\displaystyle \textbf{Question 1: }\text{Express the following speeds in m/s:}
\displaystyle \text{(i) }72\text{ km/hr}\qquad \text{(ii) }117\text{ km/hr}
\displaystyle \text{(iii) }5.4\text{ km/hr}\qquad \text{(iv) }12.6\text{ km/hr}
\displaystyle \text{Answer:}
\displaystyle \text{To convert km/hr into m/s, multiply by }\frac{5}{18}.

\displaystyle \text{(i) }72\text{ km/hr} =72\times\frac{5}{18}\text{ m/s}=20\text{ m/s}.

\displaystyle \text{(ii) }117\text{ km/hr} =117\times\frac{5}{18}\text{ m/s}=32.5\text{ m/s}.

\displaystyle \text{(iii) }5.4\text{ km/hr} =5.4\times\frac{5}{18}\text{ m/s}=1.5\text{ m/s}.

\displaystyle \text{(iv) }12.6\text{ km/hr} =12.6\times\frac{5}{18}\text{ m/s}=3.5\text{ m/s}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Express the following speeds in km/hr:}
\displaystyle \text{(i) }18\text{ m/s}\qquad \text{(ii) }2\text{ m/s}
\displaystyle \text{(iii) }3\frac{1}{3}\text{ m/s}\qquad \text{(iv) }12.5\text{ m/s}
\displaystyle \text{Answer:}
\displaystyle \text{To convert m/s into km/hr, multiply by }\frac{18}{5}.

\displaystyle \text{(i) }18\text{ m/s} =18\times\frac{18}{5}\text{ km/hr}=64.8\text{ km/hr}.

\displaystyle \text{(ii) }2\text{ m/s} =2\times\frac{18}{5}\text{ km/hr}=7.2\text{ km/hr}.

\displaystyle \text{(iii) }3\frac{1}{3}\text{ m/s} =\frac{10}{3}\times\frac{18}{5}\text{ km/hr}=12\text{ km/hr}.

\displaystyle \text{(iv) }12.5\text{ m/s} =12.5\times\frac{18}{5}\text{ km/hr}=45\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{An athlete covers a distance of }1200\text{ metres in }4
\displaystyle \text{minutes }48\text{ seconds. Find his speed in km/hr.}
\displaystyle \text{Answer:}
\displaystyle \text{Total time}=4\times60+48=288\text{ seconds}.
\displaystyle \text{Speed}=\frac{1200}{288}\text{ m/s}=\frac{25}{6}\text{ m/s}.
\displaystyle \text{Speed in km/hr} =\frac{25}{6}\times\frac{18}{5}=15\text{ km/hr}.
\displaystyle \therefore \text{the athlete's speed is }15\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Walking at the rate of }4\text{ km/hr, a man covers a}
\displaystyle \text{certain distance in }2\frac{1}{2}\text{ hours. How much time will he take to}
\displaystyle \text{cover the same distance if he cycles at }12\text{ km/hr?}
\displaystyle \text{Answer:}
\displaystyle \text{Distance covered while walking} =4\times2\frac{1}{2}=10\text{ km}.
\displaystyle \text{Time taken to cover }10\text{ km by cycle} =\frac{10}{12}=\frac{5}{6}\text{ hour}.
\displaystyle \frac{5}{6}\text{ hour} =\frac{5}{6}\times60=50\text{ minutes}.
\displaystyle \therefore \text{the man will take }50\text{ minutes}.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{A car can finish a certain journey in }10\text{ hours at a}
\displaystyle \text{speed of }48\text{ km/hr. By how much must the speed of the car be increased}
\displaystyle \text{to cover the same distance in }8\text{ hours?}
\displaystyle \text{Answer:}
\displaystyle \text{Distance covered by the car} =48\times10=480\text{ km}.
\displaystyle \text{Required speed} =\frac{480}{8}=60\text{ km/hr}.
\displaystyle \text{Increase in speed} =60-48=12\text{ km/hr}.
\displaystyle \therefore \text{the speed of the car must be increased by }12\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{A bus covers a certain distance in }50\text{ minutes if it}
\displaystyle \text{runs at a speed of }54\text{ km/hr. What must be the speed of the bus to}
\displaystyle \text{reduce the journey time to }40\text{ minutes?}
\displaystyle \text{Answer:}
\displaystyle 50\text{ minutes}=\frac{50}{60}\text{ hour}=\frac{5}{6}\text{ hour}.
\displaystyle \text{Distance covered by the bus} =54\times\frac{5}{6}=45\text{ km}.
\displaystyle 40\text{ minutes}=\frac{40}{60}\text{ hour}=\frac{2}{3}\text{ hour}.
\displaystyle \text{Required speed} =\frac{45}{2/3}=45\times\frac{3}{2}=67.5\text{ km/hr}.
\displaystyle \therefore \text{the required speed of the bus is }67.5\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{A motor car starts at a speed of }70\text{ km/hr, and its}
\displaystyle \text{speed increases by }10\text{ km/hr after every }2\text{ hours. In how much time}
\displaystyle \text{will it cover a distance of }345\text{ km?}
\displaystyle \text{Answer:}
\displaystyle \text{Speed during the first }2\text{ hours}=70\text{ km/hr}.
\displaystyle \text{Distance covered in the first }2\text{ hours} =70\times2=140\text{ km}.
\displaystyle \text{Speed during the next }2\text{ hours} =70+10=80\text{ km/hr}.
\displaystyle \text{Distance covered during the 3rd and 4th hours} =80\times2=160\text{ km}.
\displaystyle \text{Distance covered in the first }4\text{ hours} =140+160=300\text{ km}.
\displaystyle \text{Remaining distance}=345-300=45\text{ km}.
\displaystyle \text{Speed after }4\text{ hours}=80+10=90\text{ km/hr}.
\displaystyle \text{Time taken to cover the remaining }45\text{ km} =\frac{45}{90}=\frac{1}{2}\text{ hour}.
\displaystyle \text{Total time}=4+\frac{1}{2}=4\frac{1}{2}\text{ hours}.
\displaystyle \therefore \text{the car will cover }345\text{ km in }4\frac{1}{2}\text{ hours}.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{A man takes }150\text{ steps in walking }75\text{ metres. If he}
\displaystyle \text{takes }3\text{ steps in }1\text{ second, find his speed in (i) m/s and (ii) km/hr.}
\displaystyle \text{Answer:}
\displaystyle \text{Length of one step} =\frac{75}{150}=0.5\text{ m}.
\displaystyle \text{Distance covered in }1\text{ second} =3\times0.5=1.5\text{ m}.

\displaystyle \text{(i) Speed}=1.5\text{ m/s}.

\displaystyle \text{(ii) Speed in km/hr} =1.5\times\frac{18}{5}=5.4\text{ km/hr}.
\displaystyle \therefore \text{his speed is }1.5\text{ m/s or }5.4\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{A man walks at }5\text{ km/hr for }6\text{ hours and at }4
\displaystyle \text{km/hr for }12\text{ hours. Find his average speed.}
\displaystyle \text{Answer:}
\displaystyle \text{Distance covered in the first }6\text{ hours} =5\times6=30\text{ km}.
\displaystyle \text{Distance covered in the next }12\text{ hours} =4\times12=48\text{ km}.
\displaystyle \text{Total distance}=30+48=78\text{ km}.
\displaystyle \text{Total time}=6+12=18\text{ hours}.
\displaystyle \text{Average speed} =\frac{\text{Total distance}}{\text{Total time}}.
\displaystyle =\frac{78}{18} =\frac{13}{3}=4\frac{1}{3}\text{ km/hr}.
\displaystyle \therefore \text{his average speed is }4\frac{1}{3}\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{A man covers a distance of }144\text{ km at a speed of}
\displaystyle 36\text{ km/hr and another }256\text{ km at a speed of }64\text{ km/hr. Find his}
\displaystyle \text{average speed for the whole journey.}
\displaystyle \text{Answer:}
\displaystyle \text{Time taken to cover }144\text{ km} =\frac{144}{36}=4\text{ hours}.
\displaystyle \text{Time taken to cover }256\text{ km} =\frac{256}{64}=4\text{ hours}.
\displaystyle \text{Total distance}=144+256=400\text{ km}.
\displaystyle \text{Total time}=4+4=8\text{ hours}.
\displaystyle \text{Average speed} =\frac{\text{Total distance}}{\text{Total time}}.
\displaystyle =\frac{400}{8}=50\text{ km/hr}.
\displaystyle \therefore \text{his average speed for the whole journey is }50\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Two buses travel to a place at }45\text{ km/hr and }60
\displaystyle \text{km/hr respectively. If the second bus takes }5\frac{1}{2}\text{ hours less than}
\displaystyle \text{the first for the same journey, find the length of the journey.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the length of the journey be }x\text{ km}.
\displaystyle \text{Time taken by the first bus}=\frac{x}{45}\text{ hours}.
\displaystyle \text{Time taken by the second bus}=\frac{x}{60}\text{ hours}.
\displaystyle \text{The difference in their journey times is }5\frac{1}{2} =\frac{11}{2}\text{ hours}.
\displaystyle \therefore \frac{x}{45}-\frac{x}{60}=\frac{11}{2}.
\displaystyle \frac{4x-3x}{180}=\frac{11}{2}.
\displaystyle \frac{x}{180}=\frac{11}{2}.
\displaystyle x=180\times\frac{11}{2}=990\text{ km}.
\displaystyle \therefore \text{the length of the journey is }990\text{ km}.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{A boy goes to school from his village at }3\text{ km/hr}
\displaystyle \text{and returns at }2\text{ km/hr. If he takes }5\text{ hours in all, what is the}
\displaystyle \text{distance between the village and the school?}
\displaystyle \text{Answer:}
\displaystyle \text{Let the distance between the village and the school be }x\text{ km}.
\displaystyle \text{Time taken to reach the school}=\frac{x}{3}\text{ hours}.
\displaystyle \text{Time taken to return to the village}=\frac{x}{2}\text{ hours}.
\displaystyle \text{Since the total time taken is }5\text{ hours,}
\displaystyle \frac{x}{3}+\frac{x}{2}=5.
\displaystyle \frac{2x+3x}{6}=5.
\displaystyle \frac{5x}{6}=5.
\displaystyle x=6\text{ km}.
\displaystyle \therefore \text{the distance between the village and the school is }6\text{ km}.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{A bus completes a journey of }420\text{ km in }6\frac{1}{2}
\displaystyle \text{hours. The first }\frac{3}{4}\text{ of the journey is covered at }63\text{ km/hr.}
\displaystyle \text{Calculate the speed for the rest of the journey.}
\displaystyle \text{Answer:}
\displaystyle \text{Distance covered in the first }\frac{3}{4}\text{ of the journey} =\frac{3}{4}\times420=315\text{ km}.
\displaystyle \text{Time taken to cover }315\text{ km} =\frac{315}{63}=5\text{ hours}.
\displaystyle \text{Remaining distance}=420-315=105\text{ km}.
\displaystyle \text{Remaining time}=6\frac{1}{2}-5 =1\frac{1}{2}\text{ hours}.
\displaystyle \text{Speed for the remaining journey} =\frac{105}{3/2}=70\text{ km/hr}.
\displaystyle \therefore \text{the speed for the rest of the journey is }70\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{A man drives }150\text{ km from town A to town B in }3
\displaystyle \text{hours }20\text{ minutes and returns from town B to town A in }4\text{ hours}
\displaystyle 10\text{ minutes. Find his average speed for the whole journey.}
\displaystyle \text{Answer:}
\displaystyle \text{Total distance travelled}=150+150=300\text{ km}.
\displaystyle 3\text{ hours }20\text{ minutes} =3+\frac{20}{60}=3\frac{1}{3}\text{ hours}.
\displaystyle 4\text{ hours }10\text{ minutes} =4+\frac{10}{60}=4\frac{1}{6}\text{ hours}.
\displaystyle \text{Total time} =3\frac{1}{3}+4\frac{1}{6} =\frac{10}{3}+\frac{25}{6} =\frac{15}{2}\text{ hours}.
\displaystyle \text{Average speed} =\frac{\text{Total distance}}{\text{Total time}}.
\displaystyle =\frac{300}{15/2} =300\times\frac{2}{15}=40\text{ km/hr}.
\displaystyle \therefore \text{his average speed for the whole journey is }40\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{A car completed a journey in }7\text{ hours. One-third of}
\displaystyle \text{the journey was covered at }20\text{ km/hr and the rest at }30\text{ km/hr.}
\displaystyle \text{Find the total length of the journey.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the total length of the journey be }x\text{ km}.
\displaystyle \text{Distance covered at }20\text{ km/hr}=\frac{x}{3}\text{ km}.
\displaystyle \text{Distance covered at }30\text{ km/hr}=\frac{2x}{3}\text{ km}.
\displaystyle \text{Time taken for the first part} =\frac{x/3}{20}=\frac{x}{60}\text{ hours}.
\displaystyle \text{Time taken for the remaining part} =\frac{2x/3}{30}=\frac{x}{45}\text{ hours}.
\displaystyle \text{Since the total journey takes }7\text{ hours,}
\displaystyle \frac{x}{60}+\frac{x}{45}=7.
\displaystyle \frac{3x+4x}{180}=7.
\displaystyle \frac{7x}{180}=7.
\displaystyle x=180\text{ km}.
\displaystyle \therefore \text{the total length of the journey is }180\text{ km}.
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{A cyclist covered a certain distance in }3\frac{1}{2}
\displaystyle \text{hours. His speed for the first half of the distance was }15\text{ km/hr and}
\displaystyle \text{for the second half it was }20\text{ km/hr. Find the total distance covered.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the total distance covered be }x\text{ km}.
\displaystyle \text{Distance covered in each half}=\frac{x}{2}\text{ km}.
\displaystyle \text{Time taken for the first half} =\frac{x/2}{15}=\frac{x}{30}\text{ hours}.
\displaystyle \text{Time taken for the second half} =\frac{x/2}{20}=\frac{x}{40}\text{ hours}.
\displaystyle \text{Since the total time taken is }3\frac{1}{2}\text{ hours,}
\displaystyle \frac{x}{30}+\frac{x}{40}=3\frac{1}{2}.
\displaystyle \frac{4x+3x}{120}=\frac{7}{2}.
\displaystyle \frac{7x}{120}=\frac{7}{2}.
\displaystyle x=60\text{ km}.
\displaystyle \therefore \text{the total distance covered by the cyclist is }60\text{ km}.
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{A person travels equal distances at speeds of }3\text{ km/hr,}
\displaystyle 4\text{ km/hr and }5\text{ km/hr, and takes a total time of }47\text{ minutes. Find}
\displaystyle \text{the total distance travelled.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the distance travelled at each speed be }x\text{ km}.
\displaystyle \text{Time taken at }3\text{ km/hr}=\frac{x}{3}\text{ hours}.
\displaystyle \text{Time taken at }4\text{ km/hr}=\frac{x}{4}\text{ hours}.
\displaystyle \text{Time taken at }5\text{ km/hr}=\frac{x}{5}\text{ hours}.
\displaystyle 47\text{ minutes}=\frac{47}{60}\text{ hour}.
\displaystyle \therefore \frac{x}{3}+\frac{x}{4}+\frac{x}{5}=\frac{47}{60}.
\displaystyle \frac{20x+15x+12x}{60}=\frac{47}{60}.
\displaystyle \frac{47x}{60}=\frac{47}{60}.
\displaystyle x=1\text{ km}.
\displaystyle \text{Total distance}=3x=3\times1=3\text{ km}.
\displaystyle \therefore \text{the total distance travelled is }3\text{ km}.
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{A farmer travelled a distance of }61\text{ km in }9\text{ hours.}
\displaystyle \text{He travelled partly on foot at }4\text{ km/hr and partly by bicycle at }9
\displaystyle \text{km/hr. Find the distance travelled by him on foot.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the distance travelled on foot be }x\text{ km}.
\displaystyle \text{Distance travelled by bicycle}=(61-x)\text{ km}.
\displaystyle \text{Time taken on foot}=\frac{x}{4}\text{ hours}.
\displaystyle \text{Time taken by bicycle}=\frac{61-x}{9}\text{ hours}.
\displaystyle \text{Since the total time taken is }9\text{ hours,}
\displaystyle \frac{x}{4}+\frac{61-x}{9}=9.
\displaystyle 9x+4(61-x)=324.
\displaystyle 9x+244-4x=324.
\displaystyle 5x=80.
\displaystyle x=16\text{ km}.
\displaystyle \therefore \text{the farmer travelled }16\text{ km on foot}.
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{Rohan cycles to his office at }12\frac{1}{2}\text{ km/hr and}
\displaystyle \text{is late by }3\text{ minutes. However, if he travels at }15\text{ km/hr, he reaches}
\displaystyle 5\text{ minutes earlier than the usual time. What is the distance of his office}
\displaystyle \text{from his residence?}
\displaystyle \text{Answer:}
\displaystyle \text{Let the distance from his residence to his office be }x\text{ km}.
\displaystyle \text{Difference between the journey times} =3+5=8\text{ minutes}.
\displaystyle 8\text{ minutes}=\frac{8}{60}=\frac{2}{15}\text{ hour}.
\displaystyle \therefore \frac{x}{12\frac{1}{2}}-\frac{x}{15}=\frac{2}{15}.
\displaystyle \frac{2x}{25}-\frac{x}{15}=\frac{2}{15}.
\displaystyle \frac{6x-5x}{75}=\frac{2}{15}.
\displaystyle \frac{x}{75}=\frac{2}{15}.
\displaystyle x=10\text{ km}.
\displaystyle \therefore \text{the distance from his residence to his office is }10\text{ km}.
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{Robert is travelling on his cycle and has calculated that}
\displaystyle \text{he will reach point A at 2 p.m. if he travels at }10\text{ km/hr. However, he}
\displaystyle \text{will reach there at 12 noon if he travels at }15\text{ km/hr. At what speed}
\displaystyle \text{must he travel to reach A at 1 p.m.?}
\displaystyle \text{Answer:}
\displaystyle \text{Let the distance to point A be }x\text{ km}.
\displaystyle \text{The difference between the two arrival times is }2\text{ hours}.
\displaystyle \therefore \frac{x}{10}-\frac{x}{15}=2.
\displaystyle \frac{3x-2x}{30}=2.
\displaystyle \frac{x}{30}=2.
\displaystyle x=60\text{ km}.
\displaystyle \text{At }10\text{ km/hr, time taken}=\frac{60}{10}=6\text{ hours}.
\displaystyle \text{Therefore, Robert starts his journey at }8\text{ a.m.}.
\displaystyle \text{To reach point A at }1\text{ p.m., the available time is }5\text{ hours}.
\displaystyle \text{Required speed}=\frac{60}{5}=12\text{ km/hr}.
\displaystyle \therefore \text{Robert must travel at }12\text{ km/hr}.
\displaystyle \\

\displaystyle \textbf{Question 21: }\text{If a train runs at }40\text{ km/hr, it reaches its destination}
\displaystyle \text{late by }11\text{ minutes, but if it runs at }50\text{ km/hr, it is late by only}
\displaystyle 5\text{ minutes. Find the correct time for the train to complete its journey.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the distance travelled by the train be }x\text{ km}.
\displaystyle \text{Difference between the journey times} =11-5=6\text{ minutes}.
\displaystyle 6\text{ minutes}=\frac{6}{60}=\frac{1}{10}\text{ hour}.
\displaystyle \therefore \frac{x}{40}-\frac{x}{50}=\frac{1}{10}.
\displaystyle \frac{5x-4x}{200}=\frac{1}{10}.
\displaystyle \frac{x}{200}=\frac{1}{10}.
\displaystyle x=20\text{ km}.
\displaystyle \text{Time taken at }50\text{ km/hr} =\frac{20}{50}\text{ hour}=24\text{ minutes}.
\displaystyle \text{At this speed, the train is still }5\text{ minutes late}.
\displaystyle \therefore \text{correct journey time}=24-5=19\text{ minutes}.
\displaystyle \therefore \text{the correct time to complete the journey is }19\text{ minutes}.
\displaystyle \\

\displaystyle \textbf{Question 22: }\text{The distance between Delhi and Hyderabad is }1800\text{ km.}
\displaystyle \text{A train leaves Delhi and proceeds towards Hyderabad at a uniform speed of}
\displaystyle 60\text{ km/hr. Another train leaves Hyderabad at the same time and proceeds}
\displaystyle \text{towards Delhi at a uniform speed of }48\text{ km/hr. When and where will}
\displaystyle \text{they meet?}
\displaystyle \text{Answer:}
\displaystyle \text{Since the trains are moving towards each other, their relative speed is}
\displaystyle 60+48=108\text{ km/hr}.
\displaystyle \text{Time taken to meet} =\frac{1800}{108}=\frac{50}{3}=16\frac{2}{3}\text{ hours}.
\displaystyle 16\frac{2}{3}\text{ hours}=16\text{ hours }40\text{ minutes}.
\displaystyle \text{Distance travelled by the train from Delhi} =60\times\frac{50}{3}=1000\text{ km}.
\displaystyle \text{Distance travelled by the train from Hyderabad} =48\times\frac{50}{3}=800\text{ km}.
\displaystyle \therefore \text{the trains will meet after }16\text{ hours }40\text{ minutes,}
\displaystyle \text{at a point }1000\text{ km from Delhi, or }800\text{ km from Hyderabad}.
\displaystyle \\

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