# Aptitude::Boats and Streams

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Example 1 / 16

Man's speed with the current = 20 km/hr

speed of the man + speed of the current = 20 km/hr

speed of the current is 3 km/hr

Hence, speed of the man = 20-3 = 17 km/hr

man's speed against the current = speed of the man - speed of the current

= 17-3 = 14 km/hr

speed of the man + speed of the current = 20 km/hr

speed of the current is 3 km/hr

Hence, speed of the man = 20-3 = 17 km/hr

man's speed against the current = speed of the man - speed of the current

= 17-3 = 14 km/hr

**Ans.**
Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where,

u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where,

u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

we have to first take out the speed of man. Then, put the formula and take out man’s speed against the current.

Example 2 / 16

Let speed upstream = x

Then, speed downstream = 2x

Speed in still water =\(\frac{2x+x}{2}\)=\(\frac{3x}{2}\)

Speed of the stream =\(\frac{2x−x}{2}\)=\(\frac{x}{2}\)

Speed in still water : Speed of the stream =\(\frac{3x}{2}\):\(\frac{x}{2}\)=3:1

Then, speed downstream = 2x

Speed in still water =\(\frac{2x+x}{2}\)=\(\frac{3x}{2}\)

Speed of the stream =\(\frac{2x−x}{2}\)=\(\frac{x}{2}\)

Speed in still water : Speed of the stream =\(\frac{3x}{2}\):\(\frac{x}{2}\)=3:1

**Ans.**
Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

using the formulae, take out the speeds and find the ratio.

Example 3 / 16

Let the rate along with the current is x km/hr

\(\frac{x+3}{2}\)=5

⇒x+3=10 ⇒x=7 kmph

\(\frac{x+3}{2}\)=5

⇒x+3=10 ⇒x=7 kmph

**Ans.**
Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where,

u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where,

u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

based on the formula.

Example 4 / 16

Speed in still water = 6 kmph

Speed of the stream = 4 kmph

Speed upstream = (6-4)= 2 kmph

Speed downstream = (6+4)= 10 kmph

Total time = 90 minutes = \(\frac{120}{60}\) hour = 2 hour

Let L be the distance. Then

\(\frac{L}{10}\)+\(\frac{L}{2}\)=2

\(\frac{12L}{20}\)=2

L= 3.33 km

Speed of the stream = 4 kmph

Speed upstream = (6-4)= 2 kmph

Speed downstream = (6+4)= 10 kmph

Total time = 90 minutes = \(\frac{120}{60}\) hour = 2 hour

Let L be the distance. Then

\(\frac{L}{10}\)+\(\frac{L}{2}\)=2

\(\frac{12L}{20}\)=2

L= 3.33 km

**Ans.**
Distance= speed*time

1.Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

1.Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

take out the speeds and time and put in the formula, distance = speed*time and find the distance.

Example 5 / 16

Let the speed of Rahul in still water be x mph

and the speed of the current be y mph

Then, Speed upstream =(x−y)=(x−y) mph

Speed downstream =(x+y)=(x+y) mph

Distance = 12 miles

Time taken to travel upstream - Time taken to travel downstream = 6 hours

⇒\(\frac{12}{x−y}\)−\(\frac{12}{x+y}\)=6

⇒12(x+y)−12(x−y)=6(\({x}^{2}\)−\({y}^{2}\))

⇒24y=6(\({x}^{2}\)−\({y}^{2}\))

⇒4y=(\({x}^{2}\)−\({y}^{2}\))

⇒\({x}^{2}\)=(\({y}^{2}\)+4y)-----(i)

Now he doubles his speed. i.e., his new speed =2x

Now, Speed upstream =(2x−y) mph

Speed downstream =(2x+y) mph

In this case, Time taken to travel upstream - Time taken to travel downstream = 1 hour

⇒\(\frac{12}{2x−y}\)−\(\frac{12}{2x+y}\)=1

⇒12(2x+y)−12(2x−y)=4(\({x}^{2}\)−\({y}^{2}\))

⇒24y=4(\({x}^{2}\)−\({y}^{2}\))

⇒4\({x}^{2}\)=\({y}^{2}\)+24y----- (ii)

(Equation i× iv)=> 4\({x}^{2}\)=4(\({y}^{2}\)+4y)⋯(iii)

From Equation ii and iii, we have,

\({y}^{2}\)+24y=4(\({y}^{2}\)+4y)

⇒\({y}^{2}\)+24y=4\({y}^{2}\)+16y

⇒3\({y}^{2}\)=8y

⇒3y=8

⇒y=\(\frac{8}{3}\) mph

i.e., speed of the current =\(\frac{8}{3}\) mph=2\(\frac{2}{3}\) mph

and the speed of the current be y mph

Then, Speed upstream =(x−y)=(x−y) mph

Speed downstream =(x+y)=(x+y) mph

Distance = 12 miles

Time taken to travel upstream - Time taken to travel downstream = 6 hours

⇒\(\frac{12}{x−y}\)−\(\frac{12}{x+y}\)=6

⇒12(x+y)−12(x−y)=6(\({x}^{2}\)−\({y}^{2}\))

⇒24y=6(\({x}^{2}\)−\({y}^{2}\))

⇒4y=(\({x}^{2}\)−\({y}^{2}\))

⇒\({x}^{2}\)=(\({y}^{2}\)+4y)-----(i)

Now he doubles his speed. i.e., his new speed =2x

Now, Speed upstream =(2x−y) mph

Speed downstream =(2x+y) mph

In this case, Time taken to travel upstream - Time taken to travel downstream = 1 hour

⇒\(\frac{12}{2x−y}\)−\(\frac{12}{2x+y}\)=1

⇒12(2x+y)−12(2x−y)=4(\({x}^{2}\)−\({y}^{2}\))

⇒24y=4(\({x}^{2}\)−\({y}^{2}\))

⇒4\({x}^{2}\)=\({y}^{2}\)+24y----- (ii)

(Equation i× iv)=> 4\({x}^{2}\)=4(\({y}^{2}\)+4y)⋯(iii)

From Equation ii and iii, we have,

\({y}^{2}\)+24y=4(\({y}^{2}\)+4y)

⇒\({y}^{2}\)+24y=4\({y}^{2}\)+16y

⇒3\({y}^{2}\)=8y

⇒3y=8

⇒y=\(\frac{8}{3}\) mph

i.e., speed of the current =\(\frac{8}{3}\) mph=2\(\frac{2}{3}\) mph

**Ans.**2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

based on the formula. Double the rate and find the speed.

Example 6 / 16

Speed upstream = \(\frac{3}{\frac{20}{60}}\) = 9 km/hr

Speed downstream = \(\frac{3}{\frac{18}{60}}\) = 10 km/hr

Rate of current =\(\frac{10−9}{2}\)=12 km/hr

Speed downstream = \(\frac{3}{\frac{18}{60}}\) = 10 km/hr

Rate of current =\(\frac{10−9}{2}\)=12 km/hr

**Ans.**2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

based on formula to find the rate of current.

Example 7 / 16

The velocity of the stream = 8 kmph

Speed of the boat in still water is 16 kmph

Speed downstream = (16+8) = 24 kmph

Speed upstream = (16-8) = 8 kmph

Let the distance between A and B be x km

Time taken to travel downstream from A to B + Time taken to travel upstream from B to C(mid of A and B) = 40 hours

⇒\(\frac{x}{24}\)+\(\frac{\frac{x}{2}}{8}\)=40

⇒\(\frac{x}{24}\)+\(\frac{x}{16}\)=40

⇒\(\frac{40x}{224}\)=40

X=224 km

i.e., distance between A and B = 224km

Speed of the boat in still water is 16 kmph

Speed downstream = (16+8) = 24 kmph

Speed upstream = (16-8) = 8 kmph

Let the distance between A and B be x km

Time taken to travel downstream from A to B + Time taken to travel upstream from B to C(mid of A and B) = 40 hours

⇒\(\frac{x}{24}\)+\(\frac{\frac{x}{2}}{8}\)=40

⇒\(\frac{x}{24}\)+\(\frac{x}{16}\)=40

⇒\(\frac{40x}{224}\)=40

X=224 km

i.e., distance between A and B = 224km

**Ans.**
distance = speed*time

1. Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

1. Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

take out the speeds and time and put in the formula, distance = speed*time and find the distance.

Example 8 / 16

Speed downstream = 15 km/hr

Rate of the current= 1\(\frac{1}{2}\) km/hr

Speed in still water = 15 - 1\(\frac{1}{2}\) = 13\(\frac{1}{2}\)km/hr

Rate against the current = 13\(\frac{1}{2}\) km/hr - 1\(\frac{1}{2}\)= 12 km/hr

Rate of the current= 1\(\frac{1}{2}\) km/hr

Speed in still water = 15 - 1\(\frac{1}{2}\) = 13\(\frac{1}{2}\)km/hr

Rate against the current = 13\(\frac{1}{2}\) km/hr - 1\(\frac{1}{2}\)= 12 km/hr

**Ans.**2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

directly based on the formula of speed in still water and rate of the current.

Example 9 / 16

Speed of the boat in still water = 6 km/hr

Speed downstream =\(\frac{21}{3}\) = 7 km/hr

Speed of the stream = 7-6 = 1 km/hr

Speed upstream = 6-1 = 5 km/hr

Time taken to cover 21 km upstream =\(\frac{21}{5}\)= 4.2 hours

Speed downstream =\(\frac{21}{3}\) = 7 km/hr

Speed of the stream = 7-6 = 1 km/hr

Speed upstream = 6-1 = 5 km/hr

Time taken to cover 21 km upstream =\(\frac{21}{5}\)= 4.2 hours

**Ans.**2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

find the speed, distance is given, take out the time as distance=speed * time.

Example 10 / 16

Let the speed of the boat in still water =x km/hr

Speed of the current = 2 km/hr

Then, speed downstream =(x+2) km/hr

speed upstream =(x−2) km/hr

Total time taken to travel 10 km upstream and back = 55 minutes =\(\frac{55}{60}\) hour =\(\frac{11}{12}\) hour

⇒\(\frac{10}{x−2}\)+\(\frac{10}{x+2}\)=\(\frac{11}{12}\)

=> 120(x+2)+120(x−2)=11(\({x}^{2}\)−4)

240x=11\({x}^{2}\)−44

11\({x}^{2}\)−240x−44=0

11\({x}^{2}\)−242x+2x−44=0

11x(x−22)+2(x−22)=0

(x−22)(11x+2)=0 x=22 or \(\frac{−2}{11}\)

Since x cannot be negative, x = 22

i.e., speed of the boat in still water = 22 km/hr

Speed of the current = 2 km/hr

Then, speed downstream =(x+2) km/hr

speed upstream =(x−2) km/hr

Total time taken to travel 10 km upstream and back = 55 minutes =\(\frac{55}{60}\) hour =\(\frac{11}{12}\) hour

⇒\(\frac{10}{x−2}\)+\(\frac{10}{x+2}\)=\(\frac{11}{12}\)

=> 120(x+2)+120(x−2)=11(\({x}^{2}\)−4)

240x=11\({x}^{2}\)−44

11\({x}^{2}\)−240x−44=0

11\({x}^{2}\)−242x+2x−44=0

11x(x−22)+2(x−22)=0

(x−22)(11x+2)=0 x=22 or \(\frac{−2}{11}\)

Since x cannot be negative, x = 22

i.e., speed of the boat in still water = 22 km/hr

**Ans.**2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

take out the speeds, then by using formula time=\(\frac{distance}{speed}\), add both the time and find x.

Example 11 / 16

Speed of the boat in still water = 10 km/hr

Speed upstream = \(\frac{2}{1}\)= 2 km/hr

Speed of the stream = 10-2 = 8 km/hr

Speed downstream = (10+2) = 12 km/hr

Time taken to travel 2 km downstream =\(\frac{2}{12}\) hr = \(\frac{2×60}{12}\)

= 10 minutes

Speed upstream = \(\frac{2}{1}\)= 2 km/hr

Speed of the stream = 10-2 = 8 km/hr

Speed downstream = (10+2) = 12 km/hr

Time taken to travel 2 km downstream =\(\frac{2}{12}\) hr = \(\frac{2×60}{12}\)

= 10 minutes

**Ans.**
distance=speed*time

1.Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

1.Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

find the speed, distance is given, use the formula, distance=speed*time.

Example 12 / 16

Speed downstream = 12 km/hr

Rate of the current= 1\(\frac{1}{2}\) km/hr

Speed in still water = 12 - 1\(\frac{1}{2}\)= 10\(\frac{1}{2}\)km/hr

Rate against the current = 10\(\frac{1}{2}\) km/hr - 1\(\frac{1}{2}\)

= 9 km/hr

Rate of the current= 1\(\frac{1}{2}\) km/hr

Speed in still water = 12 - 1\(\frac{1}{2}\)= 10\(\frac{1}{2}\)km/hr

Rate against the current = 10\(\frac{1}{2}\) km/hr - 1\(\frac{1}{2}\)

= 9 km/hr

**Ans.**2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

based on direct formula of boats and streams.

Example 13 / 16

Speed in still water= 15 kmph

Speed of stream= 5 kmph

Speed of downstream= speed in still water+ speed of a stream

= 15+5= 20 kmph.

Speed of stream= 5 kmph

Speed of downstream= speed in still water+ speed of a stream

= 15+5= 20 kmph.

**Ans.**2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

based on direct formula of boats and streams.

Example 14 / 16

Speed of the boat downstream = 24+12= 24 kmph

= 36 * 5/18 = 10 m/s

Hence time taken to cover 60 m = 60/10 = 6 seconds.

= 36 * 5/18 = 10 m/s

Hence time taken to cover 60 m = 60/10 = 6 seconds.

**Ans.**
distance=speed*time.

1.Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

1.Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

find the speed, distance is given, use the formula, distance=speed*time.

Example 15 / 16

Let the distance between A and B be x km.

Total time = \(\frac{x}{(9 + 1)\) +\(\frac{x}{(9 - 1)}\) = 4.5

=> \(\frac{x}{10}\) + \(\frac{x}{8}\) = \(\frac{9}{2}\)

=>\(\frac{(4x + 5x)}{40}\) = \(\frac{9}{2}\)

=> x = 20 km.

Total time = \(\frac{x}{(9 + 1)\) +\(\frac{x}{(9 - 1)}\) = 4.5

=> \(\frac{x}{10}\) + \(\frac{x}{8}\) = \(\frac{9}{2}\)

=>\(\frac{(4x + 5x)}{40}\) = \(\frac{9}{2}\)

=> x = 20 km.

**Ans.**
distance=speed*time

1.Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

1.Speed downstream = (u + v) km/hr.

2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

take the distance as x. find the speed and add up in terms of time and use the formula, time=\(\frac{distance}{speed}\)

Example 16 / 16

Let Speed upstream = x km/hr

Speed downstream = 3x km/hr

Speed in still water = \(\frac{1}{2}\)(x+3x) km/hr = 2x km/hr

Speed of current = \(\frac{1}{2}\) (3x-x) km/hr = x km/hr

2x = \(\frac{28}{3}\) or x = \(\frac{14}{3}\) = 4\(\frac{2}{3}\) km/hr.

Speed downstream = 3x km/hr

Speed in still water = \(\frac{1}{2}\)(x+3x) km/hr = 2x km/hr

Speed of current = \(\frac{1}{2}\) (3x-x) km/hr = x km/hr

2x = \(\frac{28}{3}\) or x = \(\frac{14}{3}\) = 4\(\frac{2}{3}\) km/hr.

**Ans.**2. Speed upstream = (u - v) km/hr.

3. Speed in still water = 1(a + b)/2 km/hr.

4. Rate of stream = 1(a - b)/2 km/hr.

Where, u= speed in still water

v = Rate of stream

a = speed of downstream

b= speed of upstream

NA

based on direct formula of boats and streams.