Problems on Boats and Streams(Set 1) - Solved Examples

Problems on Boats and Streams(Set 1) - Solved Examples

1. A man's speed with the current is 15 km/hr and the speed of the current is 2.5 km/hr. The man's speed against the current is:
A. 8.5 km/hr	B. 10 km/hr.
C. 12.5 km/hr	D. 9 km/hr

answer with explanation

Answer: Option B
Explanation:
Man's speed with the current = 15 km/hr
=> speed of the man + speed of the current = 15 km/hr

speed of the current is 2.5 km/hr
Hence, speed of the man = 15 - 2.5 = 12.5 km/hr

man's speed against the current = speed of the man - speed of the current
= 12.5 - 2.5 = 10 km/hr

2. A motorboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:
A. 10	B. 6
C. 5	D. 4

answer with explanation

Answer: Option C
Explanation:
Speed of the motor boat $=15$ km/hr
Let speed of the stream $=v$

Speed downstream $=(15+v)$ km/hr
Speed upstream $=(15-v)$ km/hr

Time taken downstream = $\frac{30}{(15+v)}$

Time taken upstream $=\frac{30}{(15-v)}$

Total time = $\frac{30}{(15+v)}+\frac{30}{(15-v)}$

Given that total time is 4 hours 30 minutes $=4\frac{1}{2}\text{hour =}\frac{9}{2}\text{hour}$

$\text{i.e.,}\frac{30}{(15+v)}+\frac{30}{(15-v)}=\frac{9}{2}\Rightarrow \frac{1}{(15+v)}+\frac{1}{(15-v)}=\frac{9}{2\times 30}\Rightarrow \frac{1}{(15+v)}+\frac{1}{(15-v)}=\frac{3}{20}\Rightarrow \frac{15-v+15+v}{(15+v)(15-v)}=\frac{3}{20}\Rightarrow \frac{30}{{15}^2-v^2}=\frac{3}{20}\Rightarrow \frac{30}{225-v^2}=\frac{3}{20}\Rightarrow \frac{10}{225-v^2}=\frac{1}{20}\Rightarrow 225-v^2=200\Rightarrow v^2=225-200=25\Rightarrow v=5\text{km/hr}$

3. In one hour, a boat goes 14 km/hr along the stream and 8 km/hr against the stream. The speed of the boat in still water (in km/hr) is:
A. 12 km/hr	B. 11 km/hr
C. 10 km/hr	D. 8 km/hr

answer with explanation

Answer: Option B
Explanation:
Solution 1 : Using Formula

Let the speed downstream be a km/hr and the speed upstream be b km/hr, then

Speed in still water $=\frac{1}{2}(a+b)$ km/hr
Rate of stream $=\frac{1}{2}(a-b)$ km/hr

[Read more ...]

Speed in still water = $\frac{1}{2}(14+8)\text{kmph}$ = 11 kmph.

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Solution 2 : Using Principles
Let speed of the boat in still water = a and speed of the stream = b

Then
a + b = 14
a - b = 8

Adding these two equations, we get 2a = 22
=> a = 11

ie, speed of boat in still water = 11 km/hr

4. A man rows to a place 48 km distant and come back in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. The rate of the stream is:
A. 1 km/hr.	B. 2 km/hr.
C. 1.5 km/hr.	D. 2.5 km/hr.

answer with explanation

Answer: Option A
Explanation:
Assume that he moves 4 km downstream in $x$ hours

Then, speed downstream $=\frac{\text{distance}}{\text{time}}$ $=\frac{4}{x}\text{km/hr}$

Given that he can row 4 km with the stream in the same time as 3 km against the stream

i.e., speed upstream $=\frac{3}{4}$ of speed downstream

=> speed upstream = $\frac{3}{x}\text{km/hr}$

He rows to a place 48 km distant and comes back in 14 hours
$\Rightarrow \frac{48}{\left(\frac{4}{x}\right)}+\frac{48}{\left(\frac{3}{x}\right)}=14\Rightarrow 12x+16x=14\Rightarrow 6x+8x=7\Rightarrow 14x=7\Rightarrow x=\frac{1}{2}$

Hence, speed downstream $=\frac{4}{x}=\frac{4}{\left(\frac{1}{2}\right)}$ = 8 km/hr

speed upstream = $\frac{3}{x}=\frac{3}{\left(\frac{1}{2}\right)}$ = 6 km/hr

Now we can use the below formula to find the rate of the stream

Let the speed downstream be a km/hr and the speed upstream be b km/hr, then

Speed in still water $=\frac{1}{2}(a+b)$ km/hr
Rate of stream $=\frac{1}{2}(a-b)$ km/hr
[Read more ...]

Hence, rate of the stream = $\frac{1}{2}(8-6)=1$ km/hr

5. A boatman goes 2 km against the current of the stream in 2 hour and goes 1 km along the current in 20 minutes. How long will it take to go 5 km in stationary water?
A. 2 hr 30 min	B. 2 hr
C. 4 hr	D. 1 hr 15 min

answer with explanation

Answer: Option A
Explanation:
Speed upstream = $\frac{2}{2}=1\text{km/hr}$

Speed downstream = $\frac{1}{\left(\frac{20}{60}\right)}=3\text{km/hr}$

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

Time taken to travel 5 km in still water $=\frac{5}{2}=2\frac{1}{2}\text{hours}$ = 2 hour 30 minutes

6. Speed of a boat in standing water is 14 kmph and the speed of the stream is 1.2 kmph. A man rows to a place at a distance of 4864 km and comes back to the starting point. The total time taken by him is:
A. 700 hours	B. 350 hours
C. 1400 hours	D. 1010 hours

answer with explanation

Answer: Option A
Explanation:
Speed downstream = (14 + 1.2) = 15.2 kmph

Speed upstream = (14 - 1.2) = 12.8 kmph

Total time taken $=\frac{4864}{15.2}+\frac{4864}{12.8}$ = 320 + 380 = 700 hours

7. The speed of a boat in still water in 22 km/hr and the rate of current is 4 km/hr. The distance travelled downstream in 24 minutes is:
A. 9.4 km	B. 10.2 km
C. 10.4 km	D. 9.2 km

answer with explanation

Answer: Option C
Explanation:
Speed downstream = (22 + 4) = 26 kmph

Time = 24 minutes $=\frac{24}{60}\text{hour =}\frac{2}{5}\text{hour}$

Distance travelled = Time × speed $=\frac{2}{5}\times 26$ = 10.4 km

8. A boat covers a certain distance downstream in 1 hour, while it comes back in 11⁄2hours. If the speed of the stream be 3 kmph, what is the speed of the boat in still water?
A. 14 kmph	B. 15 kmph
C. 13 kmph	D. 12 kmph

answer with explanation

Answer: Option B
Explanation:
Let the speed of the water in still water $=x$
Given that speed of the stream = 3 kmph

Speed downstream $=(x+3)$ kmph
Speed upstream $=(x-3)$ kmph

He travels a certain distance downstream in 1 hour and come back in 1¹⁄₂ hour.
i.e., distance travelled downstream in 1 hour = distance travelled upstream in 1¹⁄₂ hour

Since distance = speed × time, we have
$(x+3)\times 1=(x-3)\frac{3}{2}\Rightarrow 2(x+3)=3(x-3)\Rightarrow 2x+6=3x-9\Rightarrow x=6+9=15\text{kmph}$

9. A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current respectively?
A. 5 : 6	B. 6 : 5
C. 8 : 3	D. 3 : 8

answer with explanation

Answer: Option C
Explanation:
Let the rate upstream of the boat $=x$ kmph
and the rate downstream of the boat $=y$ kmph

Distance travelled upstream in 8 hrs 48 min = Distance travelled downstream in 4 hrs.

Since distance = speed × time, we have
$x\times 8\frac{4}{5}=y\times 4x\times \frac{44}{5}=y\times 4x\times \frac{11}{5}=y\cdots \text{(equation 1)}$

Now consider the formula given below

Let the speed downstream be a km/hr and the speed upstream be b km/hr, then

Speed in still water $=\frac{1}{2}(a+b)\text{km/hr}$
Rate of stream $=\frac{1}{2}(a-b)\text{km/hr}$
[Read more ...]

Hence, speed of the boat $=\frac{y+x}{2}$
speed of the water $=\frac{y-x}{2}$

Required Ratio
$=\left(\frac{y+x}{2}\right):\left(\frac{y-x}{2}\right)=(y+x):(y-x)$

$=(\frac{11x}{5}+x):(\frac{11x}{5}-x)$ $\text{(∵ Substituted value of}y$ $\text{from equation 1)}$

$=(11x+5x):(11x-5x)=16x:6x=8:3$

10. A boat can travel with a speed of 22 km/hr in still water. If the speed of the stream is 5 km/hr, find the time taken by the boat to go 54 km downstream
A. 5 hours	B. 4 hours
C. 3 hours	D. 2 hours

answer with explanation

Answer: Option D
Explanation:
Speed of the boat in still water = 22 km/hr
speed of the stream = 5 km/hr

Speed downstream = (22+5) = 27 km/hr
Distance travelled downstream = 54 km

Time taken $=\frac{\text{distance}}{\text{speed}}=\frac{54}{27}\text{= 2 hours}$