Boats and Streams : Important Formulas and Shortcuts

Boats and Streams is Based On Concepts of Speed,Time And Distance. Speed of river flowing either aides a swimmers ( boats ) while travelling with the direction of stream of river and opposes it when travelling against the direction of stream of river.

Important Terms Used in Boats and Streams Problems :

Still Water : If the speed of streams of river is zero

Downstream Motion : If the motion of swimmers ( boats ) is along the direction of stream of river

Upstream Motion : If the motion of swimmers ( boats ) is against the direction of stream of river

Important Formulas Used in Boats and Streams Problems : 

Suppose If speed of boat or swimmer is x km/h and the speed of stream is y km/h then,

• speed of boat downstream = ( x + y ) km/h

• speed of boat upstream = ( x - y ) km/h

• speed of boat in still water ( x ) = 1 / 2 ( speed downstream + speed upstream ) km/h

• speed of stream ( y ) = 1 / 2 ( speed downstream - speed upstream ) 

Shortcuts to Solve Questions on Boats and Streams :

Type 1 : If speed of stream is x km/hr and a boat takes n times as long to row up to row down the river then,

Speed of boat in still water = x ( n + 1 ) / ( n+1) km / hr

Note : this is applicable only for equal distances

Type 2 : When the distance covered by boat in downstream is same as the distance covered by boat upstream. 

The speed of boat in still water is x and speed of stream is y then ratio of time taken in going upstream and downstream is

Time taken in upstream : Time taken in Downstream = ( x + y ) / ( x - y )

Type 3 : A boat cover certain distance downstream in t1 hours and returns the same distance upstream in t2 hours. 

If the speed of stream is y km/h, then the speed of the boat in still water is

Speed of Boat = y [ ( t2 + t1 ) / ( t2 – t1 ) ] km/ hr

Type 4 : A boat speed in still water is x km/hr and speed of stream is y km/hr it takes t time to travel to a place and come back again to starting position, then 

Distance between two places is t ( x2 – y2 ) / 2x km

Type 5 :  A boat’s speed in still water at x km/h. In a stream flowing at y km/h, if it takes t hours more in upstream than to go downstream for the same distance, then 

Distance is  [ t (x2 – y2 ) ] / 2y km

Type 6 : If A boat speed in still water is a km/hr and river is flowing with a speed of b km/hr,then average speed in going to a place and coming back is [ ( a + b )( a - b) / a ] km / hr

Type 7 : If a person can row d₁ km upstream and e₁ km downstream in t₁ hours and also he can row d₂ km upstream and e₂ km downstream in t₂ hours then,

• Upstream speed of person = ( d₁ e₂ - d₂ e₁ ) / ( e₂ t₁ - e₁ t₂ ) km / hr

• Downstream speed of person = ( d₁ e₂ - d₂ e₁ ) / ( d₁ t₂ - d₂ t₁ ) km / hr

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