Question 1: A train $150$ m long is running at a uniform speed of $90$ km/ hr. Find:

i) The time taken by it to cross a man standing on the platform.

ii) The time taken by it to cross a platform $250$ m long.

i) The time taken by it to cross a man standing on the platform.

Length of the Train $= 150 \ m$

Speed of the Train $= 90 \ \frac{km}{hr}$

Time Taken $=$ $\frac{150}{90 \times \frac{1000}{3600}}$ $= 6 sec$

ii) The time taken by it to cross a platform $250$ m long.

Length of the Train $= 150\hspace{2pt} m$

Speed of the Train $= 90 \hspace{2pt}\frac{km}{hr}$

Length of the platform $= 250\hspace{2pt} m$

Time Taken $=$ $\frac{(250 + 150)}{90 \times \frac{1000}{3600}}$ $= 16 sec$

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Question 2: A train $280$ m long is running at a uniform speed, of $63$ km / hr. Find:

i) The time taken by it to cross a telephonic pole.

ii) The time taken by it to cross a bridge $210$ m long.

i) The time taken by it to cross a telephonic pole.

Length of the Train $= 280\hspace{2pt} m$

Speed of the Train $=63 \hspace{2pt}\frac{km}{hr}$

Time Taken $=$ $\frac{280}{63 \times \frac{1000}{3600}}$ $= 16 sec$

ii) The time taken by it to cross a bridge $210$ m long.

Length of the bridge $= 250 \hspace{2pt}m$

Time Taken $=$ $\frac{(210 + 280)}{63 \times \frac{1000}{3600}}$ $= 28 sec$

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Question 3: A train $180$ m long passes a telegraph post in $12$ seconds. Find:

i) Its speed in km/hr.

ii) The time taken by it to pass a platform $135$ meters long

i) Its speed in km/hr.

Length of the Train $= 180 \hspace{2pt}m$

Time taken by the Train $= 12 s$

Let the speed be $x \hspace{2pt}\frac{m}{s}$

$12 =$ $\frac{180}{x}$ $\hspace{5pt} or \hspace{5pt} x = 15\frac{m}{s} \hspace{5pt} or \hspace{5pt} 54 \hspace{2pt}\frac{km}{hr}$

ii) The time taken by it to pass a platform $135$ meters long

Length of the Platform $= 135 \hspace{2pt}m$

Time Taken $\frac{180+135}{15}$ $= 21 \hspace{2pt}s$

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Question 4: With a speed of $60$ km/hr, a train crosses a pole in $24$ seconds. Find the length of the train.

Let the length of the train $= x \hspace{2pt}m$

$\Rightarrow$ $24 =$ $\frac{x}{60 \times \frac{1000}{3600}}$

$\Rightarrow$ $x = 400 \hspace{2pt}m$

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Question 5: A train $700$ m long is running at $72$ km/hr. If it crosses a tunnel in one minute, find the length of the tunnel.

Let the length of the train $= x \hspace{2pt}m$

$\Rightarrow$ $60 =$ $\frac{700 + x}{72 \times \frac{1000}{3600}}$

$\Rightarrow$ $x = 500 \hspace{2pt}m$

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Question 6: A train $270$ m long takes $20$ seconds to cross a bridge $330$ m long. Find:

i) The speed of the train in km/hr

Let the speed be $x \hspace{2pt} \frac{m}{s}$

$\Rightarrow$ $20 =$ $\frac{(270+330)}{x}$ $\hspace{5pt} or x = 30 \hspace{2pt} \frac{m}{s} \hspace{5pt} or \hspace{5pt} 108 \hspace{2pt} \frac{km}{hr}$

ii) Time taken by it to cross an electric pole.

Time Taken $=$ $\frac{270}{30}$ $= 9\hspace{2pt}s$

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Question 7: A train, $225$ m in length, crosses a man standing on a platform in $10$ seconds and a bridge in $28$ seconds. Find:

i) The speed of the train in km/hr and

Let the speed be $x \hspace{2pt} \frac{m}{s}$

$\Rightarrow$ $10 =$ $\frac{225}{x}$ $\hspace{5pt} or \hspace{5pt} x = 22.5 \frac{m}{s} \hspace{5pt} or \hspace{5pt} 81 \frac{km}{hr}$

ii) The length of the bridge

Let the length of the bridge be $x \hspace{2pt}m$

$\Rightarrow$ $28 =$ $\frac{(225+x)}{22.5}$ $\hspace{5pt} or \hspace{5pt} x = 405 \hspace{2pt}m$

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Question 8: A train running at $54$ km/hr crosses a telegraph post in $16$ seconds and a platform in $40$ seconds. Find

i) The length of the train and

Let the length of the train $x \hspace{2pt}m$

$\Rightarrow$ $16 =$ $\frac{x}{54 \times \frac{1000}{3600}}$ $\hspace{5pt} or \hspace{5pt} x = 240 \hspace{2pt}m$

ii) The length of the platform.

Let the length of the platform be $x \hspace{2pt}m$

$\Rightarrow$ $40 =$ $\frac{240+x}{54 \times \frac{1000}{3600}}$ $\hspace{5pt} or \hspace{5pt} x = 360 \hspace{2pt}m$

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Question 9: Two cars are $351$ km apart. They start at the same time and drive towards each other. One travels at $70$ km/hr and the other travels at $65$ km/hr. How much time do they take to meet each other?

Time taken $=$ $\frac{351 km}{(70+65) \frac{km}{hr}}$ $= 2.6 hr \hspace{5pt} or \hspace{5pt} 156 minutes$

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Question 10: In how much time will a train $250$ m long, running at $50$ km/hr pass a man, running at $5$ km/hr in the same direction in which the train is going?

Time taken $=$ $\frac{250 m}{(50-5) \times \frac{1000}{3600} \frac{m}{s}}$ $\hspace{5pt} or \hspace{5pt} 20 seconds$

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Question 11: In how much time will a train $180$ m long, running at $66$ km/hr pass a man, running at $6$ km/hr in a direction opposite to that in which the train is going?

Time taken $=$ $\frac{180 m}{(66+6) \times \frac{1000}{3600} \frac{m}{s}}$ $\hspace{5pt} or \hspace{5pt} 9 seconds$

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Question 12: A and B are two trains of lengths $250$ m and $200$ m respectively. They are running on parallel rails at $45$ km/hr and $36$ km/hr respectively in opposite directions. In how much time will they be clear of each other from the moment they meet?

Time taken $=$ $\frac{(250+200) m}{(45+36) \times \frac{1000}{3600} \frac{m}{s}}$ $\hspace{5pt} = \hspace{5pt} 20 \hspace{2pt}seconds$

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Question 13: A and B are two trains of lengths $160$ m and $140$ m. They are running on parallel rails in the same direction at $72$ km/hr and $27$ km/hr respectively. In how much time will A pass B completely, from the moment they meet?

Time taken $=$ $\frac{(160+140) m}{(72-27) \times \frac{1000}{3600} \frac{m}{s}}$ $\hspace{5pt} = \hspace{5pt} 24 \hspace{2pt}seconds$

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Question 14: A train $120$ m long, travelling at $45$ km/hr, overtakes another train travelling in the same direction at $36$ km/hr and passes it completely in $80$ seconds. Find the length of the second train.

Let the length of the second train $x \hspace{2pt}m$

$\Rightarrow$ $80 =$ $\frac{(120+x)m}{(45-36) \times \frac{1000}{3600} \frac{m}{s} }$ $\hspace{5pt} or \hspace{5pt} x = 80\hspace{2pt}m$

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Question 15: The speed of a boat in still water is $8$ km/hr and the speed of the stream is $2.5$ km/hr Find:

i) The time taken by the boat to go $63$ km downstream

Time taken $=$ $\frac{63 km}{(8+2.5) \frac{km}{hr}}$ $\hspace{5pt} = \hspace{5pt} 6 \hspace{2pt}hr$

ii) The time taken by the boat to go 22 km, upstream.

Time taken $=$ $\frac{22 km}{(8-2.5) \frac{km}{hr} }$ $\hspace{5pt} = \hspace{5pt} 4 \hspace{2pt}hr$

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Question 16: A stream is flowing at $3$ km/hr. A boat with a speed of $10$ km/hr in still water is rowed upstream for $13$ hours. Find the distance rowed. How long will it take to return to the starting point?

Speed of Stream  $= 3 \hspace{5pt} \frac{km}{hr}$

Speed of boat in still water   $= 10 \hspace{5pt} \frac{km}{hr}$

Time rowed upsteram $= 13 \hspace{5pt} hr$

Let the distance covered $x \hspace{5pt} km$

$\Rightarrow$ $13 hr =$ $\frac{x}{(10-3) \frac{km}{hr}}$ $= 91 \hspace{5pt} km$

Let the time taken to reach back $t \hspace{5pt} hr$

$\Rightarrow$ Time taken $=$ $\frac{91 km}{(10+3) \frac{km}{hr}}$ $= 7 \hspace{5pt} hr$

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Question 17: The speed of a boat in still water is $10$ km/hr. It is rowed upstream for a distance of $45$ km in $6$ hours. Find the speed of the stream.

Let the speed of Stream $= x \hspace{2pt} \frac{km}{hr}$

Speed of boat in still water $= 10 \hspace{2pt} \frac{km}{hr}$

Time rowed upsteram $= 6 \hspace{5pt} hr$

Distance covered $= 45 \hspace{5pt} km$

$6 \hspace{2pt} hr =$ $\frac{45 km}{(10-x)}$ $\hspace{5pt} or \hspace{5pt} x = 2.5 \hspace{2pt} \frac{km}{hr}$

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Question 18: A stream is flowing at $4.8$ km/hr. A boat is rowed downstream for a distance of $49$ km in $3.5$ hours. Find the speed of the boat in still water.

Speed of Stream $= 4.8 \hspace{2pt} \frac{km}{hr}$

Let the Speed of boat in still water $= x \hspace{2pt} \frac{km}{hr}$

Time rowed down stream $= 3 \hspace{5pt} hr$

Distance covered $= 49 \hspace{5pt} km$

$3.5 \hspace{2pt} hr =$ $\frac{49 km}{(4+4.8)}$ $\hspace{5pt} or \hspace{5pt} x = 9.2 \hspace{2pt} \frac{km}{hr}$

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Question 19: The speed of a boat in still water is $5$ km/hr and the speed of the stream is $1$ km/hr. The boat is rowed upstream for a certain distance and taken back to the starting point. Find the average speed for the whole journey.

Speed of Stream $= 1 \hspace{2pt} \frac{km}{hr}$

Speed of boat in still water  $= 5 \hspace{2pt} \frac{km}{hr}$

Let the distance covered upstream $= x \hspace{5pt} km$

Total time taken to cover the journey $=$ $\frac{x}{5+1} + \frac{x}{5-1} = \frac{x}{6}+\frac{x}{4} = \frac{10}{24}$ $x \hspace{2pt} hr$

Total  distance covered $= 2x \hspace{5pt} km$

$Average Speed =$ $\frac{ total \hspace{2pt}distance}{total \hspace{2pt}time} = \frac{2x}{\frac{10}{24}x}$ $= 4.8 \hspace{2pt}\frac{km}{hr}$

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Question 20: The speed of a boat downstream is $16$ km/hr and its speed upstream is $10$ km/hr. Find the speed of the boat in still water and the rate of the stream.

Let the Speed of Stream  $= y \hspace{2pt} \frac{km}{hr}$

Let the Speed of boat in still water  $= x \hspace{2pt} \frac{km}{hr}$

Therefore

$x+y=16$

$x-y=10$

Solving for $x$ and $y$ we get

$x=13 \frac{km}{hr}$

$y=3 \frac{km}{hr}$