Problems on Boats and Streams(Set 3) - Solved Examples

Problems on Boats and Streams(Set 3) - Solved Examples

21. If a man rows at the rate of 5 kmph in still water and his rate against the current is 3 kmph, then the man's rate along the current is:
A. 5 kmph	B. 7 kmph
C. 12 kmph	D. 8 kmph

answer with explanation

Answer: Option B
Explanation:
Let the rate along with the current is $x$ km/hr

$\frac{x+3}{2}=5\Rightarrow x+3=10\Rightarrow x=7\text{kmph}$

22. A man can row 8 km/hr in still water. If the river is running at 3 km/hr, it takes 3 hours more in upstream than to go downstream for the same distance. How far is the place?
A. 32.5 km	B. 25 km
C. 27.5 km	D. 22.5 km

answer with explanation

Answer: Option C
Explanation:
Solution 1

Let the speed downstream $=x$ and speed upstream $=y$. Then,

$\frac{x+y}{2}=8$
$\Rightarrow x+y=16\cdots$(Equation 1)

$\frac{x-y}{2}=3$
$\Rightarrow x-y=6\cdots$(Equation 2)

(Equation 1 + Equation 2)
=> $2x=22$
=> $x$ = 11 km/hr

(Equation 1 - Equation 2)
=> $2y=10$
=> $y$ = 5 km/hr

Time taken to travel upstream = Time taken to travel downstream + 3

Let distance be $d$ km. Then,
$\frac{d}{5}=\frac{d}{11}+3\Rightarrow 11d=5d+165\Rightarrow 6d=165\Rightarrow 2d=55\Rightarrow d=27.5$

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Solution 2

Let the speed of a man in still water be $x$ km/hr and the speed of a stream be $y$km/hr. If he takes t hours more in upstream than to go downstream for the same distance, the distance

$=\frac{(x^2-y^2)t}{2y}\text{km}$

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$x$ = 8 km/hr
$y$ = 3 km/hr
$t$ = 3 hours

As per the formula, we have

distance $=\frac{(8^2-3^2)3}{2\times 3}$

$=\frac{55\times 3}{2\times 3}=\frac{55}{2}$ = 27.5 km

23. A man can row 4 kmph is still water. If the river is running at 2 kmph it takes 90 min to row to a place and back. How far is the place?
A. 2 km	B. 4 km
C. 5 km	D. 2.25 km

answer with explanation

Answer: Option D
Explanation:
Speed in still water = 4 kmph
Speed of the stream = 2 kmph

Speed upstream = (4-2)= 2 kmph
Speed downstream = (4+2)= 6 kmph

Total time = 90 minutes = $\frac{90}{60}$ hour = $\frac{3}{2}$ hour

Let $L$ be the distance. Then

$\frac{L}{6}+\frac{L}{2}=\frac{3}{2}\Rightarrow L+3L=9\Rightarrow 4L=9\Rightarrow L=\frac{9}{4}=2.25\text{km}$

24. At his usual rowing rate, Rahul can travel 12 miles downstream in a certain river in 6 hours less than it takes him to travel the same distance upstream. But if he could double his usual rowing rate for his 24-mile round trip, the downstream 12 miles would then take only one hour less than the upstream 12 miles. What is the speed of the current in miles per hour?
A. $2\frac{1}{3}$ mph	B. $1\frac{1}{3}$ mph
C. $1\frac{2}{3}$ mph	D. $2\frac{2}{3}$ mph

answer with explanation

Answer: Option D
Explanation:
Let the speed of Rahul in still water be $x$ mph
and the speed of the current be $y$ mph

Then, Speed upstream $=(x-y)$ mph
Speed downstream $=(x+y)$ mph

Distance = 12 miles

Time taken to travel upstream - Time taken to travel downstream = 6 hours
$\Rightarrow \frac{12}{x-y}-\frac{12}{x+y}=6\Rightarrow 12(x+y)-12(x-y)=6(x^2-y^2)\Rightarrow 24y=6(x^2-y^2)\Rightarrow 4y=x^2-y^2\Rightarrow x^2=(y^2+4y)\cdots \text{(Equation 1)}$

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
$\Rightarrow \frac{12}{2x-y}-\frac{12}{2x+y}=1\Rightarrow 12(2x+y)-12(2x-y)=4x^2-y^2\Rightarrow 24y=4x^2-y^2\Rightarrow 4x^2=y^2+24y\cdots \text{(Equation 2)}$

(Equation 1 × 4)=> $4x^2=4(y^2+4y)\cdots \text{(Equation 3)}$

From Equation 2 and 3, we have,
$y^2+24y=4(y^2+4y)\Rightarrow y^2+24y=4y^2+16y\Rightarrow 3y^2=8y\Rightarrow 3y=8\Rightarrow y=\frac{8}{3}\text{mph}$

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

25. A man can row 40 kmph in still water and the river is running at 10 kmph. If the man takes 1 hr to row to a place and back, how far is the place?
A. 16.5 km	B. 12.15 km
C. 2.25 km	D. 18.75 km

answer with explanation

Answer: Option D
Explanation:
Let the distance be $x$

Speed upstream = (40 - 10) = 30 kmph
Speed downstream = (40 + 10) = 50 kmph

Total time taken = 1 hr
$\Rightarrow \frac{x}{50}+\frac{x}{30}=1\Rightarrow \frac{8x}{150}=1\Rightarrow x=\frac{150}{8}\text{= 18.75 km}$

26. A boatman can row 96 km downstream in 8 hr. If the speed of the current is 4 km/hr, then find in what time will be able to cover 8 km upstream?
A. 6 hr	B. 2 hr
C. 4 hr	D. 1 hr

answer with explanation

Answer: Option B
Explanation:
Speed downstream $=\frac{96}{8}$ = 12 kmph

Speed of current = 4 km/hr

Speed of the boatman in still water = 12-4 = 8 kmph

Speed upstream = 8-4 = 4 kmph

Time taken to cover 8 km upstream $=\frac{8}{4}$ = 2 hours

27. The speed of a boat in still water is 10 km/hr. If it can travel 78 km downstream and 42 km upstream in the same time, the speed of the stream is
A. 3 km/hr	B. 12 km/hr
C. 1.5 km/hr	D. 4.4 km/hr

answer with explanation

Answer: Option A
Explanation:
Let the speed of the stream be $x$ km/hr. Then

Speed upstream $=(10-x)$ km/hr
Speed downstream $=(10+x)$ km/hr

Time taken to travel 78 km downstream = Time taken to travel 42 km upstream
$\Rightarrow \frac{78}{10+x}=\frac{42}{10-x}\Rightarrow \frac{26}{10+x}=\frac{14}{10-x}\Rightarrow \frac{13}{10+x}=\frac{7}{10-x}\Rightarrow 130-13x=70+7x\Rightarrow 20x=60\Rightarrow x=3\text{km/hr}$

28. A man can row at a speed of 12 km/hr in still water to a certain upstream point and back to the starting point in a river which flows at 3 km/hr. Find his average speed for total journey.
A. $12\frac{3}{4}$ km/hr	B. $11\frac{3}{4}$ km/hr
C. $12\frac{1}{4}$ km/hr	D. $11\frac{1}{4}$ km/hr

answer with explanation

Answer: Option D
Explanation:

Assume that a man can row at the speed of $x$ km/hr in still water and he rows the same distance up and down in a stream which flows at a rate of $y$ km/hr. Then his average speed throughout the journey

$=\frac{\text{Speed downstream × Speed upstream}}{\text{Speed in still water}}$ $=\frac{(x+y)(x-y)}{x}\text{km/hr}$

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Speed of the man in still water = 12 km/hr
Speed of the stream = 3 km/hr

Speed downstream = (12+3) = 15 km/hr
Speed upstream = (12-3) = 9 km/hr

Average Speed $=\frac{\text{Speed downstream × Speed downstream}}{\text{Speed in still water}}$

$=\frac{15\times 9}{12}=\frac{15\times 3}{4}=\frac{45}{4}=11\frac{1}{4}\text{km/hr}$

29. A boatman can row 3 km against the stream in 20 minutes and return in 18 minutes. Find the rate of current
A. 1⁄2 km/hr	B. 1 km/hr
C. 1⁄3 km/hr	D. 2⁄3 km/hr

answer with explanation

Answer: Option A
Explanation:
Speed upstream $=\frac{3}{\left(\frac{20}{60}\right)}$ = 9 km/hr
Speed downstream $=\frac{3}{\left(\frac{18}{60}\right)}$ = 10 km/hr

Rate of current $=\frac{10-9}{2}=\frac{1}{2}\text{km/hr}$

30. A boat takes 38 hours for travelling downstream from point A to point B and coming back to point C midway between A and B. If the velocity of the stream is 4 kmph and the speed of the boat in still water is 14 kmph, what is the distance between A and B?
A. 240 km	B. 120 km
C. 360 km	D. 180 km

answer with explanation

Answer: Option C
Explanation:
velocity of the stream = 4 kmph
Speed of the boat in still water is 14 kmph

Speed downstream = (14+4) = 18 kmph
Speed upstream = (14-4) = 10 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) = 38 hours
$\Rightarrow \frac{x}{18}+\frac{\left(\frac{x}{2}\right)}{10}=38\Rightarrow \frac{x}{18}+\frac{x}{20}=38\Rightarrow \frac{19x}{180}=38\Rightarrow \frac{x}{180}=2\Rightarrow x=360$

i.e., distance between A and B = 360 k
m