72 km/h so the time taken to cover the remaining distance = (180-60)/72*60 = 100 minutes. Total time taken by bus = 60 + 100 = 160 minutes.

So statement 1 alone is also sufficient. Then IMO correct answer should be D.

Please let me know if I am missing something.

Thanks

Bunuel · Answer

How did you deduce the red part? Do we know that it took the bus 1 hour to cover 1/3 of the distance?

Mdabhi1 · Answer

Hi Bunuel,

Thanks, I got the gap in my inference. It only says the first third of a distance at v km/h. So let's say maybe the bus could have travelled 120 miles at 60 km/h or 180 km at 60 km/h speed. In the first case, the total distance will be 120*3 = 360 km and in the second case, the total distance will be 180*3=540 km. Therefore, only statement 1 is not sufficient.

Is my understanding correct?

Thanks.

Bunuel · Answer

Right. We don't know how long the bus was traveling at 60 kilometers per hour.

wolfof6thstreet · Answer

I took a slightly different approach. I looked at it by a ratio of time relative to the original planned time.

Original Planned: VT=D T =D/V
1/3 Segment Time: (1/3)D / V = (1/3) (D/V) = (1/3)T
2/3 Segment = (2/3)D / (1.2)V = (5/9)T

So 1/3 Time Segment + 2/3 Time Segment = Actual Total Time

(1/3) T + (5/9) T = (8/9) T

Planned Time - Actual Time = X minutes

T - (8/9) T = (1/9)T

Therefore

(1/9) T = X Minutes

You can breakout T in the above equation when you're given V and realize that it is insufficient.

renzosm93 · Answer

Bunuel, can you explain a little bit this part please?
"We also know that if the speed throughout the journey had been vv kilometers per hour, the bus would have needed x/60 hours more time to cover the same distance".

Bunuel · Answer

Is x/60 part unclear? We havce x/60 becasue x minutes is x/60 hours.

sarkarshubham97 · Answer

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

Adit_ · Answer

Well shucks I realised there was a trick here once i ended up with two equations and 3 variables only to forget that D/v can be directly substituted as t and just ignored that. Time pressure really does things.

Advait01 · Answer

I like the solution - it’s helpful.

M10-08

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