Time and Distance Tutorial III: Boats & Streams, Escalators, Journey with Stoppages

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Journey with stoppage

Time taken per hour for stoppage
If the speed of a car with a certain minutes of stoppages is `q " kmph"` and without stoppage the speed is `p " kmph"`, then time taken per hour for stoppage `= ((p - q)/p ) xx 60 min`
`=("Difference between speeds"/"Speed without stoppage") xx 60 min`

Concept review questions

A car travels at an average speed of 50 kmph without any stoppages and with stoppage the speed of the car reduced by 20 kmph. Find the time of the rest per hour?
Speed without stoppage = 50 kmph
50 km covered without stoppage in 60 minutes. Speed with stoppage = 30 kmph
`-> 30 km` covered with stoppage in 60 minutes. 50 km covered in effective time of 60 minutes.
`:.` 30 km is covered in `(60/50 ) xx 30 = 36min`, that is with stoppage the car is travelled in an effective time of 36 minutes per hour.
Hence the time for rest = `60 min - 36 min = 24 min`

Alternate Method I:
Speed without stoppage `= 50 kmph`
Speed with stoppage `= 30 kmph`
Time spent for stoppage `= ((50 - 30)/50) xx 60 = 24 min`

Alternate Method II:
Speed without stoppage `= 50 kmph `
Speed with stoppage `= 30 kmph`
Decrease in speed `= ((50 - 30)/50) xx 100 = 40%`
Therefore stoppage `= 40%` of time
`= 40%` of 60 mins
`= 24 min`

Bhanu travels from A to B at a constant speed and B to A at the same speed, but in the return journey he took an average of 30 minutes per hour for rest. If his average speed for the entire trip is 40 kmph. Find his speed from A to B?
Speed from A to B `= P kmph`
Stoppage time `= 30 min/hr = 1/2` of an hr
I.e. the variation happened in the return speed `= 1/2` of the initial speed `= 1/2 P kmph`
Average Speed `= (2 xx P xx 1/2P)/(P + 1/2P)`
`-> 2/3P = 40 kmph`
`P = 60kmph`; ie: Speed from A to B `= 60kmph`

Speed of moving bodies as per the speed of sound (Report of Guns- problem)

It is an old pattern of questions from time and distance. Initial period of GMAT exam GMAC (the counsel which conduct GMAT exam) started to ask it is in a 700+ range bin of questions, even though the concept is very simple.

Concept:

Two guns were fired from same place at an interval of 'x' minutes. If a person who is approaching the shooting place by train/ car and hears the shooting sound at an interval of 'y' minutes, then it is possible to find the speed of the train/car.

Speed of the sound in air is `330 "m/s"` in a fixed value.
Distance covered by the train in y minutes = Distance covered by the sound in (x-y) minutes
`= 330 xx (x-y) xx 60` minutes
Speed of the train `= (330 xx (x-y) xx 60)/y = (330 xx 60 xx (x - y))/1000 xx 60/y = 1188[(x-y)/y] ` kmph

Speed of the train `= 1188[(x-y)/y] ` kmph

Concept review questions

Two bullets were fired at interval of 14 minutes from the same shooting range. While in the journey towards the shooting range, Mr. Kennedy hears the sound of both the shots at an interval of 13 mins 30 secs. If the speed of the sound is 330 m/s then at what speed Mr.Kennedy was travelling?
Time difference `= 14.00 - 13.30 = 30 s`
In 30 seconds sound covers the same distance as Mr.Kennedy covers in 13 mins 30 sec `330xx30 = 9900 m = 9.9 km`
Distance covered by Kennedy in 13.30 mins `= 9.9 km`
Speed of Mr.Kennedy `= 90 xx (60/13.5) = 44km` (13 min 30 sec = 13.5 min)
Alternate Method:
As per the above discussed rule, speed of Kennedy `= 1188[(14 - 27/2)/13.5] = 44km`
Two guns were fired from the same place at an interval of 8 minutes, but a person travelling in a train approaching the firing place hears the second shot 7 minutes 12 seconds after the first. Supposing the speed of sound travels at 330 m/s, find the distance covered by the train in the interval of two shots?
As per rule, `X = 8 min` and `Y = 7:12 min = 36/5 min`
Speed of train `= 1188[(8 - 36/5)/(36/5)] = 132 km`
Interval of shots `= 8 min`
Distance covered `= 132 xx (8/60) = 17.6km`
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