Aptitude - Time and Distance - Discussion

Let me try it in my way
A Particular Distance is travelled by the Bus at 54kmph

When stoppages are included, 54 kmph turns to 45kmph in order to compensate the increase in time.

Let Distance = D, time = t, increase in time = x

D = 54 * t ---> (1)
D = 45 * (t + x) ---> (2)

Equating (1) & (2)

54t = 45 (t+x).
54t = 45t + 45x,
9t = 45x,
9t/45 = x,
(1/5) t = x,
t = 5x.


We need to find x for 1 hr (60 mins)
i.e t+x = 60.
5x+x = 60.
6x = 60.
X = 10.

(Some friends are getting 12mins instead of 10 mins because they are taking (t = 60 min) but (t +x = 60) is the correct one.

Here, the confusion is how come 45/54?

However, the main thing to be noted is that the bus travels at 54kmph in both cases. However, due to stopping for some time, the average speed becomes 45 in case 2. Whereas in the first case, the 54kmph average is the same as its speed because it never stops.

So considering one hour of time, the bus itself covers 45km instead of 54km. We have a difference of nine kilometers in the distance traveled. But we cannot used the average speed of 45 since it travelled at 54 irrespective of the case.
So 9/54 becomes 1/6 hours or 10 minutes.

Given in the question to find out the number of minutes the bus stops per every hour.

So let us consider that the bus travels for 1 hr. The distance travelled without stopping = 1 * 54 = 54 kms.

Let the bus stops for x mins in 1 hr. So the 54 kms are covered in 1 + (x/60) hrs (converting x mins to hrs).

So, 54 = 45 * (1 + (x/60) ) => x = 12. Now the total time taken = 1 hr + 12 min = 72 min.

This means that the bus stops for 12 min in every 72 mins. We have to calculate the time the bus stops in 60 mins.

So, (12 * 60) / 72 = 10 mins.

As speed of bus without stoppage is 54 kms per hour (i.e) in 1 hr bus travels 54 kms.(as Distance = speed *time).

Similarly as speed of bus with stoppage is 45 kms per hour (i.e)in 1 hr bus travels 45 kms(as Distance = speed *time).

Comparing the cases we can see that bus with stoppage covers 9 km less than the bus without stoppages.

Now we know that bus covers,

54 kms in 60 mins (1 hr)
1km in 60/54 mins
9 kms ---------------------- 60/54*9 = 10mins.

So with original speed of 54 km/h, bus stops for 10 mins.

To all those who have doubt why 54 kmph is considered. The problem is to find how many minutes does the bus stops in stoppages.

54 kmph means it travels 54 km per a single hour (i.e 60 min) excluding stoppages.

Including stoppages speed reduced to 45 km per a single hour. It means per one hour due to stoppages there is 9 km loss per a single hour.

We should take 54 kmph because we should know what time it takes for bus for a km. So 54 km-->60 min then 1 km-->60/54 min.

For 9 km--> (9*60) /54 = 10 min.

Bus A travels 54km in 60 min that means to travel 1km it takes(60/54) 1.11min.
Bus B travels 45km in 60min with stoppages.

Here, the speed of both the buses are same but bus B gives stoppages that's it.
i.e. both A & B bus covers 1 km in 1.11min only. So if bus B takes 1.11min to travel 1km then to cover 45km it will take 45 * 1.11 = 50min(actually travelled time without stoppages ).

So, the time T = travelled time with stoppages - travelled time without stoppages.
T = 60 - 50 = 10 mins.

Speed is 54 km/hr, here clarity is speed is never reduced.

So in one hour, it covers 54 km without stopping, now since it has stopped it covered only 45 km at speed of 54 km/hr. had it not stopped it would have covered 54 kms, but it stopped hence it covered only 45 km.
Distance not covered is 54 - 45 = 9kms.
Now the time it would have taken to cover 9kms is (time = distance/speed)
Time reqd to cover 9 km @ 54km/hr = 9/54 = 1/6 hours i.e 10 minutes.

The average speed (including stoppage) is 45 km/hr.
Distance = (rate)(time).
So, in ONE hour, the distance traveled = (45)(1) = 45 km.

The average speed (excluding stoppage) is 54 km/hr.
At this speed, HOW long will it take the bus travel 45 km?
time = distance/rate.
So, time = 45/54.
= 5/6 hours,
= 50 minutes.

At a speed of 54 km/hr, the bus can travel 45 km every 50 minutes.
So, it must be motionless for the other 10 minutes.

D = S*T.
D = 54/1 * T ....(1) T is time taken by bus without stoppages.
D = 45/1 *( T + S ) ...(2) where S is time taken by stoppages.
By (1)& (2)

54*T = 45*T + 45*S
54*T - 45*T = 45*S
9*T = 45*S or 9/T = 45/S so ratio of T/S = 45/9

T/S = 5/1 {T=time ,S= time of stoppage }

T:S ::5:1 shows that

In 6 minute(5+1) bus takes 1 minutes for stoppage.
So In 60 minutes bus will take 10 minutes stoppage.

There is another easier method.

54 km is traveled in 1 hr
1 km------------- 1/54 hr
45 km---------- 1/54*45= 5/6 hrs.

Now, if we subtract the times, i.e 1 - 5/6 = 1/6 hr (we are subtracting this because 45 km should have been traveled in 5/6 hrs and not 1 hr. Hence, the time taken is more so subtracting, we get the no. of minutes the bus stopped).

converting to minutes 1/6 * 60 = 10 min.