Aptitude - Time and Distance - Discussion

Consider there are two different buses for each case going through the same route, one stops and the other does not.

1. The time taken by the length of the journey is NOT necessary to calculate, eventually question is only asking for a given period of an hour, which means the distance would be different between the 2 cases.

2. The last line of the question says "For how many minutes does the bus stop per hour", this means stopping time is a part of "an hour",
and the two cases are as follows:

i) 45 km/hr=> 60 min = stopping time + travel time.
ii) 54 km/hr => 60 min = travel time.

3. In 1 hr, the distance covered in the first case is less, as the travel time is < 60 min and accordingly the speed is less(=45 km/hr). d1 = 45 km/hr * 1hr = 45 km.

Similarly, in 1 hr, the distance covered in the second case is more, as the time travelled is =60 min and therefore the speed is higher (=54 km/hr).
d2 = 54 km/hr * 1hr.
= 54 km.

4. Now, d2 - d1 = 54 - 45 = 9 km is the extra distance that bus 2 travels in 1 hour, which bus 1 could have travelled if it had no stopping time.

Now, stopping time can be easily calculated by determining how much extra time bus 2 takes to travel 9 km at its speed.

Stopping time = Distance/Speed = (9 km/54 km/hr) = (1/6) hr = 10 min.

I don't know whether my answer would satisfy but here are they to help, (though lenghty).

Somewhere asking why one hour is considered or 54kmph - 45kmph = 9kmph speed how it is 9km?

Ans: Read question completely it asks within one hour. So we get it by calculating for one hour.
(this is same reason why answer is not 12 if you consider 45 you will get 12 but it is wrong)

Second question is why 54 and not 45 its because 9km was not travelled in 45 its included (I'm not referring to stops included) in 54 so what's the time taken for that so we calculate the time for that extra 9km travelled that was done with 54 and not with 45.

Now distances with speed 54kmph and 45kmph within one hour (mentioned in the question) is 54km and 45km respectively. Now it could have travelled 9km additional if stops excluded (54km - 45km).

(calculating time taken to travel is same as calculating time spent for stoppages).

As its asking in one hour and answer are in minutes (mentioned in the question) we use 60minutes.

For 1km it takes {60min/54km = 1.11minutes per km. Now for 9km it takes 9km * 1 11111min/km = 10minutes}.

This is why people wrote : time= ( (60/54) * 9) =10.

Answer 12 is not correct.

@Ravi.

We have been given speeds with stoppages and without stoppages,i.e., 45KM/h(with stoppages) and 54 Km/h( without stoppages).

Now, the question wants us to find that how much time is consumed in those stoppages.

Here the value of speed is based upon the time it took to reach a particular distance.

So naturally with stoppages the time will be more so for same distance the average speed is reduced i.e., 45Km/h.

Without stoppages, speed will be 54km/h.

Now consider the time to be one hour.Why one hour, because question wants us to find the minutes the bus stop per hour. Per hour.

So what is the distance for One hour at 54km/h and 45km/h?

Distance = 54KM and 45KM.
Now, what is the difference between these two?
9Km.
This is done to find what stoppages cost in term of distance between two buses.

Now, why 54Km?
Since you already know the difference in distance due to stoppages you need to use the speed with which the bus is covering that stoppage distance.

So,
9/54 = 1/6hr.
To convert this into minute just multiply it with 60 and you will have a number of minutes.,i.e., 10 Mins.

Is it clear now Ravi?

See I will reframe the question to understand it in a better way.. Bus has started from point A. there are stops in between till point B so its speed is less i.e. 45km/hr. Now after this point there are no stops so it increases its speed to 54km/hr to reach point C. We have to find the time lapse of distance covered between these two speeds. lets find the distance 1st.
s=d/t.
let's say time taken be 1hr. so acc to this distance covered till point B is 45km
n from B to C is 54km
So, the distance gap is of 9km.

Now the time lapse for this distance we have to find. the extra distance from B to C is covered with speed of 54km/hr.
Thus, t= 9/ 54 is 1/6 hr.
So 1/6hr * 60 = 10 mins.

Suppose bus will take t hours to cover certain distance, if there are stoppages, as we know speed = d/t = = 45kmph.

Total distance covered by bus = speed*time = 45t....equation(1).

If bus did not stop at those stoppages, where stoppage time was s hours, then bus actually saved s hours to cover same distance.

It means that due to travel time of (t-s) hours, the speed was increased, as we know speed = d/(t-s) = 54kmph.

Total distance covered by bus = speed*time = 54kmph*(t-s) = 54t - 54s...equation(2).

From equation (1) and (2),

54t-54s = 45t.

=> 54t-45t = 54s.
=> 9t = 54s.
=> 9t/54 = s.
=> s = t/6.

If t = 1 hour = 60 minutes,

Then, s = 60/6 = 10 minutes.

There's a formula for this type of ques:

Speed of bus (excluding stoppage) - Speed of bus (including stoppage) * time/ speed of bus (excluding stoppage).

If you put values in this formula:
Speed of bus without stoppages= 54-45 = 9km/hr,
Time = 1 hr.
So, distance without stoppages = 9 *1= 9 km.

[(54-45) / 54] * 60 = 10.

We have taken the difference of speeds of excl and incl stoppages because we want distance without stoppages. We have taken 54 as denominator because the bus has stopped. We have multiplied it by 60 because the options are in minutes and we were getting answer in hours. If we had options in hours, so we won't have needed to multiply 60.

The LCM approach here makes our work easy. This can even be applied in problems on pipes and cisterns, work and time, etc.

So, first, take the LCM of 54 and 45 to be the distance covered i.e. 270 km.

The time taken without stoppage is 270/54 = 5 hrs. The time taken with stoppage is 270/45 = 6 hrs. This implies that out of 6 hrs, 1 hour accounts for the stoppage because 6-5 = 1.
Next, apply the unitary method.

6hrs ---------- 1 hr of stoppage.
1 hr ----------- (1/6)*60 = 10 mins stoppage.

Analogously, in case of pipes and cisterns, the lcm of the time taken by the pipes to fill/empty can be taken as the total capacity of the cistern.

@All.

As per my knowledge, to solve this question, we can use the concept of average speed.

Let's denote the time the bus stops per hour as (T) minutes.

The speed without stoppages is 54 kmph, and the speed with stoppages is 45 kmph.

The time taken to cover the same distance without stoppages = 60 minutes.
The time taken to cover the same distance with stoppages = 60 + (T) minutes.

Using the formula for average speed: Average Speed = Total distance/Total time.

We can set up the equations:
54 = Total distance/60
45 = Total distance/(60 + (T)).
Now, solve for (T) minutes to find out how many minutes the bus stops per hour.

@All.

Consider the distance is only 54 kms. At 54 kmph it will take only 1 hr.
But since due to stoppages it takes 10 minutes extra (the time of stoppages).
So, shouldn't we get 45 kmph if we divide 54 km the distance by 1 hr 10 minutes (70 minutes)? We don't get that, of course. The bus runs at 54 and with the stoppage, rit reaches 45 km. 9 km is remaining. The average speed with stoppages was 45kmph. the remaining distance of 9 km has to be completed by that average speed as there are no more stoppages to get the correct time taken by the bus and not by 54kmph.

So, it will work out to be 12 minutes, not 10 minutes.

Consider A and B two bus,

A with stoppages km/hr - 45
B without stoppage km/hr - 54

1 hour travel.
B far away from A.
that distance is 9 km.

Just think 9 km A don't stop in anywhere there speed is increased. Because A's is the Avg speed 45 km/hr.

Here we can't find the stoppage from A's speed
Go next option B,
from B the bus,
Easy to find out the time taken from 9 min travel that 9 min travel is A's stoppage time

So,
distance = time * speed,
54 km = 1 hour * 54 km/hrs,
if 1 km = time ? * 54km/hrs,
9 = t* 54,
1/6 = t (in hours),

In min,
1/6*60 =10min,