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

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This is a continuation of article on tricks to solve questions related to time and distance in quantitative aptitude examinations. In this part of tutorial we’ll learn about some of the important concepts for solving problems related to boats and streams, escalators and journey with stoppage.

Objectives

  • Boats and streams
  • Escalators
  • Journey with stoppages
  • Speed of moving bodies as per the speed of sound
Aptitude questions related to boats and streams/escalator tests your skills in following areas;
  • Understanding and visualizing the situation.
  • Effective practical application of the concept of relative speed.

Important Terminologies

Speed of boat in still water (b): Speed of the boat in the still water is the real speed of the boat without any additional supportive or resistive forces.

Speed of current/ stream / wave (c): This is the speed of water flow.

Downstream speed (v): If the boat is moving along with the direction of the flow then the effective speed of the boat is called downstream speed.
Downstream speed = sped of the boat in still water + speed of the stream; `v = b + c`

Upstream speed (u): If the boat is moving against the direction of the stream, then the effective speed of boat is called the upstream speed.
Upstream speed = speed of boat in still water - speed of the stream; `u = b - c`

Shortcuts and formula
`(u + v)/2 = ((b - c) + (b + c))/2 = b`
`-> "Speed of boat in still water" = ("upstream speed" + "downstream speed")/2`

`(u - v)/2 = ((b - c) - (b + c))/2 = c`
`-> "Speed of stream" = ("downstream speed" - "upstream speed")/2`

A very useful illustration of the relationship between basic parameters

As per the above explanations, it is easy to observe that the Upstream Speed (u), Speed of the boat in still water (b) and the Downstream speed (v) are the three consecutive terms in an Arithmetic Progression in the mentioned increasing order, with a common difference which is the Speed of the Stream (c). image

`"Upstream Speed " + " Speed of Stream " = " Speed of Boat in Still Water"`
`"Speed of Boat in Still Water " + " Speed of Stream " = " Downstream Speed"`

Escalators

This is most repeating type of questions in quant area of CAT, MAT and CMAT exams. Concept of escalators is same as the concept of downstream and upstream.

Let the speed of a person when he is walking with the escalator (both moving up or down) ` = s` and speed of the moving escalator `= e`

Case I: An escalator is moving up at a speed of 'e'

Effective speed of a person who is moving upon escalator and walking with the escalator
` = "Normal speed of the person" + "speed of escalator"`
`= "Normal speed of the person" + "speed of escalator"`
`= s + e`
If he is moving down against the movement of the escalator
`"Then his effective speed"``= "Normal speed of the person" - "speed of escalator"`
`= s - e`

Case II: An escalator is moving down at a speed of 'e'.

If the person is walking up against the direction of escalator, then his effective speed
`= "Normal speed of the person" - "speed of the escalator"`
`= s - e`
If he is walking down on escalator with the downward moving escalator, then his effective speed
` = "Normal speed of the person" + "speed of escalator"`
`= s + e`

Concept review questions

Speed of a boat in still water is 20 kmph. If it's upstream and downstream speeds are in the ratio of `4:5`. Find the speed of the stream?
Speed of boat in still water `= 20 kmph`
Let the speed of stream `= c kmph`
Upstream speed `= (20 - c) kmph`
Downstream speed `= (20 + c) kmph`
`"Upstream speed"/"Downstream speed" = (20 -c)/(20 + c) = 4/5`
On solving this, we'll get `c = 20/9 = 2.22 kmph`

Alternate Method:
`"Speed of boat in still water"/"Speed of stream" = (u + v)/(u - v)`
`-> 20/c = (5 + 4)/ (5 - 4)`
`-> c = 20/9 = 2.22 kmph`
A boat goes 24 Km upstream in 48 minutes. If the speed of the stream is 5 kmph then find the downstream speed of the boat?
Upstream speed `= 24 xx 60/48 = 30kmph`
Speed of current `= 5 kmph`
Speed of boat in still water `= 30 + 5 = 35 kmph`
Downstream speed `= 35 + 5 = 40 kmph`
An escalator of length 32 meters is moving upward. When Abhijith is walking on the escalator at a constant speed, he covered escalator from bottom to top in 10 seconds. And he walks down against the direction of the escalator with the same speed, travelled from top to bottom of the escalator in 16 seconds. Find the speed of Abijith?
Length of escalator `= 32 m`
Let the speed of Abijith `= s "m/s"`
And the speed of escalator `= e "m/s"`
Time taken by Abijith for moving upward `= 10 sec`
`s + e = 32/10 = 3.2 "m/s" -> (1)`

Note that the speed of Abijith is more than that of escalator, and then only he can move against the escalator and reach down.

Time taken by Abijith for moving downward `= 16 sec`
`s - e = 32/16 = 2 "m/s" -> (2)`
From equation (1) and (2), `s = 2.6 "m/s"`
Hence speed of Abhjit is `2.6 "m/s"`
John can row up from a point A to point B in a river at a speed of 8 kmph. Once he reached B, immediately start his return trip from B to A. Find his average speed of the entire trip if the speed of the stream is 2 kmph.
Let the speed of John in still water `= s kmph`
Upstream speed `= s - 2 kmph = 8 kmph`
`-> s = 10 kmph`
Downstream speed `= s + 2 = 12 kmph`
Average speed `= (2 xx 8 xx 12)/(8 + 12) = 9.6kmph`
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