 ## Boat and Stream Questions:

Downstream:
In water, the direction of a boat along with the stream is called Downstream.

Upstream:
The direction of a boat against the stream is called Upstream.

### Formulas:

Speed Downstream = ( u + v ) km/hr
Speed Upstream = ( u – v ) km/hr

If the speed downstream is x km/hr and the speed upstream is y km/hr then,
Speed in Still Water = 1 / 2 ( x + y ) km/hr
Rate of Stream = 1 / 2 ( x – y ) km/hr

### Type 1:

When the distance covered by boat downstream is the same as the distance covered by boat upstream. The speed of the boat in still water is x and the speed of the stream is y then the ratio of time taken in going upstream and downstream is,

#### Short Trick:

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

Example:
A man can row 9km/h in still water. It takes him twice as long as to row up as to row down. Find the rate of the stream of the river.
Sol:
Time taken in upstream: Time is taken in Downstream = 2: 1
Downstream speed : Upstream speed = 2 : 1
Let the speed of man = B, & speed of stream = S
B + S : B – S = 2/1
By using Componendo & Dividendo
B/R = 3/1, R = B/3
R = 9/3 = 3km/h

### Type 2:

A boat covers a certain distance downstream in t hours and returns the same distance upstream in t hours. If the speed of the stream is y km/h, then the speed of the boat in still water is:

#### Short Trick:

Speed of Boat = y [(t + t ) / (t – t )]

Example:
distance upstream in 6 hours. If the speed of the stream is 1.5 km/h, then the speed of man in still water is
Sol:
By using the above formulae
= 1.5 [(6+2) / (6-2)] = 1.5 * (8/4) = 1.5 * 2 = 3km/h

### Type 3:

A boat’s speed in still water at x km/h. In a stream owing at y km/h, if it takes it t hours to row to a place and come back, then the distance between two places is

Short Trick: Distance = [t*(x – y )]/2x

Example:
A motorboat can move with a speed of 7 km/h. If the river is owing at 3 km/h, it takes him 14 hours for a round trip. Find the distance between two places?
Sol: By using the above formulae
= [14 * (7 – 3 )]/2* 7 = [14 * (49-9)]/2*7
= 14*40/2*7 = 40km

### Type 4:

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

Short Trick: Distance = [t*(x – y )]/2y

Example
A professional swimmer challenged himself to cross a small river and back. His speed in the swimming pool is 3km/h. He calculated the speed of the river that day was 1km/h. If it took him 15 mins more to cover the distance upstream than downstream, then find the width of the river?
Sol:
By using the above formulae
Distance = [t*(x – y )]/2y
= [(15/60) (3 – 1 )]/2*1
= [(1/4) * 8] / 2 = 2/2 = 1 km.

### Type 5:

A boat’s speed in still water at x km/h. In a stream owing at y km/h, if it covers the same distance up and down the stream, then its average speed is

Example
Find the average speed of a boat in a round trip between two places 18 km apart. If the speed of the boat in still water is 9km/h and the speed of the river is 3km/h?
Sol:
Average speed = upstream * downstream / man’s speed in still water
Average speed = 6 * 12 / 9 = 8km/h

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