Question 1: A train $\displaystyle 150$ m long is running at a uniform speed of $\displaystyle 90$ 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 $\displaystyle 250$ m long.

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

$\displaystyle \text{Length of the Train } = 150 \text{ m }$

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

$\displaystyle \text{Time taken }= \frac{150}{90 \times \frac{1000}{3600}} = 6 \text{ sec }$

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

$\displaystyle \text{Length of the Train } = 150 \ \text{ m }$

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

$\displaystyle \text{Length of the platform } = 250 \ \text{ m }$

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

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

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

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

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

$\displaystyle \text{Length of the Train } = 280 \ \text{ m }$

$\displaystyle \text{Speed of the Train } =63 \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Time taken }= \frac{280}{63 \times \frac{1000}{3600}} = 16 \text{ sec }$

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

$\displaystyle \text{Length of the bridge } = 250 \ \text{ m }$

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

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Question 3: A train $\displaystyle 180$ m long passes a telegraph post in $\displaystyle 12 \text{ seconds }$. Find:

i) Its speed in km/hr.

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

i) Its speed in km/hr.

$\displaystyle \text{Length of the Train } = 180 \ \text{ m }$

$\displaystyle \text{Time taken by the Train } = 12\text{ sec }$

$\displaystyle \text{ Let the speed be } x \frac{\text{m}}{\text{s}}$

$\displaystyle 12 = \frac{180}{x} \text{ or } x = 15\frac{\text{m}}{\text{s}} \text{ or } 54 \ \frac{\text{km}}{\text{hr}}$

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

$\displaystyle \text{Length of the Platform } = 135 \ \text{ m }$

$\displaystyle \text{Time taken }\frac{180+135}{15} = 21 \text{ sec }$

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

$\displaystyle \text{Let the length of the train } = x \ \text{ m }$

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

$\displaystyle \Rightarrow x = 400 \ \text{ m }$

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

$\displaystyle \text{Let the length of the train } = x \ \text{ m }$

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

$\displaystyle \Rightarrow x = 500 \ \text{ m }$

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Question 6: A train $\displaystyle 270$ m long takes $\displaystyle 20 \text{ seconds }$ to cross a bridge $\displaystyle 330$ m long. Find:

i) The speed of the train in km/hr

$\displaystyle \text{Let the speed be } x \frac{\text{m}}{\text{s}}$

$\displaystyle \Rightarrow 20 = \frac{(270+330)}{x} \text{ or } x = 30 \frac{\text{m}}{\text{s}} \text{ or } 108 \ \frac{\text{km}}{\text{hr}}$

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

$\displaystyle \text{Time taken }= \frac{270}{30} = 9\text{ sec }$

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

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

$\displaystyle \text{Let the speed be } x \frac{\text{m}}{\text{s}}$

$\displaystyle \Rightarrow 10 = \frac{225}{x} \text{ or } x = 22.5 \frac{\text{m}}{\text{s}} \text{ or } 81 \ \frac{\text{km}}{\text{hr}}$

ii) The length of the bridge

$\displaystyle \text{Let the length of the bridge be } x \ \text{ m }$

$\displaystyle \Rightarrow 28 = \frac{(225+x)}{22.5} \text{ or } x = 405 \ \text{ m }$

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

i) The length of the train and

$\displaystyle \text{Let the length of the train } x \ \text{ m }$

$\displaystyle \Rightarrow 16 = \frac{x}{54 \times \frac{1000}{3600}} \text{ or } x = 240 \ \text{ m }$

ii) The length of the platform.

$\displaystyle \text{Let the length of the platform be } x \ \text{ m }$

$\displaystyle \Rightarrow 40 = \frac{240+x}{54 \times \frac{1000}{3600}} \text{ or } x = 360 \ \text{ m }$

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

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

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

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

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

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

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

$\displaystyle \text{Time taken }= \frac{(250+200) m}{(45+36) \times \frac{1000}{3600} \frac{\text{m}}{\text{s}}} = 20 \text{ seconds }$

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

$\displaystyle \text{Time taken }= \frac{(160+140) m}{(72-27) \times \frac{1000}{3600} \frac{\text{m}}{\text{s}}} = 24 \text{ seconds }$

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

Let the length of the second train $\displaystyle x \ \text{ m }$

$\displaystyle \Rightarrow 80 = \frac{(120+x)m}{(45-36) \times \frac{1000}{3600} \frac{\text{m}}{\text{s}} } \text{ or } x = 80\ \text{ m }$

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

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

$\displaystyle \text{Time taken }= \frac{63 km}{(8+2.5) \ \frac{\text{km}}{\text{hr}}} = 6 \text{ hr }$

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

$\displaystyle \text{Time taken }= \frac{22 km}{(8-2.5) \ \frac{\text{km}}{\text{hr}} } = 4 \text{ hr }$

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

$\displaystyle \text{ Speed of Stream } = 3 \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Speed of boat in still water } = 10 \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Time rowed upsteram } = 13 \text{ hr }$

$\displaystyle \text{Let the distance covered } x \text{ km }$

$\displaystyle \Rightarrow 13 hr = \frac{x}{(10-3) \ \frac{\text{km}}{\text{hr}}} = 91 \text{ km }$

Let the time taken to reach back $\displaystyle t \text{ hr }$

$\displaystyle \Rightarrow \text{Time taken }= \frac{91 km}{(10+3) \ \frac{\text{km}}{\text{hr}}} = 7 \text{ hr }$

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

$\displaystyle \text{Let the speed of Stream } = x \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Speed of boat in still water } = 10 \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Time rowed upsteram } = 6 \text{ hr }$

$\displaystyle \text{Distance covered } = 45 \text{ km }$

$\displaystyle 6 hr = \frac{45 km}{(10-x)} \text{ or } x = 2.5 \ \frac{\text{km}}{\text{hr}}$

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

$\displaystyle \text{ Speed of Stream } = 4.8 \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Let the Speed of boat in still water } = x \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{ Time rowed down stream } = 3 \text{ hr }$

$\displaystyle \text{ Distance covered } = 49 km$

$\displaystyle 3.5 hr = \frac{49 km}{(4+4.8)} \text{ or } x = 9.2 \ \frac{\text{km}}{\text{hr}}$

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Question 19: The speed of a boat in still water is $\displaystyle 5$ km/hr and the speed of the stream is $\displaystyle 1$ 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.

$\displaystyle \text{ Speed of Stream } = 1 \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Speed of boat in still water } = 5 \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Let the distance covered upstream } = x \text{ km }$

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

$\displaystyle \text{Total distance covered } = 2x km$

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

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

$\displaystyle \text{ Let the Speed of Stream } = y \ \frac{\text{km}}{\text{hr}}$

$\displaystyle \text{Let the Speed of boat in still water } = x \ \frac{\text{km}}{\text{hr}}$

Therefore

$\displaystyle x+y=16$

$\displaystyle x-y=10$

Solving for $\displaystyle x$ and $\displaystyle y$ we get

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

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