\displaystyle \textbf{Question 1: }\text{A train }150\text{ m long is running at a uniform speed}
\displaystyle \text{of }90\text{ km/hr. Find:}
\displaystyle \text{(i) The time taken by it to cross a man standing on the platform.}
\displaystyle \text{(ii) The time taken by it to cross a platform }250\text{ m long.}
\displaystyle \text{Answer:}
\displaystyle \text{Speed of the train} =90\times\frac{5}{18}=25\text{ m/s}.

\displaystyle \text{(i) To cross a stationary man, the train must cover its own length}.
\displaystyle \text{Distance to be covered}=150\text{ m}.
\displaystyle \text{Time taken}=\frac{150}{25}=6\text{ seconds}.

\displaystyle \text{(ii) To cross the platform, the train must cover the combined length}.
\displaystyle \text{Distance to be covered}=150+250=400\text{ m}.
\displaystyle \text{Time taken}=\frac{400}{25}=16\text{ seconds}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{A train }280\text{ m long is running at a uniform speed}
\displaystyle \text{of }63\text{ km/hr. Find:}
\displaystyle \text{(i) The time taken by it to cross a telephone pole.}
\displaystyle \text{(ii) The time taken by it to cross a bridge }210\text{ m long.}
\displaystyle \text{Answer:}
\displaystyle \text{Speed of the train} =63\times\frac{5}{18}=17.5\text{ m/s}.

\displaystyle \text{(i) To cross the pole, the train must cover its own length}.
\displaystyle \text{Distance to be covered}=280\text{ m}.
\displaystyle \text{Time taken}=\frac{280}{17.5}=16\text{ seconds}.

\displaystyle \text{(ii) To cross the bridge, the train must cover the combined length}.
\displaystyle \text{Distance to be covered}=280+210=490\text{ m}.
\displaystyle \text{Time taken}=\frac{490}{17.5}=28\text{ seconds}.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{A train }180\text{ m long passes a telegraph post in }12
\displaystyle \text{seconds. Find:}
\displaystyle \text{(i) Its speed in km/hr.}
\displaystyle \text{(ii) The time taken by it to pass a platform }135\text{ m long.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) To pass the post, the train covers its own length of }180\text{ m}.
\displaystyle \text{Speed of the train} =\frac{180}{12}=15\text{ m/s}.
\displaystyle \text{Speed in km/hr} =15\times\frac{18}{5}=54\text{ km/hr}.
\displaystyle \therefore \text{the speed of the train is }54\text{ km/hr}.

\displaystyle \text{(ii) Distance to be covered} =180+135=315\text{ m}.
\displaystyle \text{Time taken} =\frac{315}{15}=21\text{ seconds}.
\displaystyle \therefore \text{the train takes }21\text{ seconds to pass the platform}.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{At a speed of }60\text{ km/hr, a train crosses a pole in}
\displaystyle 24\text{ seconds. Find the length of the train.}
\displaystyle \text{Answer:}
\displaystyle \text{Speed of the train} =60\times\frac{5}{18}=\frac{50}{3}\text{ m/s}.
\displaystyle \text{When crossing a pole, the distance covered equals the train's length}.
\displaystyle \text{Length of the train} =\text{Speed}\times\text{Time}.
\displaystyle =\frac{50}{3}\times24=400\text{ m}.
\displaystyle \therefore \text{the length of the train is }400\text{ m}.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{A train }700\text{ m long is running at }72\text{ km/hr.}
\displaystyle \text{If it crosses a tunnel in one minute, find the length of the tunnel.}
\displaystyle \text{Answer:}
\displaystyle \text{Speed of the train} =72\times\frac{5}{18}=20\text{ m/s}.
\displaystyle \text{Time taken}=1\text{ minute}=60\text{ seconds}.
\displaystyle \text{Distance covered while crossing the tunnel} =20\times60=1200\text{ m}.
\displaystyle \text{Let the length of the tunnel be }x\text{ m}.
\displaystyle 700+x=1200.
\displaystyle x=1200-700=500\text{ m}.
\displaystyle \therefore \text{the length of the tunnel is }500\text{ m}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{A train }270\text{ m long takes }20\text{ seconds to cross a}
\displaystyle \text{bridge }330\text{ m long. Find:}
\displaystyle \text{(i) The speed of the train in km/hr.}
\displaystyle \text{(ii) The time taken by it to cross an electric pole.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) Distance covered while crossing the bridge} =270+330=600\text{ m}.
\displaystyle \text{Speed of the train} =\frac{600}{20}=30\text{ m/s}.
\displaystyle \text{Speed in km/hr} =30\times\frac{18}{5}=108\text{ km/hr}.
\displaystyle \therefore \text{the speed of the train is }108\text{ km/hr}.

\displaystyle \text{(ii) To cross a pole, the train must cover its own length of }270\text{ m}.
\displaystyle \text{Time taken}=\frac{270}{30}=9\text{ seconds}.
\displaystyle \therefore \text{the train takes }9\text{ seconds to cross the pole}.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{A train }225\text{ m long crosses a man standing on a}
\displaystyle \text{platform in }10\text{ seconds and a bridge in }28\text{ seconds. Find:}
\displaystyle \text{(i) The speed of the train in km/hr.}
\displaystyle \text{(ii) The length of the bridge.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) To cross the man, the train covers its own length of }225\text{ m}.
\displaystyle \text{Speed of the train} =\frac{225}{10}=22.5\text{ m/s}.
\displaystyle \text{Speed in km/hr} =22.5\times\frac{18}{5}=81\text{ km/hr}.
\displaystyle \therefore \text{the speed of the train is }81\text{ km/hr}.

\displaystyle \text{(ii) Distance covered in }28\text{ seconds} =22.5\times28=630\text{ m}.
\displaystyle \text{Length of the bridge} =630-225=405\text{ m}.
\displaystyle \therefore \text{the length of the bridge is }405\text{ m}.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{A train running at }54\text{ km/hr crosses a telegraph post}
\displaystyle \text{in }16\text{ seconds and a platform in }40\text{ seconds. Find:}
\displaystyle \text{(i) The length of the train.}
\displaystyle \text{(ii) The length of the platform.}
\displaystyle \text{Answer:}
\displaystyle \text{Speed of the train} =54\times\frac{5}{18}=15\text{ m/s}.

\displaystyle \text{(i) To cross the post, the train covers its own length}.
\displaystyle \text{Length of the train} =15\times16=240\text{ m}.
\displaystyle \therefore \text{the length of the train is }240\text{ m}.

\displaystyle \text{(ii) Distance covered in }40\text{ seconds} =15\times40=600\text{ m}.
\displaystyle \text{Length of the platform} =600-240=360\text{ m}.
\displaystyle \therefore \text{the length of the platform is }360\text{ m}.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Two cars are }351\text{ km apart. They start at the same}
\displaystyle \text{time and drive towards each other. One travels at }70\text{ km/hr and the}
\displaystyle \text{other at }65\text{ km/hr. How much time do they take to meet each other?}
\displaystyle \text{Answer:}
\displaystyle \text{Since the cars are moving towards each other, their relative speed is}
\displaystyle 70+65=135\text{ km/hr}.
\displaystyle \text{Time taken to meet} =\frac{351}{135}=2.6\text{ hours}.
\displaystyle 2.6\text{ hours}=2\text{ hours }36\text{ minutes}=156\text{ minutes}.
\displaystyle \therefore \text{the cars will meet after }2\text{ hours }36\text{ minutes}.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{In how much time will a train }250\text{ m long, running at}
\displaystyle 50\text{ km/hr, pass a man running at }5\text{ km/hr in the same direction as}
\displaystyle \text{the train?}
\displaystyle \text{Answer:}
\displaystyle \text{Relative speed}=50-5=45\text{ km/hr}.
\displaystyle \text{Relative speed in m/s} =45\times\frac{5}{18}=12.5\text{ m/s}.
\displaystyle \text{To pass the man, the train must cover its own length of }250\text{ m}.
\displaystyle \text{Time taken}=\frac{250}{12.5}=20\text{ seconds}.
\displaystyle \therefore \text{the train will pass the man in }20\text{ seconds}.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{In how much time will a train }180\text{ m long, running at}
\displaystyle 66\text{ km/hr, pass a man running at }6\text{ km/hr in the direction opposite}
\displaystyle \text{to that in which the train is travelling?}
\displaystyle \text{Answer:}
\displaystyle \text{Since they are moving in opposite directions, their relative speed is}
\displaystyle 66+6=72\text{ km/hr}.
\displaystyle \text{Relative speed in m/s} =72\times\frac{5}{18}=20\text{ m/s}.
\displaystyle \text{To pass the man, the train must cover its own length of }180\text{ m}.
\displaystyle \text{Time taken}=\frac{180}{20}=9\text{ seconds}.
\displaystyle \therefore \text{the train will pass the man in }9\text{ seconds}.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{A and B are two trains of lengths }250\text{ m and }200
\displaystyle \text{m respectively. They are running on parallel tracks at }45\text{ km/hr and}
\displaystyle 36\text{ km/hr respectively, in opposite directions. In how much time will they}
\displaystyle \text{be clear of each other from the moment they meet?}
\displaystyle \text{Answer:}
\displaystyle \text{Distance to be covered for the trains to cross completely}
\displaystyle =250+200=450\text{ m}.
\displaystyle \text{Relative speed}=45+36=81\text{ km/hr}.
\displaystyle \text{Relative speed in m/s} =81\times\frac{5}{18}=22.5\text{ m/s}.
\displaystyle \text{Time taken}=\frac{450}{22.5}=20\text{ seconds}.
\displaystyle \therefore \text{the trains will be clear of each other in }20\text{ seconds}.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{A and B are two trains of lengths }160\text{ m and }140
\displaystyle \text{m respectively. They are running on parallel tracks in the same direction}
\displaystyle \text{at }72\text{ km/hr and }27\text{ km/hr respectively. In how much time will A}
\displaystyle \text{pass B completely, from the moment they meet?}
\displaystyle \text{Answer:}
\displaystyle \text{Distance to be covered for A to pass B completely}
\displaystyle =160+140=300\text{ m}.
\displaystyle \text{Since the trains move in the same direction, their relative speed is}
\displaystyle 72-27=45\text{ km/hr}.
\displaystyle \text{Relative speed in m/s} =45\times\frac{5}{18}=12.5\text{ m/s}.
\displaystyle \text{Time taken}=\frac{300}{12.5}=24\text{ seconds}.
\displaystyle \therefore \text{A will pass B completely in }24\text{ seconds}.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{A train }120\text{ m long, travelling at }45\text{ km/hr,}
\displaystyle \text{overtakes another train travelling in the same direction at }36\text{ km/hr}
\displaystyle \text{and passes it completely in }80\text{ seconds. Find the length of the second train.}
\displaystyle \text{Answer:}
\displaystyle \text{Relative speed}=45-36=9\text{ km/hr}.
\displaystyle \text{Relative speed in m/s} =9\times\frac{5}{18}=2.5\text{ m/s}.
\displaystyle \text{Distance covered relative to the second train} =2.5\times80=200\text{ m}.
\displaystyle \text{Let the length of the second train be }x\text{ m}.
\displaystyle 120+x=200.
\displaystyle x=200-120=80\text{ m}.
\displaystyle \therefore \text{the length of the second train is }80\text{ m}.
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{The speed of a boat in still water is }8\text{ km/hr and}
\displaystyle \text{the speed of the stream is }2.5\text{ km/hr. Find:}
\displaystyle \text{(i) The time taken by the boat to go }63\text{ km downstream.}
\displaystyle \text{(ii) The time taken by the boat to go }22\text{ km upstream.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) Downstream speed} =8+2.5=10.5\text{ km/hr}.
\displaystyle \text{Time taken} =\frac{63}{10.5}=6\text{ hours}.
\displaystyle \therefore \text{the boat takes }6\text{ hours to travel }63\text{ km downstream}.

\displaystyle \text{(ii) Upstream speed} =8-2.5=5.5\text{ km/hr}.
\displaystyle \text{Time taken} =\frac{22}{5.5}=4\text{ hours}.
\displaystyle \therefore \text{the boat takes }4\text{ hours to travel }22\text{ km upstream}.
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{A stream is flowing at }3\text{ km/hr. A boat with a speed}
\displaystyle \text{of }10\text{ km/hr in still water is rowed upstream for }13\text{ hours. Find}
\displaystyle \text{the distance rowed. How long will it take to return to the starting point?}
\displaystyle \text{Answer:}
\displaystyle \text{Upstream speed} =10-3=7\text{ km/hr}.
\displaystyle \text{Distance rowed upstream} =7\times13=91\text{ km}.
\displaystyle \text{For the return journey, the boat travels downstream}.
\displaystyle \text{Downstream speed} =10+3=13\text{ km/hr}.
\displaystyle \text{Time taken to return} =\frac{91}{13}=7\text{ hours}.
\displaystyle \therefore \text{the distance rowed upstream is }91\text{ km, and the boat will}
\displaystyle \text{take }7\text{ hours to return to the starting point}.
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{The speed of a boat in still water is }10\text{ km/hr.}
\displaystyle \text{It is rowed upstream for a distance of }45\text{ km in }6\text{ hours. Find}
\displaystyle \text{the speed of the stream.}
\displaystyle \text{Answer:}
\displaystyle \text{Upstream speed of the boat} =\frac{45}{6}=7.5\text{ km/hr}.
\displaystyle \text{Speed of the stream} =10-7.5=2.5\text{ km/hr}.
\displaystyle \therefore \text{the speed of the stream is }2.5\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{A stream is flowing at }4.8\text{ km/hr. A boat is rowed}
\displaystyle \text{downstream for a distance of }49\text{ km in }3.5\text{ hours. Find the speed}
\displaystyle \text{of the boat in still water.}
\displaystyle \text{Answer:}
\displaystyle \text{Downstream speed of the boat} =\frac{49}{3.5}=14\text{ km/hr}.
\displaystyle \text{Speed of the boat in still water} =14-4.8=9.2\text{ km/hr}.
\displaystyle \therefore \text{the speed of the boat in still water is }9.2\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{The speed of a boat in still water is }5\text{ km/hr and}
\displaystyle \text{the speed of the stream is }1\text{ km/hr. The boat is rowed upstream for}
\displaystyle \text{a certain distance and then back to the starting point. Find the average}
\displaystyle \text{speed for the whole journey.}
\displaystyle \text{Answer:}
\displaystyle \text{Upstream speed}=5-1=4\text{ km/hr}.
\displaystyle \text{Downstream speed}=5+1=6\text{ km/hr}.
\displaystyle \text{Let the distance travelled upstream be }x\text{ km}.
\displaystyle \text{The distance travelled downstream is also }x\text{ km}.
\displaystyle \text{Total time taken} =\frac{x}{4}+\frac{x}{6} =\frac{3x+2x}{12}=\frac{5x}{12}\text{ hours}.
\displaystyle \text{Total distance travelled}=x+x=2x\text{ km}.
\displaystyle \text{Average speed} =\frac{\text{Total distance}}{\text{Total time}}.
\displaystyle =\frac{2x}{5x/12} =\frac{24}{5}=4.8\text{ km/hr}.
\displaystyle \therefore \text{the average speed for the whole journey is }4.8\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{The speed of a boat downstream is }16\text{ km/hr and}
\displaystyle \text{its speed upstream is }10\text{ km/hr. Find the speed of the boat in still}
\displaystyle \text{water and the speed of the stream.}
\displaystyle \text{Answer:}
\displaystyle \text{Speed of the boat in still water} =\frac{16+10}{2}=13\text{ km/hr}.
\displaystyle \text{Speed of the stream} =\frac{16-10}{2}=3\text{ km/hr}.
\displaystyle \therefore \text{the speed of the boat in still water is }13\text{ km/hr and the}
\displaystyle \text{speed of the stream is }3\text{ km/hr}.
\displaystyle \\

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