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Time and Distance Questions

21. A pedestrian and a cyclist start simultaneously towards each other from Aurangabad and Paithan which are 40 km apart and meet 2 hours after the start. Then they resume their trips and the cyclist arrives at Aurangabad 7 hours 30 minutes earlier than the pedestrian arrives at Paithan. What of these could be the speed of the pedestrian?




SHOW ANSWER
Correct Ans:4 km/hr
Explanation:
Let the Speed of pedestrian = x km/h
Speed of cyclist = y km/h
Given, Distance between Aurangabad and Paithan = 40 km

Since they are moving in opposite direction, speed can be added.
=> Relative speed = Speed of pedestrian + Speed of cyclist = (x + y) km/h
Given, They meet 2 hours after the start.
W.K.T: Time * Relative Speed = Distance
---> 2 * (x + y) = 40
---> x + y = 20 -------> eqn (1)

According to second condition, i.e., cyclist arrives at Aurangabad 7 hours 30 minutes earlier than the pedestrian arrives at Paithan
---> Time taken by cyclist to complete his journey = Time taken by pedestrian to complete his journey - 7 hours 30 minutes
---> (40/y) = (40/x) - [7 + (30/60)] [Since, 30 minutes is converted into (30/60) hours]
---> (40/x) - (40/y) = [7 + (30/60)]
---> 40 [(1/x) - (1/y)] = 15/2
---> 8 [(y - x) / xy] = 3/2
---> 16 (y - x) = 3xy
From eqn (1), y = 20 - x, substitute this x value in the above equation, we get
---> 16 (20 - x - x) = 3x(20 - x)
---> 16 (20 - 2x) = 60x - 3x2
---> 320 - 32x - 60x + 3x2 = 0
---> 3x2 - 92x + 320 = 0
By using the formula, x = {-b ± √[b2 - 4ac]} / 2a
We can find the value of x,
Where, b = - 92
a = 3
c = 320
---> x = {92 ± √[(-92)2 - 4 * 3 * 320]} / 2 * 3
= {92 ± √[8464 - 3840]} / 6
= {92 ± √[4624]} / 6
= {92 ± 68} / 6
= {(92 + 68)/6}, {(92 - 68)/6}
= {160/6}, {24/6}
---> x = 26.667, 4
Now, going through options, we get x = 4 km/h
---> Speed of pedestrian = 4 km/h
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22. A steamer covers 80 km in 4 hrs in still water. If speed of boat is 50% more than that of stream then time required for steamer to cover this distance in upstream?




SHOW ANSWER
Correct Ans:12 hrs
Explanation:
Speed of steamer in still water = 80/4
= 20 km/hr.
Speed of stream = 20*100 / 150
= 40/3 km/hr.
Speed of steamer in upstream = 20 - 40/3
= 20/3 km/hr.
Required time = (80*3)/20
= 12 hrs.
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23. A man makes four trips of equal distances. His speed on first trip was 60 km/hr and in each subsequent trip his speed was half of the previous trip.What is the average speed (in km/hr) of the man in these four trips?




SHOW ANSWER
Correct Ans:16
Explanation:
Let distance travelled in each trip = d km
Total distance travelled = 4d km
Speed in first trip = 60
Speed in second trip = 60/2 = 30
Speed in third trip = 15
Speed in fourth trip = 7.5
Time taken in each trip = distance/speed
= d/60 + d/30 + d/15 + d/7.5
= d/7.5 (1/8 + 1/4 + 1/2 + 1)
= d/7.5 * (1 + 2 + 4 + 8)/8
= d/7.5 * 15/8
= d/4
Average speed = total distance/total time
= 4d / (d/4) = (4d * 4)/d = 4*4 = 16
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24. A boat goes 16 km upstream in 4 hours and travels the same distance downstream in 96 minutes. What is the speed of the boat in still water?




SHOW ANSWER
Correct Ans:7 km/hr
Explanation:
Let the speed of boat be x km/hr
Let the speed of stream be y km/hr
Time taken to go upstream = 16/(x-y) = 4
(x-y) = 4 km/hr
Time taken to go upstream = 16/(x+y) = 96/60
(x+y) = 10 km/hr
Solving these equations give,
x = 7 km/hr, y = 3 km/hr
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25. The wheel of a car made 200 rotations, how much distance did the car travel if the diameter of the wheel is 28 inches. (1 inch = 2.54cm)




SHOW ANSWER
Correct Ans:447.04 m
Explanation:
Distance travelled in 1 rotation = 2Ï€r [circumference of the wheel]
Radius = 28/2 = 14 inches
Distance travelled in 200 rotations = 200 * 2Ï€r = (400*22*14)/7 inches
= 17600inches
Distance in cm = 17600*2.54 = 44704 cm
1 m = 100 cm
Distance in m = 447.04 m
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26. A car covers 1/5th the distance from A to B at the speed of 8 km/hr, 1/10 of the distance at 25 km per hour and the remaining at the speed of 20 km per hour. Find the average speed of the whole journey.




SHOW ANSWER
Correct Ans:15.625 km/hr
Explanation:
Time = Distance/Speed
Remaining distance = 1 - (1/5 + 1/10)
= 1 - 3/10 = 7/10
Total time = 1/(5*8) + 1/(10*25) + 7/(10*20)
= 1/40 + 1/250 + 7/200
= (25 + 4 + 35)/1000 = 64/1000
= 8/125 hrs
Average speed = Total distance/Time taken
= 125/8 = 15.625 km/hr
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27. A car crosses a man walking at 6 km/hr. The man can see the things upto 450 m only in one direction due to fog. He sees the car which was going in the same direction for 4.5 minutes. What is the speed of the car?




SHOW ANSWER
Correct Ans:12 km/hr
Explanation:
Time = 4.5/60 = 45/600
Time = Distance / Speed
45/600 = (450/1000) / Speed
Speed = (45*600)/(100*45)
Speed = 6 km/hr
peed of car - speed of man = 6
x - 6 = 6
x = 12 km/hr
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28. A motorboat went downstream for 36 km and immediately returned. It took the boat twice as long to make the return trip. If the speed of the river flow were twice as high, the trip downstream and back would take 648 minutes. Find the speed of the boat in still water and the speed of the river flow.




SHOW ANSWER
Correct Ans:12 km/hr, 4 km/hr
Explanation:
Given, Downstream Distance = 36 km
Let Speed of Boat = B km/hr
and Speed of River (stream) = R km/hr
W.K.T: Speed of Downstream = speed of boat + Speed of stream
----> Downstream Speed = B + R km/hr
W.K.T: Speed of Upstream = speed of boat - Speed of stream
----> Upstream Speed = B - R km/hr

Time taken for downstream = Distance / Downstream Speed = 36/(B + R)
Time taken for upstream = 36/(B - R)
Given, time taken for upstream is double the time taken for downstream
---> Time taken for upstream = 2 * Time taken for downstream
---> 36/(B - R) = 2 * [36/(B + R)]
---> 1/(B - R) = 2/(B + R)
---> B + R = 2(B - R)
---> B + R = 2B - 2R
---> B = 3R ---- eqn (1)

Given, if the speed of the River flow were twice as high,
---> R = 2R
Then, Downstream Speed = B + 2R
----> Substituting B = 3R (from eqn (1)) in above expression
----> Downstream Speed = 3R + 2R = 5R km/hr
And Upstream speed = B - 2R = 3R - 2R = R km/hr

Given, if R = 2R, then, Downstream time + Upstream time = 648 minutes
---> (36/5R) + (36/R) = (648/60) hr
---> [36/R] [(1/5) + 1] = (648/60)
---> [36/R] [6/5] = (648/60)
---> 216/5R = 648/60
---> R = (216 * 60) / (5 * 648)
---> R = 12960 / 3240
---> R = 4
Hence, Speed of River = R = 4 km/hr
From eqn (1), Speed of Boat = 3R = 3 * 4 = 12 km/hr
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29. A plane left 30 min later than the scheduled time and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed?




SHOW ANSWER
Correct Ans:750 km/hr
Explanation:
Let the usual speed of the plane be x km/hr.
New Speed of the plane = (x + 250) km/hr
Given, distance to reach the destination =1500 km

According to the question,
Scheduled (Usual) time - current time taken by the train to reach destination after 30 mins delay = 30 mins
----> (1500/x) - [1500/(x + 250)] = 30/60 [Since 30 mins is converted into hours]
---> {[1500(x + 250)] - 1500x} / x(x + 250) = 1/2
---> {[1500x + 375000] - 1500x} / x(x + 250) = 1/2
---> 375000 / x(x + 250) = 1/2
---> 375000 * 2 = x(x + 250)
---> 750000 = x2 + 250x
---> x2 + 250x - 750000 = 0
---> x2 + 1000x - 750x - 750000 = 0
---> x(x + 1000) - 750(x + 1000) = 0
---> (x + 1000) (x - 750) = 0
---> x = -1000, 750
Hence, Usual speed of the plane = 750 km/hr
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30. A train met with an accident 60 km away from Howrah station. It completed the remaining journey at 5/6 th of the previous speed and reached Cuttack station 1 hour 12 mins late. Had accident taken place 60 km further, it would have been only 1 hour late.
A) What is the normal speed of the train?
B) What is the distance between Howrah and Cuttack? 




SHOW ANSWER
Correct Ans:60 km/hr, 420 km
Explanation:
Let normal speed of the train be x km/hr and
distance between Howrah and Cuttack be y km.
Then, Actual Time taken by train to reach destination = Distance/Speed = y/x hrs.

Case 1:
Distance covered by the train till accident point = 60 km
Till accident point, the train runs in normal speed ie., x km/hr
Time taken by the train till accident = 60/x hours

For the remaining journey, new speed of train = (5/6) * previous speed = (5x/6)
Remaining distance = y - 60 (ie., Total distance - distance covered till accident)
Then, Time taken = Remaining distance/new speed of train = (y - 60)/ (5x/6)
---> Time taken by the train for remaining journey = 6(y - 60)/5x

According to the question,
Time taken by the train till accident + Time taken by the train for remaining journey = Actual Time taken by train to reach destination + 1 hour 12 minutes
---> (60/x) + [6(y - 60)/5x] = (y/x) + (6/5) [--- Since, converted 1 hour 12 minutes into (6/5) hours]
Solving the above eqn, we get
---> (60/x) + [6(y - 60)/5x] = (y/x) + (6/5)
---> [300 + 6(y - 60)] / 5x = (5y + 6x) / 5x
---> [300 + 6y - 360] = (5y + 6x)
---> 6y - 60 = 5y + 6x
---> y - 6x = 60 ------> eqn (1)

Case 2:
Had accident taken place 60 km further.
Now, distance covered by the train till accident point = 60 + 60 = 120 km
And Speed of the train till accident happens = x km/hr
Then, Time taken till accident = (120 km/x km/hr) = (120/x) hr

For the remaining journey, new speed of train = (5/6) * previous speed = (5x/6)
Remaining distance = y - 120
Then, Time taken by the train for remaining journey = (y - 120)/(5x/6) = 6(y - 120)/5x

According to the question,
Time taken by the train till accident + Time taken by the train for remaining journey = Actual Time taken by train to reach destination + 1 hour
---> (120/x) + [6(y - 120)/5x] = (y/x) + 1
---> [120 * 5 + 6(y - 120)]/5x = (y + x)/x
---> [600 + 6y - 720]/5x = (y + x)/x
---> [6y - 120]/5 = (y + x)
---> 6y - 120 = 5y + 5x
---> y - 5x = 120 ------> eqn (2)

On solving eqn (1) and eqn (2) we get,
---> Eqn (1) - eqn (2), we get, x = 60 km/hr (which is the normal speed of the train) and y = 420 km (which is the distance between Howrah and Cuttack)
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31. Two buses start at same time from Chennai and Bangalore, which are 250 km apart. If the two buses travel towards each other, they meet after 1 hr and if they travel in same direction they meet after 5 hrs. What is the speed of the bus starts from Chennai if it is know that the one which started from Chennai has more speed than the other one? 




SHOW ANSWER
Correct Ans:150 km/hr
Explanation:
Given: Distance - 250 km

WKT, Speed = Distance/Time
If the two buses travel towards each other, they meet after 1 hr ,
Speed of Chennai bus + Speed of Bangalore bus = 250/1 ...(i)

If they travel in same direction they meet after 5 hrs,
Speed of Chennai bus - Speed of Bangalore bus = 250/5
Speed of Chennai bus - Speed of Bangalore bus = 50 ...(ii)

By solving (i) and (ii),
Speed of Chennai bus = 150 km/hr.
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32. A boat can travel 352 km downstream and 112 km upstream in total 24 hours. If respective ratio of speed of boat in still water to speed of stream is 9: 2, then find total distance travelled by boat in 5 hours in downstream is what percent more than total distance travelled by boat in two hours in upstream? 




SHOW ANSWER
Correct Ans:292(6/7)%
Explanation:
Given:
Ratio of speed of boat in still water to speed of stream = 9: 2
Let the speed of the boat in still water and speed of stream be 9X and 2X respectively.

WKT, Time = Distance/Speed
Given that a boat can travel 352 km downstream and 112 km upstream in total 24 hours.
[352/(9X + 2X)] + [112/(9X - 2X)] = 24
(352/11X) + (112/7X) = 24
1/X[32 + 16] = 24
48 = 24X
X = 2 km/hr
Speed of boat in still water = 9X = 9*2 = 18 km/hr
Speed of stream = 2X = 2*2 = 4 km/hr

WKT, Distance = Speed x Time
Speed of downstream = (u + v)
Speed of upstream = (u - v)

Total distance travelled by boat in 5 hours in downstream = (18 + 4) *5 = 110 km
Total distance travelled by boat in 2 hours in upstream = (18 - 4)*2 = 28 km
Required percentage = [(110 - 28)/28]*100 = 292(6/7)%.
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33. A boat goes 28 km downstream and while returning covered only 75% of distance that covered in downstream. If boat takes 3 hr more to cover upstream than downstream then find the speed of boat in still water (km/hr) if speed of stream is (5/9) m/sec? 




SHOW ANSWER
Correct Ans:5 km/hr
Explanation:
Given:
Speed of the stream = (5/9)m/sec
Converting m/sec to km/hr,
Speed of the stream = (5/9) x (18/5) = 2 km/hr

Let the speed of boat in still water be 'X'.
WKT, Time = Distance/Speed
As per the question,
Distance/Speed in upstream - Distance/Speed in downstream = Difference in time
[28 x (75/100)/(X - 2)] - [28/(X + 2)] = 3
[21/(X - 2)] - [28/(X + 2)] = 3
[21X + 42 - 28X + 56]/[X2 - 4] =3
-7X + 98 = 3X2 - 12
3X2 + 7X - 110 = 0
3X2 - 15X + 22X - 110 = 0
3X(X - 5) + 22(X - 5) = 0
(3X + 22) (X - 5) = 0
X = -(22/3), 5
Therefore, speed of the boat in still water is 5 km/hr.
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34. A boat travels with the direction of the current from point A to B and then returns against the current and stop at point C. Distance between B and C is 75% of distance between A and B and boat takes 9 hours to cover total distance. If distance between A to B is 40 Km and Speed of current is 2 km/hr, then find speed of boat in still water ?




SHOW ANSWER
Correct Ans:8 km/hr
Explanation:
Let speed of boat in still water be x km/hr
Water Current speed = 2 km/hr
Distance between A and C = 40 km
Distance between B to C = 40 * 75/100 = 30 km
Speed of downstream = (x + 2) km/hr
Speed of upstream = (x - 2) km/hr
Time taken to cover total distance by boat = 9 hours
Time = Distance/Speed
9 = 40/(x + 2) + 30/(x - 2)
9 = [40(x - 2) + 30(x + 2)] / [(x + 2)(x - 2)]
9 = (40x - 80 + 30x + 60)/(x2 - 4)
9x2 - 36 = 70x -20
9x2 - 70x - 16 = 0
9x2 - 72x + 2x - 16 = 0
9x(x - 8) + 2(x - 8) = 0
x = 8, -2/9
So, the speed of boat in still water = 8 km/hr
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35. Krishna covers a certain distance by train at 25 km/hr. and the equal distance on foot at 4 km/hr. If the time taken by him for the whole journey be 5 hrs and 48 minutes, how much total distance did he cover ?




SHOW ANSWER
Correct Ans:40 km
Explanation:
Total distance = x km
Distance by train = x/2 km
Distance by Feet = x/2 km
Time taken to cover x/2 by train = Distance/Speed = x/50 hours
Time taken to cover x/2 by foot = x/8 hours
x/50 + x/8 = 5(48/60)
(8x + 50x)/400 = 5(4/5)
58x/400 = 29/5
2x/80 = 1
x = 40 km
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36. Train A crosses a pole in 25 seconds and Train B crosses the same pole in 40 seconds. The length of Train A is half the length of Train B. What is the ratio of the speed of Train A to that of Train B? 




SHOW ANSWER
Correct Ans:4 : 5
Explanation:
The length of Train A = (1/2)* The length of Train B
The length of Train A : The length of Train B = 1 : 2
Speed of Train A = Distance/Time = x/25
Speed of Train B = 2x/40 = x/20
Required ratio = (x/25) : (x/20) = 20 : 25 = 4 : 5
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37. Two trains pass each other on parallel lines. Each train is 100 metres long. When they are going in the same direction, the faster one takes 60 seconds to pass the other completely. If they are going in opposite directions they pass each other completely in 10 seconds. Find the speed of the slower train in km/hr.




SHOW ANSWER
Correct Ans:30 km/h
Explanation:
Let the speed of faster train = x and speed of slower train = y.
We know that if
Two trains are moving in the same direction at x m/s, y m/s then relative speed = (x - y) m/s
Two trains are moving in the opposite direction at x m/s, y m/s then relative speed = (x + y) m/s

Case 1: Two trains moving in same direction
Relative distance of two trains = Train 1 length + Train 2 length = 100 + 100 = 200 m
Time taken by faster train to pass slower train (t) = Relative distance/Relative speed
---> t = 200/(x - y)
Given t = 60 sec
---> 60 = 200/(x - y)
---> 60(x - y) = 200
---> 3(x - y) = 10
---> 3x - 3y = 10 ----------(1)

Case 2: Two trains moving in opposite direction
Relative distance = 200 m
Time taken by faster train to pass slower train (t) = 200/(x + y)
Given, t = 10 sec
---> 10 = 200/(x + y)
---> x + y = 20 -------------(2)


Now dividing eqn (1) by 3 and adding eqn (1) and (2), we get,
x - y = 10/3
x + y = 20
_________
2x = 70/3
x = 70/6
Substitute this x value in eqn (2), then y = 20 - (70/6)
---> y = 50/6
Hence, speed of slower train = y = 50/6 m/s

Now converting m/s to km/hr
y = (50/6)*(18/5) (since 1 m/s = 18/5 km/h)
= 30 km/h.
Therefore the speed of slower train = 30 km/h
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38. The ratio of the speed of the boat in still water to the speed of stream is 17 : 6. A boat goes 15.5 km in 45 minute upstream, find the time taken by boat to cover the distance of 18.5 km downstream.




SHOW ANSWER
Correct Ans:25.8 minutes
Explanation:
Let the speed of the boat in still water = 17x,
speed of stream = 6x
Formulae used:
Speed upstream = Speed of boat in still water - Speed of stream
Now, Speed upstream = 17x - 6x = 11x
W.K.T:- Speed = Distance/Time
So, Speed upstream = upstream Distance/upstream Time
---> 11x = (15.5/45)*60
(Here, time is converted from minutes to hour)
---> x = 1.87

Then, Speed of boat in still water = 17x = 17 * (1.87) = 31.79 km/h
Speed of stream = 6x = 6 * (1.87) = 11.22 km/h
So Speed downstream = Speed of boat in still water + Speed of stream.
---> Speed downstream = 31.79 + 11.22 = 43.01 km/h
Time taken by boat to cover downstream
= downstream distance / Speed downstream
= 18.5/43.01
= 0.43 hr
= (43/100)*60
= 25.8 min
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39. A steamer goes downstream from one port to another in 3 hours. It covers the same distance upstream in 4 hours. If the speed of stream is 5 km/h, the distance between the two ports is




SHOW ANSWER
Correct Ans:120 km
Explanation:
Time in upstream = 4 hours
Time in downstream = 3 hours
Speed of stream = 5 km/h
Let the distance from 1 port to another = d
Let the speed of boat = x km/h
We know that Speed (s) = Distance/Time taken.-----(1)
Formulae used:
Speed downstream = speed of boat in still water + speed of stream ---(2)
Speed upstream = speed of boat in still water - speed of stream---(3)

i) For downstream:
Speed downstream = x + 5 (from 2)
Speed downstream = Distance/Time in downstream (from 1)
x + 5 = d/3
d = 3(x + 5)
d = 3x + 15

ii) For upstream:
Speed upstream = x - 5 (from 3)
Speed upstream = Distance/Time in upstream (from 1)
x - 5 = d/4
d = 4(x - 5)
d = 4x - 20

Given it covers the same distance both in upstream and downstream ,so equating both the distance
3x + 15 = 4x - 20
4x - 3x = 20 + 15
x = 35
Therefore speed of boat = x = 35 km/hr.
Speed = Distance/Time taken
Either calculate with upstream or downstream equation we get distance,
Say calculating with upstream distance d = 4x - 20
= 4(35) - 20
= 120 km
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40. A train leaves Chennai at 7.30 am and travels 40 km an hour, another train leaves Chennai at noon and travels 64 km an hour, when and where will the second train overtake the first?




SHOW ANSWER
Correct Ans:480 km, 2.30 pm
Explanation:
Given:
First train: Speed - 40 km/hr; 7.30 am
Second train : Speed - 64 km/hr; 12.00 pm

WKT, Distance = Speed x Time
Distance travelled by first train by 4(1/2) hours,
= 40 x 4(1/2)
= 40 x (9/2)
= 180 km

Time at which second train cross first train = Distance covered by first train/Difference of second train and first train
= 180/(64 - 40)
= 180/24
= 7.5 hour
So, time taken by second train to cross first train is 7.5 hour that is 7 hr 30 minutes.
Exact time when second train cross first train = 12.00 pm + 7.30 = 7.30 pm
Distance from chennai = 64 x 7.5 = 480 km
Therefore, second train meets the first train at 7.30 pm, 480 km.
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