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.

Answer:

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.

Answer:

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

Answer:

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.

Answer:

\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.

Answer:

\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:

Answer:

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:

Answer:

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

Answer:

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?

Answer:

\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?

Answer:

\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?

Answer:

\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?

Answer:

\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?

Answer:

\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.

Answer:

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:

Answer:

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?

Answer:

\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.

Answer:

\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.

Answer:

\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.

Answer:

\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.

Answer:

\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}}