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Time and Speed - Train and Boat

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Time and Speed - Train and Boat

PrepBank · পাতা ৫ / ১৫ · ৪০১–৫০০ / ১,৪৩৯

৪০১.
A boat travelled upstream and then returned, stopping 4 km short of the place from which it started. Speed of the stream is 2 km/hr. If total time taken by the boat is 6 hour and its speed in still water is 4 km/hr, distance travelled upstream is :
  1. ক
    ক) 12 km
  2. খ
    খ) 8 km
  3. গ
    গ) 6 km
  4. ঘ
    ঘ) 9 km
  5. ঙ
    ঙ) 10 km
ব্যাখ্যা

Let distance travelled upstream be x km
x/(4 − 2) +( x − 4)/( 4 + 2) = 6
⇒ x/2 + (x − 4)/6 = 6
⇒ 3 x + x − 4 = 36
⇒ 4 x = 40
⇒ x = 10

৪০২.
A train crosses a platform 90 m long in 50 seconds at a speed of 36 km/hr. What is the time taken by the train to cross an electric pole?
  1. ক
    ক) 30 sec
  2. খ
    খ) 41 sec 
  3. গ
    গ) 45 sec 
  4. ঘ
    ঘ) 53 sec 
ব্যাখ্যা
Question: A train crosses a platform 90 m long in 50 seconds at a speed of 36 km/hr. What is the time taken by the train to cross an electric pole?

Solution:
Speed of the train = (36 × 1000)/(60 × 60) = 10 m/sec
So, in 50 sec it goes = (10 × 50) = 500 m

Since the platform is 90 m so, only train = (500 - 90) m = 410 m

∴ Time taken by the train to cross an electric pole = (410/10) sec = 41 sec
৪০৩.
A train without stopping travels 60 km/h and with stoppage 40 km/h. What is the time taken for stoppage on a route 300 km?
  1. ক
    ক) 10 hours
  2. খ
    খ) 20 hours
  3. গ
    গ) 5 hours
  4. ঘ
    ঘ) 2.5 hours
  5. ঙ
    ঙ) 25 hours
ব্যাখ্যা

Since, the train travels at 60 km/h, it's speed per minute is 1 km per minute. Hence, if it's speed with stoppage is 40 km/h, it will travel 40 minutes per hour i.e. train stops 20 min per hour.
Time taken to travel 300 km with stoppage,
= 300/40 = 7.5 hours
Time taken for stoppage as it stops for 20 min per hour
= (7 × 20 + 10)
= 140 + 10
= 150 minutes.
= 150/60 hours
= 2.5 hours.

৪০৪.
A 180 m long train crosses a milestone in 18 seconds and a train of same length coming from the opposite direction in 12 seconds. The speed of the other train is -
  1. ক
    ক) 56 km/hr
  2. খ
    খ) 62 km/hr
  3. গ
    গ) 64 km/hr
  4. ঘ
    ঘ) 72 km/hr
ব্যাখ্যা
Speed of first train = 180/18 m/sec
                               = 10 m/sec

Let the speed of second train be x m/sec
Relative speed = (10 + x) m/sec

Here
360/(10 + x) = 12
12(10 + x) = 360
120 + 12x = 360
12x = 360 - 120 
12x = 240 
x = 20 m/sec
x = 20 × (18/5) km/hr
x = 72 km/hr
৪০৫.
A train 900 meters long is running at a speed 78 kmph. If it crosses a tunnel in 1 min, then the length of the tunnel is-
  1. ক
    200 m
  2. খ
    300 m
  3. গ
    400 m
  4. ঘ
    500 m
ব্যাখ্যা
Question: A train 900 meters long is running at a speed 78 kmph. If it crosses a tunnel in 1 min, then the length of the tunnel is-

Solution:
৩৬০০ সেকেন্ডে অতিক্রম করে ৭৮০০০ মি.
১ সেকেন্ডে অতিক্রম করে ৭৮০০০/৩৬০০ মি.
৬০ সেকেন্ডে অতিক্রম করে (৭৮০০০ × ৬০)/৩৬০০ মি.
= ১৩০০ মি.

টানেলের দৈর্ঘ্য = ১৩০০ - ৯০০ = ৪০০ মি.
৪০৬.
A man can row three-quarters of a kilometer against the stream in 45/4 minutes and down the stream in 15/2  minutes. The speed (in km/hr) of the man in still water is:
  1. ক
    ক) 8 km/hr
  2. খ
    খ) 7 km/hr
  3. গ
    গ) 6 km/hr
  4. ঘ
    ঘ) 5 km/hr
ব্যাখ্যা
Question: A man can row three-quarters of a kilometer against the stream in 45/4 minutes and down the stream in 15/2 minutes. The speed (in km/hr) of the man in still water is:

Solution: 
three - quarters of a kilometer = 750 meters
45/4 minutes = (45/4) × 60 = 675 sec
15/2 minutes = (15/2) × 60 = 450sec


Rate upstream = 750/ 675 m/sec
                         = 10/9 
Rate downstream =750/450 m/sec
 =5/3 m/sec


∴Rate in still water= (1/2){(10/9) + (5/3)} m/sec
= (1/2){(10 + 15)/9}
= (1/2)(25/9)
= 25/18
= (25/18)(18/5) km/hr.
= 5 km/hr
৪০৭.
While going to office, Salman travels at a speed of 30 kmph and on his way back, he travels at a speed of 45 kmph. What is his average speed of the whole journey?
  1. ক
    45 kmph
  2. খ
    36 kmph
  3. গ
    32 kmph
  4. ঘ
    42 kmph
ব্যাখ্যা
Question: While going to office, Salman travels at a speed of 30 kmph and on his way back, he travels at a speed of 45 kmph. What is his average speed of the whole journey?

Solution:
When distance travelled is same, then average speed = (2ab)/(a + b); (where a and b are two different speeds)

∴ The Average Speed = (2 × 45 × 30)/(45 + 30)
= 2700/75 kmph
= 36 kmph
৪০৮.
Two trains are running in opposite directions with the same speed. If the length of each train is 180 meters and they cross each other in 9 seconds, then the speed of each train (in km/hr) is
  1. ক
    48km/hr
  2. খ
    72km/hr
  3. গ
    36km/hr
  4. ঘ
    60km/hr
  5. ঙ
    None of these
ব্যাখ্যা
Question: Two trains are running in opposite directions with the same speed. If the length of each train is 180 meters and they cross each other in 9 seconds, then the speed of each train (in km/hr) is

Solution:
Let the speed of each train be x m/sec.
Then, relative speed of the two trains = 2xm/sec

Now
⇒ 2x = (180 + 180)/9
⇒ 2x = 360/9
⇒ 2x = 40
∴ x = 20

∴ Speed of each train = 20m /sec = 20(18/5) = 72km/hr
৪০৯.
A train is moving at a speed of 132 km/hr. If the length of the train is 110 meters, how long it will take to cross a railway platform 165 meter long?
  1. ক
    ক) 4.5 seconds
  2. খ
    খ) 5.5 seconds
  3. গ
    গ) 6.5 seconds
  4. ঘ
    ঘ) 7.5 seconds
ব্যাখ্যা
Speed = 132 km/hr = 132 × 5/18 m/s = 110/3 m/s
time = distance/speed = (110 + 165)/(110/3) = 7.5 seconds
৪১০.
Two trains are running at 40 km/hr and 20 km/hr respectively in the same direction. If the faster train completely passes a man sitting in the slower train in 5 seconds, the length of the faster train is -
  1. ক
    19
  2. খ
    13(2/9)
  3. গ
    27(7/9)
  4. ঘ
    33
ব্যাখ্যা

Relative speed = 40 - 20
= 20 km/hr
= 20 × (5/18)
= 50/9 m/s
Time = 5 s
Distance = (50/9) × 5
= 250/9
= 27(7/9).

৪১১.
A and B take part in 100 m race. A runs at 5 kmph. A gives B a start of 8 m and still beats him by 8 seconds. The speed of B is-
  1. ক
    5.15 kmph
  2. খ
    4.14 kmph
  3. গ
    4.25 kmph
  4. ঘ
    4.4 kmph
ব্যাখ্যা
Question: A and B take part in 100 m race. A runs at 5 kmph. A gives B a start of 8 m and still beats him by 8 seconds. The speed of B is-

Solution:
A's speed = 5 × (5/18)m/sec = 25/18 m/sec

Time taken by A to cover 100 m = 100 × (18/25) sec = 72 sec

∴ Time taken by B to cover 92 m = (72 + 8) = 80 sec

∴ B's speed = (92/80) × (18/5) kmph = 4.14 kmph
৪১২.
If Rahim walks at 14 km/hr instead of 10 km/hr for a certain time, he would have walked 20 km more. If Rahim walks at a speed of 10 km/hr, the distance travelled by him within that time is -
  1. ক
    50 km
  2. খ
    55 km
  3. গ
    60 km
  4. ঘ
    None
ব্যাখ্যা
Question: If Rahim walks at 14 km/hr instead of 10 km/hr for a certain time, he would have walked 20 km more. If Rahim walks at a speed of 10 km/hr, the distance travelled by him within that time is-

Solution:
Let,
the actual distance travelled be x km.

Then,
x/10 = (x + 20)/14
⇒ 14x = 10x + 200
⇒ 14x - 10x = 200
⇒ 4x = 200
⇒ x = 200/4
∴ x = 50 km
৪১৩.
Rahim ran half the distance at 5 km/h and the remaining half at 10 km/h. What was his average speed for the entire run? 
  1. ক
    8.67 km/h 
  2. খ
    7.67 km/h 
  3. গ
    5.67 km/h 
  4. ঘ
    6.67 km/h
ব্যাখ্যা

Question: Rahim ran half the distance at 5 km/h and the remaining half at 10 km/h. What was his average speed for the entire run?

Solution:
Let the total distance = 2x km, so each half = x km.

Time for first half = x / 5 hours
Time for second half = x / 10 hours

∴ Total time = (x/5) + (x/10) = (2x + x)/10 = 3x/10 hours

∴Average speed = Total distance / Total time
= 2x ÷ (3x/10)
= 2x × (10/3x)
= 20/3
= 6.67 km/h 

৪১৪.
A man can row at a speed of 12 km/hr in still water to a certain upstream point and back to the starting point in a river which flows at 3 km/hr. Find his average speed for the total journey.
  1. ক
    ক) 11(3/4) km/hr
  2. খ
    খ) 12 (1/4) km/hr
  3. গ
    গ) 12(3/4) km/hr
  4. ঘ
    ঘ) 11(1/4) km/hr
ব্যাখ্যা

Speed of the man in still water = 12 km/hr.
Speed of the stream = 3 km/hr.
Speed downstream = 12 + 3
= 15 km/hr.
Speed upstream = (12 - 3)
= 9 km/hr.
Average speed = (Speed downstream × Speed upstream)/Speed in still water
= (15 × 9)/12
= (15 × 3)/4
= 45/4
= 11(1/4) km/hr.

৪১৫.
A man covers half of his journey at 10 km/h and the remaining half at 4 km/h. His average speed is:
  1. ক
    6.5 km/h
  2. খ
    7.5 km/h
  3. গ
    5.71 km/h
  4. ঘ
    6.82 km/h
ব্যাখ্যা
Question: A man covers half of his journey at 10 km/h and the remaining half at 4 km/h. His average speed is:
Solution:
let, the total road is 2x
first x distance is covered in 10 km/h
∴ time = x/10 h
second x distance is covered in 4 km/h
∴time = x/4 h

∴ average speed is = total distance/ total time
= 2x / (x/10 + x/4)
= 2x / (7x/20)
= 40/7
= 5.71 km/h
৪১৬.
A train 110 metres long is running with a speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the direction opposite to that in which the train is going?
  1. ক
    3s
  2. খ
    4s
  3. গ
    5s
  4. ঘ
    6s
ব্যাখ্যা
Speed of train relative to man = (60 + 6)km/hr
= 66km/hr
= {66 × 5/18}m/sec [ 66 km/hour = 66 × 1000 meters / (60 × 60) seconds = 66 × 5/18 m/s ]
= (55/3)m/sec∴
Time taken to pass the man = (110 × 3/55) sec
= 6sec
৪১৭.
Two trains of equal length are running on parallel lines in the same direction at 52 km/h and 40 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is-
  1. ক
    60 m
  2. খ
    50 m
  3. গ
    45 m
  4. ঘ
    55 m
ব্যাখ্যা
Question: Two trains of equal length are running on parallel lines in the same direction at 52 km/h and 40 km/h. The faster train passes the slower train in 36 seconds. The length of each train is-

Solution:
Let,
The length of each train = x metres.
Then, distance covered = 2x metres.
Relative speed = (52 - 40) km/hr
= {12 × (5/18)} m/sec
= 10/3 m/sec

ATQ,
2x/36 = 10/3
⇒ 2x = 120
∴ x = 60 m
৪১৮.
A train passes two bridges of length 1000 m and 600 m in 120 seconds and 80 seconds respectively. The length of the train.
  1. ক
    200 m 
  2. খ
    250 m 
  3. গ
    220 m 
  4. ঘ
    180 m 
ব্যাখ্যা
Question: A train passes two bridges of length 1000 m and 600 m in 120 seconds and 80 seconds respectively. The length of the train.

Solution:
Distance covered in 120 second = 1000 + length of train(l) 
Distance covered in 80 seconds = 600 + l 
So, distance covered in 40 seconds = (1000 + l) - (600 + l) 
= 400 m 

Speed = 400/40 = 10 m/s 

∴ Distance covered in 80 second = 80 × 10 = 800 m 
So, 600 + l = 800 
∴ Length of the train (l) = 200 m 
৪১৯.
Two trains are running in opposite directions at the same speed. The length of each train is 120 meter. If they cross each other in 12 seconds, the speed of each train (in km/hr) is
  1. ক
    42
  2. খ
    36
  3. গ
    28
  4. ঘ
    20
  5. ঙ
    None of the above
ব্যাখ্যা

Distance covered = 120 + 120 = 240 m
Time = 12s
Let the speed of each train = v.
Then relative speed = v + v = 2v
2v = distance/time
= 240/12
= 20 m/s
Speed of each train = v = 20/2
= 10 m/s
= 10 × 36/10 km/hr
= 36 km/hr

৪২০.
A motorboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:
  1. ক
    ক) 4 km/hr.
  2. খ
    খ) 6 km/hr.
  3. গ
    গ) 5 km/hr.
  4. ঘ
    ঘ) 10 km/hr.
ব্যাখ্যা
Question: A motorboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:

Solution: 
Let
the speed of the stream be x km/hr. Then,
Speed downstream = (15 + x) km/hr,
Speed upstream = (15 - x) km/hr.

Now,
{30/(15 + x)} + {30/(15 - x)} = 9/2
⇒ 30(15 - x) + 30(15 + x)/(15 + x)(15 - x) = 9/2
⇒ (450 - 30x + 450 + 30x)/(225 - x2) = 9/2
⇒ 900/(225 - x2) = 9/2
⇒ 100/(225 - x2) = 1/2
⇒ 200 = 225 - x2
⇒ x2 = 225 - 200
⇒ x2 = 25 
⇒ x2 = 52 
⇒ x = 5

the speed of the stream be 5 km/hr.
৪২১.
A train 240m long passed a pole in 24 seconds. How long will it take to pass a platform 650m long?
  1. ক
    ক) 65s
  2. খ
    খ) 89s
  3. গ
    গ) 100s
  4. ঘ
    ঘ) 130s
ব্যাখ্যা
প্রশ্ন: A train 240m long passed a pole in 24 seconds. How long will it take to pass a platform 650m long?

সমাধান:
ট্রেনটির মোট দূরত্ব অতিক্রম করতে হবে = (240 + 650) মিটার = 890 মিটার 

ট্রেনটি 240 মিটার অতিক্রম করতে সময় নেয় = 24 সেকেন্ড 
ট্রেনটি1 মিটার অতিক্রম করতে সময় নেয় = 24/240 সেকেন্ড 
ট্রেনটি 890 মিটার অতিক্রম করতে সময় নেয় = (24 × 890)/240 সেকেন্ড 
                                                                      = 89 সেকেন্ড
৪২২.
A Man travelled a distance of 61 km in 9 hours. He travelled partly on foot at 4 km/hr and partly on bicycle at 9 km/hr. What is the distance travelled on foot?
  1. ক
    ক) 16 km
  2. খ
    খ) 14 km
  3. গ
    গ) 12 km
  4. ঘ
    ঘ) 10 km
ব্যাখ্যা

Let the time in which he traveled on foot = x hour
Time for travelling on bicycle = (9-x) hr
Distance = Speed×Time, and Total distance = 61 km
So,
4x + 9(9-x) = 61
=> 5x = 20
=> x = 4
So distance traveled on foot = 4x4 = 16 km

৪২৩.
A boat takes 10 hours to cover a distance while traveling upstream, whereas while travelling downstream it takes 9 hours. If the speed of the current is 4 kmph, what is the speed of the boat in still water?
  1. ক
    ক) 74 kmph
  2. খ
    খ) 76 kmph
  3. গ
    গ) 78 kmph
  4. ঘ
    ঘ) 79 kmph
ব্যাখ্যা
Let the speed of the boat in still water be x kmph
Then, Speed downstream = (x+4) kmph.
Speed upstream = (x-4) kmph.
∴ (x+4)× 9 = (x-4)×10
⇒ 9x+36 = 10x-40 ⇒ x = 76 ⇒ x = 76 kmph.
৪২৪.
A train travelling at the speed of x km/h crossed a 300 m long platform in 30 seconds, and overtook a man walking in the same direction at 6 km/h in 20 seconds. What is the value of x?
  1. ক
    96 km/h
  2. খ
    112 km/h
  3. গ
    144 km/h
  4. ঘ
    210 km/h
  5. ঙ
    228 km/h
ব্যাখ্যা

Question: A train travelling at the speed of x km/h crossed a 300 m long platform in 30 seconds, and overtook a man walking in the same direction at 6 km/h in 20 seconds. What is the value of x?

Solution: 
Train speed = x km/h
Length of train = L
Length of platform = 300m
Man's speed = 6 km/h

∴ (x - 6) × (5/18) = L/20
⇒ (5x - 30)/18 = L/20
⇒ 100x - 600 =18L . . . . . . (i)

And, x × (5/18) = (L+ 300)/30
⇒ 150x = 18L + 5400
⇒ v150x - 5400 = 18L . . . . . . (ii)

From equations (i) & (ii)
⇒ 100x - 600 = 150x - 5400
⇒ 50x = 4800
∴ x = 96 

৪২৫.
Three runners P, Q and R run a race, with runner P finishing 15 metres ahead of runner Q and 24 metres ahead of runner R, in another race of same type runner Q finished 10 metres ahead of runner R. Each runner travels the entire distance at a constant speed. The length of the race is?
  1. ক
    140 metres
  2. খ
    150 metres
  3. গ
    160 metres
  4. ঘ
    180 metres
ব্যাখ্যা
Question: Three runners P, Q and R run a race, with runner P finishing 15 metres ahead of runner Q and 24 metres ahead of runner R, in another race of same type runner Q finished 10 metres ahead of runner R. Each runner travels the entire distance at a constant speed. The length of the race is?

Solution:
Let P finish the race of = x metres
Q finish the race of = (x - 15) meters.
R finish the race of = (x - 24) ............. (1)

In another race of Q & R
Q finish race of = x meters.
R finish race of = (x - 10) .............. (2)

Ratio of speeds of Q & R,
(x - 15)/(x - 24) = x/(x - 10)
⇒ (x - 15)(x - 10) = x(x - 24)
⇒ x2 - 15x - 10x + 150 = x2 - 24x
⇒ x2 - 25x + 150 = x2 - 24x
⇒ 150 = x
∴ x = 150 metres
৪২৬.
A train 150 metres long is travelling at 72 km/h. How much time will it take to completely cross a railway platform that is 250 metres long?
  1. ক
    18 sec
  2. খ
    22 sec
  3. গ
    16.5 sec
  4. ঘ
    20 sec
ব্যাখ্যা

Question: A train 150 metres long is travelling at 72 km/h. How much time will it take to completely cross a railway platform that is 250 metres long?

Solution:
Here,
Speed of the running train = 72 km/hr
= {72 × (5/18)} m/sec
= 20 m/sec

And length of the train is = 150 metres
Length of platform = 250 m

So, the time will taken by the train = (Length of train + Length of platform)/Speed
= (150 + 250)/30
= 400/20 
= 20 sec

৪২৭.
A motorboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:
  1. ক
    3 km/hr
  2. খ
    4 km/hr
  3. গ
    5 km/hr
  4. ঘ
    8 km/hr
ব্যাখ্যা
Question: A motorboat, whose speed is 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:

Solution: 
Let
The speed of the stream is x km/hr. Then,
Speed downstream = (15 + x) km/hr,
Speed upstream = (15 - x) km/hr.

Now,
{30/(15 + x)} + {30/(15 - x)} = 9/2
⇒ 30(15 - x) + 30(15 + x)/(15 + x)(15 - x) = 9/2
⇒ (450 - 30x + 450 + 30x)/(225 - x2) = 9/2
⇒ 900/(225 - x2) = 9/2
⇒ 100/(225 - x2) = 1/2
⇒ 200 = 225 - x2
⇒ x2 = 225 - 200
⇒ x2 = 25 
⇒ x2 = 52 
⇒ x = 5

the speed of the stream is 5 km/hr.
৪২৮.
Rajib is walking on a foggy road at a speed of x km/hr. Due to low visibility, he sees only up to 600 meters. If a car overtakes him from behind with the speed of 15 km/hr then he can see the car for 216 seconds. Find the speed of Rajib.
  1. ক
    4 km/hr 
  2. খ
    5 km/hr 
  3. গ
    6 km/hr 
  4. ঘ
    8 km/hr 
ব্যাখ্যা
Question: Rajib is walking on a foggy road at a speed of x km/hr. Due to low visibility, he sees only up to 600 meters. If a car overtakes him from behind with the speed of 15 km/hr then he can see the car for 216 seconds. Find the speed of Rajib.

Solution: 
216 seconds = 216/60 min 
= 3.6 min 
= 3.6/60  hr
=0.06 hr

ATQ,
(15 × 0.06) - 0.06x = 600/1000
⇒ 0.06x = 0.9 - 0.6
⇒ 0.06x = 0.3
⇒ x = 0.3/0.06 = 5 km/hr
৪২৯.
A man rows 24 km upstream in 6 hours and a distance of 35 km downstream in 7 hours. Then the speed of the man in still water is
  1. ক
    4.5 km/hr
  2. খ
    5.5 km/hr
  3. গ
    4 km/hr
  4. ঘ
    6.5 km/hr
  5. ঙ
    None of these
ব্যাখ্যা
Question: A man rows 24 km upstream in 6 hours and a distance of 35 km downstream in 7 hours. Then the speed of the man in still water is

Solution:
Speed of upstream = 24/6 = 4 km/hr.
Speed of downstream = 35/7 = 5km/hr.

∴ Speed of man in still water = (4 + 5)/2 = 4.5 km/hr.
৪৩০.
A can complete a certain job in 12 days. B is 60% more efficient than A. In how many days can B complete the same job?
  1. ক
    ক) 6
  2. খ
    খ) 6.25
  3. গ
    গ) 7
  4. ঘ
    ঘ) 7.5
  5. ঙ
    ঙ) 4.8
ব্যাখ্যা

B is 60% more efficient than A, which means that the rate of B is 1.6 times as greater as that of A.
So, if A can complete the job in 12 days,
then B completes the same job in 12/1.6 = 12/(8/5)
= 15/2
= 7.5 days.

Alternative method:
Ratio of times taken by
A and B = 160 : 100 = 8 : 5
Suppose B alone takes x days to do the job.
Then, 8 : 5 :: 12 : x
or, 8x = 5 × 12
or, x = 7.5 days

৪৩১.
Two trains starting at the same time from two stations 200 km apart and going in opposite directions cross each other at a distance of 110 km from one of the stations. What is the ratio of their speeds?
  1. ক
    9 : 20
  2. খ
    11 : 9
  3. গ
    11 : 20
  4. ঘ
    None of these
ব্যাখ্যা
Question: Two trains starting at the same time from two stations 200 km apart and going in opposite directions cross each other at a distance of 110 km from one of the stations. What is the ratio of their speeds?

Solution:
In the same time, they cover 110 km and 90 km respectively.
Therefore, Ratio of their speeds = 110 : 90 = 11 : 9
 
৪৩২.
A runner can complete a 750 m race in two and a half minutes. What is the speed of runner in km/hr?
  1. ক
    17.95 km/hr
  2. খ
    18 km/hr
  3. গ
    5 km/hr
  4. ঘ
    16 km/hr
ব্যাখ্যা
Question: A runner can complete a 750 m race in two and a half minutes. What is the speed of runner in km/hr?

Solution:
We are given that the first runner can complete a 750 m race in 2 minutes and 30 seconds or 150 seconds. 
∴ Speed of the first runner = 750/150 = 5m/sec 

We convert this speed to km/hr by multiplying it by 18/5. 
Speed of the first runner = (5 × 18)/5 = 18 km/hr
৪৩৩.
A car travels at a speed of 80 km/hr and reaches its destination in 6 hours. A bus travels at a speed of 50 km/hr and reaches its destination in 10 hours. What is the ratio of the distance covered by the car to the distance covered by the bus?
  1. ক
    7 : 8
  2. খ
    12 : 13
  3. গ
    4 : 5
  4. ঘ
    24 : 25
  5. ঙ
    None of these
ব্যাখ্যা
Question: A car travels at a speed of 80 km/hr and reaches its destination in 6 hours. A bus travels at a speed of 50 km/hr and reaches its destination in 10 hours. What is the ratio of the distance covered by the car to the distance covered by the bus?

Solution:
Car,
Speed = 80 km/hr
Time = 6 hours
∴ Distance = 80 × 6 = 480 km

And Bus,
Speed = 50 km/hr
Time = 10 hours
∴ Distance = 50 × 10 = 500 km

∴ Ratio of distances (Car : Bus) = 480 : 500 = 24 : 25
৪৩৪.
A train takes 12 seconds to cross a telegraph pole. It takes 42 seconds to cross a tunnel. What is the ratio of the length of the tunnel to that of the train?
  1. ক
    3 : 2
  2. খ
    3 : 5
  3. গ
    4 : 7
  4. ঘ
    5 : 2
ব্যাখ্যা

Question: A train takes 12 seconds to cross a telegraph pole. It takes 42 seconds to cross a tunnel. What is the ratio of the length of the tunnel to that of the train?

Solution:
Let the speed of the train be x m/s

While crossing the telegraph pole,
The train travels 12 × x meters = 12x meters, which is the length of the train.

While crossing the tunnel,
The train travels 42 × x meters = 42x meters

∴ Length of the tunnel = 42x - 12x = 30x meters

∴ Length of the tunnel : Length of the train = 30x : 12x
= 30 : 12
= 5 : 2

৪৩৫.
A train 108 m long moving at a speed of 50 km/hr crosses a train 112 m long coming from opposite direction in 6 seconds. The speed of the second train is _____
  1. ক
    ক) 48 km/hr
  2. খ
    খ) 54 km/hr
  3. গ
    গ) 82 km/hr
  4. ঘ
    ঘ) 66 km/hr
ব্যাখ্যা
Question: A train 108 m long moving at a speed of 50 km/hr crosses a train 112 m long coming from opposite direction in  6 seconds. The speed of the second train is _____

Solution:
Distance covered = (108 + 112)
= 220 meter.
Time = 6 seconds.

Relative speed = 220/6 = 110/3 m/s.
= (110 × 3600)/(3 × 1000) km/hr
= 132 km/hr.

Now,
50 + Speed of second train = 132 km/hr.
∴ Speed of second train = (132 - 50) km/hr.
= 82 km/hr.
৪৩৬.
Karim Traveled 60 miles from Dhaka to Gazipur at a certain speed. If his speed per hour were 2 miles faster, he would need 1 hour less to reach Gazipur. What was his initial speed?
  1. ক
    ক) 8 miles per hour
  2. খ
    খ) 10 miles per hour
  3. গ
    গ) 12 miles per hour
  4. ঘ
    ঘ) 15 miles per hour
  5. ঙ
    ঙ) None
ব্যাখ্যা

Let, initial speed = x
ATQ,
⇒ 60/x - 60/(x+2) = 1
⇒ {60(x+2) - 60x} / x(x+2) = 1
⇒ 60x + 120 - 60x = x2 + 2x
⇒ x2 + 2x - 120 = 0
⇒ x2 + 12x - 10x - 120 = 0
⇒ x(x + 12) - 10(x + 12) = 0
⇒ (x + 12)(x - 10) = 0
Either, x = -12 [Not acceptable] or, x = 10 

৪৩৭.
A runs twice as fast as B and B runs thrice as fast as C. The distance covered by C in 78 minutes, will be covered by A in :  
  1. ক
    ক) 12 minutes
  2. খ
    খ) 13 minutes
  3. গ
    গ) 14 minutes
  4. ঘ
    ঘ) 15 minutes
ব্যাখ্যা
The ratio of the speed of A, B and C = 6 ∶ 3 ∶ 1
The ratio of the time taken = 1/6 ∶ 1/3 ∶ 1 = 1 ∶ 2 ∶ 6

Time taken by C to cover the distance = 78 minutes

If C takes 6 min, then A takes 1 min.
If C takes 78 min, then A takes 78 × (1/6) min.
                                                = 13 minutes.

৪৩৮.
In a race of 1km, A can beat B by 100m. In a 400m, B beats C by 40m. In a race of 500m. A will beat C by-
  1. ক
    45 m
  2. খ
    75 m
  3. গ
    50 m
  4. ঘ
    95 m
ব্যাখ্যা

Question: In a race of 1km, A can beat B by 100m. In a 400m, B beats C by 40m. In a race of 500m. A will beat C by-

Solution:
We know, 
1km = 1000m

∴ While A covers 1000 B covers 900
∴ while A covers 500 B covers 450m

∴ While B covers 400, C covers 360m
∴ While B covers 450, C covers (360 × 450)/400 = 405m

∴ in a 500m race A will beat C  by = (500 - 405) = 95m

That means when A runs 500 meter then B can run 450m then C runs 405m.

৪৩৯.
A train 125m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is-
  1. ক
    ক) 45 km/hr
  2. খ
    খ) 50 km/hr
  3. গ
    গ) 54 km/hr
  4. ঘ
    ঘ) 55 km/hr
ব্যাখ্যা

Speed of the train relative to man
= (125/10) m/sec
= (25/2) m/sec
= (25/2)×(18/5) km/hr
= 45 km/hr
Let the speed of the train be x km/hr.
Then, relative speed = (x−5) km/hr
∴ x−5 = 45
⇒ x = 50km/hr

৪৪০.
If the distribution of movie durations is symmetric around the mean of 1.5 hours following a bell curve and if the standard deviation is 20 minutes, approximately what percentage of all movies are longer than 110 minutes?
  1. ক
    12
  2. খ
    16
  3. গ
    20
  4. ঘ
    25
  5. ঙ
    50
ব্যাখ্যা
Question: If the distribution of movie durations is symmetric around the mean of 1.5 hours following a bell curve and if the standard deviation is 20 minutes, approximately what percentage of all movies are longer than 110 minutes?

Solution:
Mean movie duration, µ = 1.5 hours = 90 minutes 
Standard deviation, σ = 20 minutes 

We are to find the percentage of movies longer than 110 minutes. 
So, X = 110 
Here, z = (X - µ)/σ
= (110 - 90)/20
= 1

Using standard normal distribution, 
Area to the left of z = 1 (i.e. movies ≤ 110 minutes) is approximately 84.13%. 
So, the area to the right (i.e., movies > 110 minutes) is = 100% + (- 84.13)% = 15.87% approx  16%
৪৪১.
A motor boat takes 12 hours to go downstream and it takes 24 hours to return the same distance. What is the time taken by boat in still water?
  1. ক
    15 h
  2. খ
    16 h
  3. গ
    8 h
  4. ঘ
    20 h
  5. ঙ
    None of the above
ব্যাখ্যা
If t1 and t2 are the upstream and down stream times.
Then time taken in still water is given by
(2×t1×t2)/(t1+t2)
= (2×12×24)/36
= 16h
৪৪২.
A takes 30 minutes more than B to reach a destination. The speeds of A and B are in the ratio 4 : 6. Time in which A reaches the destination?
  1. ক
    1.5 hr
  2. খ
    2 hr
  3. গ
    2.5 hr
  4. ঘ
    1.25 hr
ব্যাখ্যা
Question: A takes 30 minutes more than B to reach a destination. The speeds of A and B are in the ratio 4 : 6. Time in which A reaches the destination?

Solution: 
here, ratio of speed is 4 : 6
as the ratio of time is inversly proportional to speed,
ratio of time = 6 : 4

ATQ,
6x - 4x = 30 min
2x = 30 min
x = 15 min = 1/4 hour

A takes total time of = 6 × 1/4 = 1.5 hr
৪৪৩.
The current of a stream keeps running at 4 km in 60 minutes. A boat goes 6 km and back to the beginning stage in 2 hours. The rate of the boat in still water is:
  1. ক
    ক) 6 km/hr
  2. খ
    খ) 7.5 km/hr
  3. গ
    গ) 8 km/hr
  4. ঘ
    ঘ) 6.8 km/hr
ব্যাখ্যা

Let the speed in still water be x km/hr.
Then,
Speed downstream = (x+ 4) km/hr,
speed upstream = (x-4) km/hr.
6/(x+4) + 6 /(x-4) = 2
=>  1/(x+4) +1/(x-4)=2/6 = 1/3
=> (x+4)+(x-4)/x2-16= 1/3
=> x2-16= 6x
=>  x2 -6x-16= 0
=> (x-8) (x+2) = 0
=>  x = 8.
∴ Speed of boat in still water = 8 km/hr.

৪৪৪.
Two boats are spotted on the two sides of a lighthouse. If the angle of depression made by both the boats from top of the lighthouse is 30° and 45° and the height of the light house is 125 m then find the distance between the two boats.
  1. ক
    1 m
  2. খ
    188.56 m
  3. গ
    250 m
  4. ঘ
    197.17 m
ব্যাখ্যা

Let AB be the lighthouse and the two boats be at C and D
AB = 125 m

tan30° = BC/AB
= x/125
= 1/v3
x = 72.17 m

tan45° = BD/AB
= y/125
= 1
y = 125 m
Therefore,
the distance between the two boats is = x + y
= 72.17 + 125
= 197.17 m

৪৪৫.
A boat can travel 48 km upstream in 6 hours. If the speed of the stream is 2 km/hr, how much time will the boat take to cover a distance of 120 km downstream?
  1. ক
    20 hours
  2. খ
    18 hours
  3. গ
    10 hours
  4. ঘ
    19 hours
ব্যাখ্যা

Question: A boat can travel 48 km upstream in 6 hours. If the speed of the stream is 2 km/hr, how much time will the boat take to cover a distance of 120 km downstream?

Solution:
Distance covered by a boat in 6 hours = 48 km
Rate upstream of boat = 48/6 
= 8 km/hr

Now,
Speed of stream = 2 km/hr
∴ Speed of boat in still water = (8 + 2)
= 10 km/hr

∴ Rate downstream of boat = (10 + 2) km/hr
= 12 km/hr

∴ Time taken in covering 120 km distance = 120/12 
= 10 hours

৪৪৬.
Prova travels a distance of 9 km from her house to the school by auto-rickshaw at 18 km/hr and returns on rickshaw at 15 km/hr. Find the average speed for the whole journey.
  1. ক
    ক) 12.2 km/hr
  2. খ
    খ) 16.3 km/hr
  3. গ
    গ) 19.8 km/hr
  4. ঘ
    ঘ) 26.6 km/hr
ব্যাখ্যা
Question: Prova travels a distance of 9 km from her house to the school by auto-rickshaw at 18 km/hr and returns on rickshaw at 15 km/hr. Find the average speed for the whole journey.

Solution: 
total distance = 9 + 9 = 18 km

time taken by auto rickshaw = 9/18 hr
= 1/2 hr 

time taken by auto rickshaw = 9/15 hr
= 3/5 hr 

total time = 1/2 hr  +  3/5 hr 
= 11/10 hr

average speed = 18/11/10
= 180/11 km/hr
= 16.3 km/hr
৪৪৭.
10 minutes after a plane leaves the airport, it is reported to be 40 miles away. What is the average speed in miles per hour of the plane?
  1. ক
    ক) 560
  2. খ
    খ) 400
  3. গ
    গ) 240
  4. ঘ
    ঘ) 200
ব্যাখ্যা
Question: 10 minutes after a plane leaves the airport, it is reported to be 40 miles away. What is the average speed in miles per hour of the plane?

Solution: 
গড় বেগ = {40/(10/60)} miles/hour
= 40 × 6 miles/hour
= 240 miles/hour
৪৪৮.
What is the length of a train that takes 12 seconds to pass a pole when it runs at a speed of 24 km/h?
  1. ক
    75 m
  2. খ
    80 m
  3. গ
    85 m
  4. ঘ
    95 m
ব্যাখ্যা
Question: What is the length of a train that takes 12 seconds to pass a pole when it runs at a speed of 24 km/h?

Solution:
speed = 24 km/h
= (24 × 5/18) m/s
= 20/3 m/s

∴ The length of the train = (12 × 20/3) m = 80 m
৪৪৯.
P, Q and R are three towns on a river which flows uniformly. Q is equidistant from P and R. I row from P to Q and back in 10 hours and I can row from P to R in 4 hours. Compare the speed of my boat in still water with that of the river.
  1. ক
    ক) 4 : 3
  2. খ
    খ) 5 : 3
  3. গ
    গ) 6 : 5
  4. ঘ
    ঘ) 7 : 3
ব্যাখ্যা

Let PQ = Qr = x km
Let speed downstream = a km/hr.
and speed upstream = b km/hr.

Then,
x/a + x/b = 10
x = 10ab/(a + b) .........(i)

And,
2x/a = 4
x = 4a/2
x = 2a .............(ii)

From (i) and (ii) we have:
2a = 10ab/(a + b)
5b = a + b
a = 4b

Required ratio = Speed in the water/Speed of river
= {1/2(a + b)}/{(1/2) (a - b)}
= (a + b)/(a - b)
= (4b + b)/(4b - b)
= 5b/3b
= 5/3

৪৫০.
A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?
  1. ক
    420 meters
  2. খ
    210 meters
  3. গ
    240 meters
  4. ঘ
    280 meters
ব্যাখ্যা
Question: A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?

Solution:
Speed = (54 × 1000)/3600 
= 15 m/sec

Length of the train = (15 × 20) m = 300 m
Let
the length of the platform be x meters.

Then,
(x + 300)/36 = 15
⇒ x + 300 = 540
∴ x = 240 meters
৪৫১.
A man in a train notices that he can count 21 telephone posts in one minute. If they are known to be 25 meters apart, then at what speed is the train travelling?
  1. ক
    60 km/h
  2. খ
    50 km/h
  3. গ
    40 km/h
  4. ঘ
    30 km/h
ব্যাখ্যা
Question: A man in a train notices that he can count 21 telephone posts in one minute. If they are known to be 25 meters apart, then at what speed is the train travelling?

Solution:
Number of gaps between 21 telephone posts = 20

∴ Distance travelled in 1 minute or 60 sec = (25 × 20) meter
= 500 m

∴ Speed of the train = (500 ÷ 60) m/sec
= 25/3 m/sec
= {(25/3) × (18/5)} km/h
= 30 km/h
৪৫২.
A man covers half of his journey at 6 km/h and the remaining half at 3 km/h. His average speed is?
  1. ক
    12 kmph
  2. খ
    6 kmph
  3. গ
    5 kmph
  4. ঘ
    9 kmph
  5. ঙ
    4 kmph
ব্যাখ্যা
Question: A man covers half of his journey at 6 km/h and the remaining half at 3 km/h. His average speed is?

Solution:
Given that,
A man covers half of his journey at 6 km/h
And The remaining half at 3 km/h.

We know that,
Average speed = 2xy/(x + y)
= (2 × 6 × 3)/(6 + 3)
= 36/9
= 4 kmph
৪৫৩.
A runs twice as fast as B and B runs thrice as fast as C. The distance covered by C in 144 minutes, will be covered by A in:
  1. ক
    12 minutes
  2. খ
    24 minutes
  3. গ
    18 minutes
  4. ঘ
    15 minutes
ব্যাখ্যা
Question: A runs twice as fast as B and B runs thrice as fast as C. The distance covered by C in 144 minutes, will be covered by A in:

Solution:
Let,
The speed of C = x
So the speed of B = 3x
and the speed of A = 3x × 2
= 6x

Ratio of speed A : B : C = 6x : 3x : x
= 6 : 3 : 1

Then ratio of time taken = 1/6 : 1/3 : 1
= 1 : 2 : 6

Hence, time taken by A = 144/6 minutes
= 24 minutes
৪৫৪.
A motorcycle covers 40km with a speed of 20km/hr. Find the speed of the motorcycle for the next 40km journey so that the average speed of the whole journey will be 30km/hr.
  1. ক
    ক) 70 km/hr
  2. খ
    খ) 52.5 km/hr
  3. গ
    গ) 60 km/hr
  4. ঘ
    ঘ) 60.5 km/hr
ব্যাখ্যা
Question: A motorcycle covers 40km with a speed of 20km/hr. Find the speed of the motorcycle for the next 40km journey so that the average speed of the whole journey will be 30km/hr.

Solution:
ধরি 
পরবর্তী 40km ভ্রমণের গতিবেগ = x km/hr

প্রশ্নমতে,
(40/20) + (40/x) = 80/30
⇒ (1/20) + (1/x) = 2/30
⇒ 1/x = (1/15) - 1/20
⇒ 1/x = (4 - 3)/60
⇒ 1/x =1/60
 x = 60
৪৫৫.
A boat can travel from point A to point B and return back to point A in 10 hours. Speed of the boat in still water is 8 km/h and the speed of the stream is 4 km/h. Find the distance between A and B.
  1. ক
    ক) 25 km
  2. খ
    খ) 27 km
  3. গ
    গ) 30 km
  4. ঘ
    ঘ) 29 km
ব্যাখ্যা
Speed of boat along the stream = 8 + 4 = 12 km/h
Speed of boat against stream = 8 - 4 = 4 km/h
ATQ, 
x/12 + x/4 = 10
⇒ (x+3x)/12 = 10
⇒ 4x = 120
⇒ x = 30 km
৪৫৬.
If it is 250 miles from Dhaka to Dinajpur and 120 miles from Dhaka to Pabna, what percentage of the distance from Dhaka to Dinajpur is the distance from Dhaka to Pabna? 
  1. ক
    ক) 12
  2. খ
    খ) 24
  3. গ
    গ) 36
  4. ঘ
    ঘ) 48
ব্যাখ্যা
ঢাকা থেকে পাবনার দূরত্ব = 120  মাইল 
ঢাকা থেকে দিনাজপুরের দূরত্ব = 250 মাইল 

নির্ণেয় শতাংশ = (ঢাকা থেকে পাবনার দূরত্ব/ঢাকা থেকে দিনাজপুরের দূরত্ব) × 100%
                       = (120/250) × 100%
                        = 48%
৪৫৭.
A boat takes 6 hours to move downstream from point P to Q and to return to point P moving upstream. If the speed of the stream is 4 km/hr and speed of the boat in still water is 6 km/hr, what is the distance between point P and Q?
  1. ক
    11 km
  2. খ
    10 km
  3. গ
    9 km
  4. ঘ
    8 km
ব্যাখ্যা
Question: A boat takes 6 hours to move downstream from point P to Q and to return to point P moving upstream. If the speed of the stream is 4 km/hr and speed of the boat in still water is 6 km/hr, what is the distance between point P and Q?

Solution:
Let the distance between points P and Q = X km
Speed downstream = 6 + 4= 10 km/hr
Speed upstream = 6 - 4 = 2 km/hr

ATQ,
X/10 + X/2 = 6
⇒ (X + 5X)/10 = 6
⇒ 6X = 10
∴ X = 10
৪৫৮.
Robinson crosses a street 500 m long in 6 minutes. His speed in km, per hour is:
  1. ক
    ক) 10 km
  2. খ
    খ) 8 km
  3. গ
    গ) 4 km
  4. ঘ
    ঘ) 5 km
ব্যাখ্যা
Question: Robinson crosses a street 500 m long in 6 minutes. His speed in km, per hour is:

Solution: 
  ৬ মিনিটে যায় ৫০০ মিটার
∴ ১ মিনিটে যায় ৫০০/৬ মিটার
∴ ৬০ মিনিট বা ১ ঘণ্টায় যায় (৫০০ × ৬০)/৬ মিটার 
= ৫০০০ মিটার বা ৫ কিলোমিটার। 
৪৫৯.
A train passes a man at 110 kmph running towards the train at the speed of 10 kmph. If it took 3 seconds to cross the man, what would be the length of the train?
  1. ক
    80 m 
  2. খ
    100 m 
  3. গ
    150 m 
  4. ঘ
    200 m 
ব্যাখ্যা
Question: A train passes a man at 110 kmph running towards the train at the speed of 10 kmph. If it took 3 seconds to cross the man, what would be the length of the train?

Solution: 
As both of them are facing towards each other.
total speed will be = 110 + 10 kmph
= 120 kmph

length of the train = 120 × (3/3600) km
= 0.1 km
= 100 m
৪৬০.
The distance between cities A and B is 120 miles. A car travels from A to B at 60 miles per hour and returns from B to A along the same route at 40 miles per hour. What is the average speed for the round trip?
  1. ক
    48
  2. খ
    50
  3. গ
    52
  4. ঘ
    56
ব্যাখ্যা
Question: The distance between cities A and B is 120 miles. A car travels from A to B at 60 miles per hour and returns from B to A along the same route at 40 miles per hour. What is the average speed for the round trip?

Solution:
দেওয়া আছে,
A এবং B শহরের দূরত্ব = 120 মাইল

A থেকে B তে যাওয়ার সময়,
60 মাইল অতিক্রম করে = 1 ঘণ্টায় 
∴ 1 মাইল অতিক্রম করে = 1/60
∴ 120 মাইল অতিক্রম করে = 120/60 = 2 

B থেকে A তে ফেরার সময়,
40 মাইল অতিক্রম করে1
∴ 1 মাইল অতিক্রম করে 1/40
∴ 120 মাইল অতিক্রম করে 120/40 = 3

মোট সময় = (2 + 3) ঘণ্টা = 5 ঘণ্টা 
মোট অতিক্রান্ত দূরত্ব = (120 + 120) মাইল = 240 মাইল

∴  গড় গতিবেগ = মোট দূরত্ব/মোট সময় = (240/5) মাইল/ঘণ্টা = 48 মাইল/ঘণ্টা 

• বিকল্প সমাধান:
2xy/(x + y)
= (2 × 60 × 40)/(60 + 40)
= 4800/100 = 48 মাইল/ঘণ্টা 
৪৬১.
A train passes a man at 110 kmph running towards the train at the speed of 10 kmph. If it took 3 seconds to cross the man, what would be the length of the train?
  1. ক
    300 m 
  2. খ
    100 m 
  3. গ
    250 m 
  4. ঘ
    180 m 
ব্যাখ্যা

Question: A train passes a man at 110 kmph running towards the train at the speed of 10 kmph. If it took 3 seconds to cross the man, what would be the length of the train?

Solution: 
As both of them are facing towards each other.
total speed will be = 110 + 10 kmph
= 120 kmph

length of the train = 120 × (3/3600) km
= 0.1 km
= 100 m

৪৬২.
A man can row 7.5 kmph in still water and he finds that it takes him twice as long to row up as to row down the river. Find the rate of stream.
  1. ক
    ক) 10 km/hr.
  2. খ
    খ) 7 km/hr
  3. গ
    গ) 5 km/hr
  4. ঘ
    ঘ) 2.5 km/hr
ব্যাখ্যা

Given that,
time is taken to travel upstream = 2 × times taken to travel downstream
When the distance is constant, speed is inversely proportional to the time
Hence, 2 × speed upstream = speed downstream

Let speed upstream = x
Then speed downstream = 2x

we have,
1/2(x + 2x) = speed in still water
⇒ 1/2(3x)=7.5
⇒ 3x = 15
⇒ x = 5
∴ speed upstream = 5 km/hr

∴ Rate of stream = 1/2(2x - x)
= x/2
= 5/2
= 2.5 km/hr.

৪৬৩.
A train passes two bridges of length 800 m and 400 m in 100 seconds and 60 seconds respectively. The length of the train is-
  1. ক
    350 m
  2. খ
    450 m
  3. গ
    300 m
  4. ঘ
    200 m
ব্যাখ্যা

Question: A train passes two bridges of length 800 m and 400 m in 100 seconds and 60 seconds respectively. The length of the train is-

Solution: 
Let the length of the train is x m and speed is s.
ATQ,
s = (x + 800)/100 and,
s = (x + 400)/60

∴ (x + 800)/100 = (x + 400)/60
or, 60x + 48000 = 100x + 40000
or, 40x = 8000
or, x = 200 m

৪৬৪.
A speedboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. What is the Upstream speed in km/hr?
  1. ক
    20 km/hr
  2. খ
    10 km/hr
  3. গ
    5 km/hr
  4. ঘ
    15 km/hr
ব্যাখ্যা
Question: A speedboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. What is the Upstream speed in km/hr?

Solution:
Let the speed of the stream be x km/hr
Upstream Speed = 15 - x
Downstream Speed = 15 + x
4 hours 30 minutes = 9/2 hours

So,
⇒ {30/(15 + x)} + {30/(15 - x)} = 9/2
⇒ {30(15 - x) + 30(15 + x)}/(15 + x)(15 - x) = 9/2
⇒ (450 - 30x + 450 + 30x)/(225 - x2) = 9/2
⇒ {900/(225 -  x2)} = 9/2
⇒ 225 - x2 = 1800/9
⇒ 225 - x2 = 200
⇒ x2 = 225 - 200
⇒ x2 = 25
⇒ x = 5

∴ So the Upstream speed = 15 - 5 = 10 km/hr
৪৬৫.
A boat moves at a speed of 30 km/h in still water. The speed of the current is 5 km/h. How far will the boat travel downstream in 12 minutes? 
  1. ক
    8 km
  2. খ
    7 km
  3. গ
    6 km
  4. ঘ
    10 km
ব্যাখ্যা

Question: A boat moves at a speed of 30 km/h in still water. The speed of the current is 5 km/h. How far will the boat travel downstream in 12 minutes?

Solution:
Speed of the boat in still water = 30 km/h
Speed of the current = 5 km/h

∴ Speed downstream = 30 + 5 = 35 km/h

Time = 12 minutes = 12/60 hours = 1/5 hours

∴ Distance travelled downstream = 35 × (1/5) km
= 7 km

∴ The distance travelled downstream is 7 km.

৪৬৬.
With a uniform speed, a car covers a distance in 8 hours. Had the speed been increased by 4 km/hr, the same distance could have been covered in 7 hr and 30 min. What is the distance covered?
  1. ক
    480 km
  2. খ
    450 km
  3. গ
    467 km
  4. ঘ
    478 km
  5. ঙ
    494 km
ব্যাখ্যা
Let the speed of car be x km/hr
Distance= Speed × Time
Distance = 8x km
According to the question,
⇒ (x+4)×7.5= 8x
⇒ 7.5x+30= 8x
⇒ 8x−7.5x= 30
⇒ 0.5x= 30
⇒x= (30/0.5)= 60 km/hr
Required distance: = 8 × 60 = 480 km
৪৬৭.
Two trains of equal length are running on parallel lines in the same direction at 72 km and 54 km per hour. The faster train passes the slower train in 30 seconds. What is the length of train?
  1. ক
    50m
  2. খ
    64m
  3. গ
    75m
  4. ঘ
    80m
ব্যাখ্যা
Question: Two trains of equal length are running on parallel lines in the same direction at 72 km and 54 km per hour. The faster train passes the slower train in 30 seconds. What is the length of train?

Solution:
To cross each other, two trains have to cover a distance equal to the sum of the lengths of the train.
Let the length of the trains be = x m each.

So the distance to be covered = 2x.
Now the trains are running int he same direction.
∴ Their relative speed = (72 - 54) km/hr. =18km/hr. = 18 × (5/18) km/hr. = 5m/sec.

So, the time taken by the trains to cove 2x m distance,
= 2x ÷ 5 sec.

∴ By the given conditions,
2x ÷ 5 = 30
⇒ 2x ×  = 30
⇒ 2x = (30 × 5)
⇒ x = 150/2
∴ x = 75

So the length of each train = 75 m.
৪৬৮.
A distance is covered by a cyclist at a certain speed. If a jogger covers half of the distance in double the time, the ratio of the speed of the jogger to that of the cyclist is -
  1. ক
    ক) 1:4
  2. খ
    খ) 4:1
  3. গ
    গ) 1:2
  4. ঘ
    ঘ) 2:1
ব্যাখ্যা

Cyclist:Jogger
Ratio of distance→ 2:1
Ratio of time→ 1:2
Ratio of their speed (Jogger:Cyclist)
= (1/2):(2/1)
= 1:4

৪৬৯.
The length of the train that takes 8 second to pass a pole when it runs at a speed of 36 km/hr is-
  1. ক
    288m
  2. খ
    45m
  3. গ
    48m
  4. ঘ
    80m
ব্যাখ্যা
Question: The length of the train that takes 8 second to pass a pole when it runs at a speed of 36 km/hr is-

Solution: 
speed = 36 km/hr
= (36 × 1000)/3600  m/s
= 10 m/s

∴ The length of the train = (8 × 10) m 
= 80 m
৪৭০.
An outlet pipe can empty a cistern in 9 hours. In what time will it empty 2/3 part of the cistern?
  1. ক
    ক) 8 hours
  2. খ
    খ) 6 hours
  3. গ
    গ) 5 hours
  4. ঘ
    ঘ) 4 hours
ব্যাখ্যা
Question: An outlet pipe can empty a cistern in 9 hours. In what time will it empty 2/3 part of the cistern?

Solution: 
Time taken to empty (2/3) × 9 = 6 hours
৪৭১.
A train 108 m long moving at a speed of 50 km/h crosses a train 112 m long coming from opposite direction in 6 seconds. The speed of the second train is-
  1. ক
    108 km/h
  2. খ
    95 km/h
  3. গ
    89 km/h
  4. ঘ
    82 km/h
ব্যাখ্যা
Question: A train 108 m long moving at a speed of 50 km/h crosses a train 112 m long coming from opposite direction in 6 seconds. The speed of the second train is-

Solution:
Distance covered = (108 + 112)
= 220 meter.
Time 6 seconds.

Relative speed = 220/6 = 110/3 m/s.
= (110 × 3600)/(3 × 1000) km/h
= 132 km/h

Now,
50 + Speed of second train = 132 km/h
∴ Speed of second train = (132 - 50) km/h
= 82 km/h
৪৭২.
The ratio between the speeds of two buses is 5 : 6. If the second bus runs 450 km in 5 hours, then the speed of the first bus is:
  1. ক
    65 km/h
  2. খ
    90 km/h
  3. গ
    75 km/h
  4. ঘ
    102 km/h
ব্যাখ্যা

Question: The ratio between the speeds of two buses is 5 : 6. If the second bus runs 450 km in 5 hours, then the speed of the first bus is:

Solution:
Given that,
Ratio of speeds of first bus : second bus = 5 : 6
Second bus covers 450 km in 5 hours

Now, Speed of second bus = distance/time
= 450/5
= 90 km/h

Let speed of first bus = 5x km/h
Speed of second bus = 6x km/h
We know speed of second bus = 90 km/h
So, 6x = 90
⇒ x = 90/6
∴ x = 15

∴ Speed of first bus = 5x = 5 × 15 = 75 km/h

So the speed of the first bus is 75 km/h.

৪৭৩.
A car starts at A and reaches a destination in 8 hours. If it travels first and second half at the speed of 80 km/hr and 100 km/hr respectively then the distance between A and destination is -
  1. ক
    ক) 818.25 km
  2. খ
    খ) 979.50 km
  3. গ
    গ) 711.11 km
  4. ঘ
    ঘ) 850 km
ব্যাখ্যা

Let,
The distance between A and destination be X
The total time taken by the car to cover X = 8 hours
Since X/2 by 80km/hr and remaining X/2 by 100km/hr
Then by Time = distance/speed, we have
(X/2)/80 + (X/2)/100 = 8
⇒ 1/2 {(X/80 + X/100)} = 8
⇒ X/80 + X/100 = 16
⇒ (5X + 4X)/400 = 16
⇒ 5X + 4X = 6400
⇒ 9X = 6400
⇒ X = 6400/9
⇒ X = 711.11
Hence 711.11 km is the required answer.

৪৭৪.
A boat travels 24 km downstream in 40 minutes. If the speed of the stream is 6 km/h, what is the speed of the boat in still water?
  1. ক
    18 km/h
  2. খ
    24 km/h
  3. গ
    30 km/h
  4. ঘ
    36 km/h
ব্যাখ্যা

Question: A boat travels 24 km downstream in 40 minutes. If the speed of the stream is 6 km/h, what is the speed of the boat in still water?

Solution:
স্রোতের অনুকূলে 40 মিনিটে যায় 24 কিমি
স্রোতের অনুকূলে 1 মিনিটে যায় 24/40 কিমি
স্রোতের অনুকূলে 1 ঘণ্টা বা 60 মিনিটে যায় (24 × 60)/40 কিমি
= 36 কিমি

∴ স্রোতের অনুকূলে বেগ = 36 কিমি/ঘণ্টা

দেওয়া আছে,
স্রোতের বেগ = 6 কিমি/ঘণ্টা।

∴ স্থির পানিতে নৌকার বেগ = স্রোতের অনুকূলে বেগ - স্রোতের বেগ
= 36 - 6 = 30 কিমি/ঘণ্টা।

৪৭৫.
A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?
  1. ক
    200m
  2. খ
    220m
  3. গ
    236m
  4. ঘ
    240m
ব্যাখ্যা
Question: A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?

Solution:
Speed = {54 × (5/18)} m/sec
=15 m/sec
∴ Length of the train =(15 × 20)m
=300m

Let,
the length of the platform be = a metres
Then,
(a + 300)/36 = 15
⇒ a + 300 = 540
⇒ a = 240m
৪৭৬.
Two trains 180 m and 120 m long respectively pass each other in 54 seconds when they run in the same direction and in 18 seconds when run in opposite directions. Find the speed of first train.
  1. ক
    40 km/hr 
  2. খ
    45 km/hr 
  3. গ
    35 km/hr 
  4. ঘ
    20 km/hr 
ব্যাখ্যা
Question: Two trains 180 m and 120 m long respectively pass each other in 54 seconds when they run in the same direction and in 18 seconds when run in opposite directions. Find the speed of first train.

Solution: 
Let the speed of 1st train is S1 and speed of 2nd train is S2 
Time = total distance/ relative speed 

1) In same direction 
54 = (180 + 120)/(S1 - S2) × (5/18)
⇒ (S1 - S2)54 = (300 × 18)/5 
⇒ (S1 - S2) = 20 

2) In opposite direction 
18 = (180 + 120) / (S1 + S2) × (5/18)
⇒ (S1 + S2)18 = (300 × 18)/5 
⇒ (S1 + S2) = 60 

from 1 and 2 
S1 = 40 km/hr
৪৭৭.
A train reaches from A to B in 5 hours travelling at a speed of 63 km/h. If its speed is reduced by 18 km/h, then the time of journey is increased by-
  1. ক
    1 hours
  2. খ
    1.5 hours
  3. গ
    3 hours
  4. ঘ
    2 hours
ব্যাখ্যা
Question: A train reaches from A to B in 5 hours travelling at a speed of 63 km/h. If its speed is reduced by 18 km/h, then the time of journey is increased by-

Solution: 
Here,
Total distance = speed × time 
= 63 × 5 = 315 km

If speed reduced then new speed = (63 - 18) = 45 km/hr 

New time = Total distance/speed 
= 315/45
= 7 hours

Time increased by (7 - 5) = 2 hours

∴ The time of journey is increased by 2 hours.
৪৭৮.
Speed of motorboat in still water is 35 kmph. If the motorboat travels 100 km along the stream in 2 hour 30 min, then the time taken by it to cover the same distance against the stream is-
  1. ক
    2 hour 20 min
  2. খ
    4 hour 20 min
  3. গ
    3 hour 20 min
  4. ঘ
    5 hour 20 min
ব্যাখ্যা
Question: Speed of motorboat in still water is 35 kmph. If the motorboat travels 100 km along the stream in 2 hour 30 min, then the time taken by it to cover the same distance against the stream is-

Solution: 
The speed of the motorboat in still water is 35 km/hr. 
let the speed of the stream = x km/hr 
Downstream speed = Distance/time 
= 100/2.5 
= 40 km/hr 

Speed of stream = 35 + x = 40 
∴ x = 5 km/hr 

Upstream speed = 35 - 5 = 30 km/hr 
Time taken in upstream = 100/30 = 3 hour 20 min
৪৭৯.
A train overtakes two persons who are walking in the same direction to that of the train at 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. What is the length of the train?
  1. ক
    ক) 45
  2. খ
    খ) 50
  3. গ
    গ) 55
  4. ঘ
    ঘ) 65
  5. ঙ
    ঙ) 60
ব্যাখ্যা

Let length and speed of the train be x metre and v kmph respectively.
x/9 = (v − 2) × 5/18 ⋯ ( 1 )
x/10 = (v − 4) × 5/18 ⋯ ( 2 )
Dividing (1) by (2) gives,
10/9 = (v − 2)/(v − 4)
⇒ 10v − 40 = 9v − 18
⇒ v = 22
Substituting the value of v in (1)
x/9 = 100/18
⇒ x = 50

৪৮০.
A train 165 m long is running at a uniform speed of 54 km/hr. How much time will it take to cross a pole?
  1. ক
    ক) 10 seconds
  2. খ
    খ) 11 seconds
  3. গ
    গ) 12 seconds
  4. ঘ
    ঘ) 13 seconds
ব্যাখ্যা
Question: A train 165 m long is running at a uniform speed of 54 km/hr. How much time will it take to cross a pole?

Solution: 
Speed of train = 54 km/hr
= 54 × 5/18 m/sec
= 15 m/sec

Length of the train = 165 meters

Therefore, time taken by the train to cross a pole = length of train/speed of train
= 165/15 sec
= 11 sec

Thus, train takes 11 seconds to cross the pole.
৪৮১.
A train travels 10 miles at a speed of 50 miles per hour. How fast must the train travel on the return trip if the round-trip travel time is to be 20 minutes?
  1. ক
    ক) 55 miles/hours
  2. খ
    খ) 60 miles/hours
  3. গ
    গ) 65 miles/hours
  4. ঘ
    ঘ) 75 miles/hours
  5. ঙ
    ঙ) None
ব্যাখ্যা

To travel 10 miles at a speed of 50 mph, the train needs = 10/50 × 60 = 12 minutes
As the given time is 20 minutes
Then for the round trip the train has to travel 10 miles within 8 minutes 
∴ Required speed = 10 × 8/60 = 75 mph

৪৮২.
Two trains start at the same time from Dhaka and Cumilla and proceed towards each other at 70 kmph and 90 kmph respectively. When they meet, it is found that one train has travelled 60 km more than the other. Find the distance between Dhaka and Cumilla.
  1. ক
    390 km
  2. খ
    480 km
  3. গ
    350 km
  4. ঘ
    450 km
ব্যাখ্যা
Question: Two trains start at the same time from Dhaka and Cumilla and proceed towards each other at 70 kmph and 90 kmph respectively. When they meet, it is found that one train has travelled 60 km more than the other. Find the distance between Dhaka and Cumilla.

Solution:
Let,
They meet after t time.

ATQ,
(90t) = (70t) + 60
⇒ (90t) - (70t) = 60
⇒ 20t = 60
∴ t = 3 hours

So the distance between Dhaka and Cumilla = (90 × 3) + (70 × 3)
= 270 + 210
= 480 km
৪৮৩.
A person swimming in a stream that flows 3 km/hr finds that in a given time, he can swim four times as far with the stream as he can against it. At what rate does he swim?
  1. ক
    3 km/hr
  2. খ
    5 km/hr
  3. গ
    4.5 km/hr
  4. ঘ
    6 km/hr
ব্যাখ্যা

Question: A person swimming in a stream that flows 3 km/hr finds that in a given time, he can swim four times as far with the stream as he can against it. At what rate does he swim?

Solution: 
ধরি,
স্রোতের প্রতিকূলে গতিবেগ = x কিমি/ঘন্টা
এবং স্রোতের অনুকূলে গতিবেগ = 4x কিমি/ঘন্টা।

স্রোতের গতিবেগ = (স্রোতের অনুকূলে গতিবেগ - স্রোতের প্রতিকূলে গতিবেগ)/2
= (4x - x)/2
= 3x/2 কিমি/ঘন্টা

প্রশ্নমতে, স্রোতের গতিবেগ 3 কিমি/ঘন্টা।
⇒ 3x/2 = 3
⇒ 3x = 6
⇒ x = 2

স্থির পানিতে সাঁতার কাটার গতিবেগ = (স্রোতের অনুকূলে গতিবেগ + স্রোতের প্রতিকূলে গতিবেগ)/2
= (4x + x)/2
= 5x/2
= (5 × 2)/2   [x এর মান বসিয়ে]
= 10/2
= 5 কিমি/ঘন্টা

সুতরাং, লোকটি 5 কিমি/ঘন্টা গতিতে সাঁতার কাটে।

৪৮৪.
A train 240 m long is running at a speed of 90 km/hr. In what time will it pass a bridge 260 m long?
  1. ক
    20 seconds
  2. খ
    30 seconds
  3. গ
    40 seconds
  4. ঘ
    50 seconds
  5. ঙ
    55 seconds
ব্যাখ্যা
Speed = 90 km/hr
            = 90 × 5/18 m/s
            = 25 m/s
Total distance = (240 + 260) m
                        = 500 m
Required time = (500m) / (25m/s)
                        = 20 seconds
[ Time = Distance / Speed ]
৪৮৫.
৫০ কি.মি./ঘণ্টা গতিতে চলে, একটি ট্রেন সঠিক সময়ে তার গন্তব্যে পৌঁছে। ট্রেনটি ৪০কি.মি. /ঘণ্টা গতিবেগে চললে তাহলে ২৪ মিনিট দেরি হয়। মোট দূরত্ব কত কিলোমিটার? 
  1. ক
    ক) ৬০ কিলোমিটার 
  2. খ
    খ) ৮০ কিলোমিটার 
  3. গ
    গ) ৮৫ কিলোমিটার 
  4. ঘ
    ঘ) ৯০ কিলোমিটার 
ব্যাখ্যা
মনেকরি
মোট দূরত্ব = ক কিলোমিটার 

প্রশ্নমতে,
ক /৪০ - ক/৫০ = ২৪/৬০
(৫ক - ৪ক)/২০০ = ২৪/৬০ 
ক/২০০ = ২৪/৬০
ক  = (২৪/৬০) × ২০০
ক = ৮০ কিলোমিটার 
৪৮৬.
A boat travel 240 km downstream in 3 hours and the time is taken by the boat to travel the same distance in upstream in 6 hours. Find the speed of stream (in kmph).
  1. ক
    28 km/h.
  2. খ
    24 km/h.
  3. গ
    10 km/h.
  4. ঘ
    12 km/h.
  5. ঙ
    20 km/h.
ব্যাখ্যা
Question: A boat travel 240 km downstream in 3 hours and the time is taken by the boat to travel the same distance in upstream in 6 hours. Find the speed of stream (in kmph).

Solution:
Given that,
Distance (both ways) = 240 km
Time downstream = 3 hours
Time upstream = 6 hours

Now,
Downstream speed = 240/3 = 80 km/h
Upstream speed = 240/6 = 40 km/h

Let,
Speed of boat in still water = B
Speed of stream = S

Then,
B + S = 80 .........(1)
B - S = 40 ...........(2)
Now, (1) - (2) then we get,
⇒ B + S - B + S = 80 - 40
⇒ 2S = 40
∴ S = 40/2 = 20 km/h 

Therefore, the speed of the stream is 20 km/h.
৪৮৭.
A man walk a certain distance and rides back in 4 hours 30 minutes. He could ride both ways in 3 hours. The time required by the man to walk both ways is:
  1. ক
    ক) 5 hours
  2. খ
    খ) 6 hours 30 minutes.
  3. গ
    গ) 5 hours 45 minutes.
  4. ঘ
    ঘ) 6 hours
ব্যাখ্যা
Question: A man walk a certain distance and rides back in 4 hours 30 minutes. He could ride both ways in 3 hours. The time required by the man to walk both ways is:

Solution:
Time taken to ride one way = 3/2
= 1.5 hrs
Time taken to walk one way = 4.5 - 1.5
= 3 hrs
Time taken to walk both way = 3 × 2
= 6 hours
৪৮৮.
Two horses start trotting towards each other, one from A to B and another from B to A. They cross each other after one hour and the first horse reaches B, 5/6 hour before the second horse reaches A. If the distance between A and B is 50 km. what is the speed of the slower horse?
  1. ক
    30 km/h
  2. খ
    20 km/h
  3. গ
    25 km/h
  4. ঘ
    15 km/h
ব্যাখ্যা
Question: Two horses start trotting towards each other, one from A to B and another from B to A. They cross each other after one hour and the first horse reaches B, 5/6 hour before the second horse reaches A. If the distance between A and B is 50 km. what is the speed of the slower horse?

Solution:
If the speed of the faster horse be f and that of slower horse be s 
Then,
f + s = 50/1 = 50 km/h
∴ f = 50 - s

ATQ,
50/s - 50/f = 5/6
⇒ (50f - 50s)/(sf) = 5/6
⇒ 50(f - s) = (5/6)(sf)
⇒ 50(50 - s - s) = (5/6)(sf)
⇒ 6(2500 - 100s) = 5 × s × (50 - s)
⇒ 15000 - 600s = 250s - 5s2
⇒ 5s2 - 600s - 250s + 15000 = 0
⇒ 5s2 - 850s + 15000 = 0
⇒ s2 - 170s + 3000 = 0
⇒ s2 - 150s - 20s + 3000 = 0
⇒ s(s - 150) - 20(s - 150) = 0
⇒ (s - 150)(s - 20) = 0
∴ s = 150, 20 [ 150 not be acceptablr]

∴ The speed of the slower horse is 20 km/h
৪৮৯.
In covering a distance of 48 km, Robin takes 2 hours more than Karim. If Robin triples his speed, he would take 2 hours less than Karim. What is Robin's original speed in km/h?
  1. ক
    8 km/h
  2. খ
    10 km/h
  3. গ
    12.5 km/h
  4. ঘ
    15 km/h
ব্যাখ্যা

Question: In covering a distance of 48 km, Robin takes 2 hours more than Karim. If Robin triples his speed, he would take 2 hours less than Karim. What is Robin's original speed in km/h?

সমাধান:
ধরি, Robin-এর মূল গতিবেগ = x কিমি/ঘন্টা
∴ Robin-এর 48 কিমি অতিক্রম করতে সময় লাগে = 48/x ঘন্টা

ধরি, Karim-এর 48 কিমি অতিক্রম করতে সময় লাগে = t ঘন্টা

প্রথম শর্ত অনুযায়ী:
Robin, Karim-এর চেয়ে 2 ঘন্টা বেশি সময় নেয়,
⇒ 48/x = t + 2 ........ (i)

দ্বিতীয় শর্ত অনুযায়ী:
Robin যদি তার গতিবেগ তিনগুণ করে (3x কিমি/ঘন্টা), তাহলে সে Karim-এর চেয়ে 2 ঘন্টা কম সময় নেয়,
⇒ 48/(3x) = t - 2 ........ (ii)

সমীকরণ (i) থেকে: t = 48/x - 2

সমীকরণ (ii)-তে বসিয়ে পাই,
48/(3x) = (48/x - 2) - 2
⇒ 16/x = 48/x - 4
⇒ 48/x - 16/x = 4
⇒ 32/x = 4
⇒ x = 32/4
⇒ x = 8 কিমি/ঘন্টা

সুতরাং, Robin-এর মূল গতিবেগ হলো 8 কিমি/ঘন্টা।

৪৯০.
Arif travels 1/3rd of the distance at an average speed of 5 km/hr, 2/5the of the distance at an average speed of 4 km/hr and the rest 12 miles in 2 hours. What is the total distance traveled by Arif?
  1. ক
    45
  2. খ
    48
  3. গ
    52
  4. ঘ
    54
  5. ঙ
    None of these
ব্যাখ্যা

Distance = X
Distance covered at 5km and 4 km = x/3 + 2x/5
= (5x + 6x)/15
= 11x/15
Rest of the distance = 1 - 11x/15 = 4x/15
ATQ,
4x/15 = 12
or, x = (12 × 15)/4
or, x = 45.

৪৯১.
A man decided to cover a distance of 6 km in 84 minutes. He decided to cover two-thirds of the distance at 4 km/hr and the remaining at some different speed. Find the speed after the two third distance has been covered -
  1. ক
    ক) 5 kmph
  2. খ
    খ) 7 kmph
  3. গ
    গ) 9 kmph
  4. ঘ
    ঘ) 3 kmph
ব্যাখ্যা

Given that,
Two thirds of the 6 km was covered at 4 km/hr
i.e. 4 km distance was covered at 4 km/hr.
Time taken to cover 4 km = 4 km/(4 km/hr)
= 1 hr
= 60 minutes
Time left = (84 – 60) minutes.
= 24 minutes
Now,
The man has to cover remaining 2 km in 24 minutes
or 24/60
= 2/5 hours
Speed required for remaining 2 km
= 2 km/(2/5)hr
= 5 km/hr.

৪৯২.
A train 180 meters long passes a pole in 12 seconds. How long will it take to pass a platform that is 420 meters long?
  1. ক
    30 seconds
  2. খ
    40 seconds
  3. গ
    80 seconds
  4. ঘ
    60 seconds
ব্যাখ্যা

Question: A train 180 meters long passes a pole in 12 seconds. How long will it take to pass a platform that is 420 meters long?

Solution:
Train's speed = Distance/Time
= 180/12 = 15 m/s

Total distance to pass the platform,
= Length of train + Length of platform
= 180 m + 420 m
= 600 m

∴ Required time = Distance/Speed
= 600/15
= 40 seconds

∴ The train will take 40 seconds to pass platform.

৪৯৩.
Two trains, each 50 m long, moving in opposite directions, cross each other in 4 seconds. If one is moving twice as fast the other, then the speed of the faster train is:
  1. ক
    ক) 16.67 m/sec
  2. খ
    খ) 33.33 m/sec
  3. গ
    গ) 35.5 m/sec
  4. ঘ
    ঘ) 40 m/sec
ব্যাখ্যা
Question: Two trains, each 50 m long, moving in opposite directions, cross each other in 4 seconds. If one is moving twice as fast the other, then the speed of the faster train is:

Solution:
Let the speed of the slower train be x m/sec.
∴ speed of the faster train = 2x m/sec.

Relative speed = (x + 2x) m/sec = 3x m/sec.

Now,
(50 + 50)/4 = 3x
⇒ 12x = 100
⇒ x = 100/12
∴ x = 8.333

∴ speed of the faster train = 2 × 8.333 = 16.67 m/sec.
৪৯৪.
A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is:
  1. ক
    5 : 4
  2. খ
    3 : 2
  3. গ
    5 : 2
  4. ঘ
    3 : 1
ব্যাখ্যা
Question: A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is:

Solution:
Let man's rate upstream be x kmph.
Then, his rate downstream = 2x kmph.

∴ (Speed in still water) : (Speed of stream) = {(2x + x)/2} : {(2x - x)/2}
= (3x/2) : (x/2)
= 3 : 1
৪৯৫.
A man takes 3 hours 45 minutes to row a boat 22.5 km downstream of a river and 2 hours 30 minutes to cover a distance of 10 km upstream . Find the speed of the river current in km /hr.
  1. ক
    ক) 1 km/hr
  2. খ
    খ) 2 km/ hr
  3. গ
    গ) 3 km /hr
  4. ঘ
    ঘ) 4km/ hr
ব্যাখ্যা

We have to find the speed of a current.
t (downstream) = 3 h 45 min = 3.75 h
t (upstream) = 2 h 30 min = 2.5 h
Speed downstream = 22.5 km / 3.75 h = 6 km/h
Speed upstream = 10 km / 2.5 h = 4 km/h

So, Speed of the current = (Speed downstream - speed upstream) / 2
= (6 - 4) / 2
= (2 / 2) km/h
= 1 km/h

৪৯৬.
Masum performs 6/15 of the total journey by rail, 8/20 by bus and the remaining 12 km on foot, his total journey is: 
  1. ক
    60 km
  2. খ
    58 km
  3. গ
    30 km
  4. ঘ
    50 km
ব্যাখ্যা
Question: Masum performs 6/15 of the total journey by rail, 8/20 by bus and the remaining 12 km on foot, his total journey is:

Solution: 
Let, the total journey be x km.

ATQ,
(6x/15) + (8x/20) + 12 = x
or,  (24x + 24x + 720)/60 = x
or, 24x + 24x + 720 = 60x
or, 60x - 48x = 720
or, 12x = 720
or, x = 60

∴ Total journey = 60 km.
৪৯৭.
A man travels at 3 mph in still water. If the current’s velocity is 1 mph, it takes 3 hours to row to a place and come back. How far is the place?
  1. ক
    ক) 2 miles
  2. খ
    খ) 3 miles
  3. গ
    গ) 4 miles
  4. ঘ
    ঘ) 5 miles
ব্যাখ্যা
প্রশ্ন: A man travels at 3 mph in still water. If the current’s velocity is 1 mph, it takes 3 hours to row to a place and come back. How far is the place?

সমাধান: 
Let,
the distance be d miles.

Time taken to cover the distance upstream + Time taken to cover the distance downstream = 3 hours 
Here,
Speed upstream = 3 - 1 = 2 mph

Speed downstream = 3 + 1 = 4 mph

So, our equation would be,
d/2 + d/4 = 3 
⇒ (2d + d)/4 = 3
⇒ 3d = 12
⇒ d = 12/3 
∴ d = 4 

The distance is 4 miles.
৪৯৮.
A man rows downstream at 20 km/hr and rows upstream at 15 km/hr. At what speed he can row in still water?
  1. ক
    22 km/hr
  2. খ
    20.5 km/hr
  3. গ
    17.5 km/hr
  4. ঘ
    18 km/hr
ব্যাখ্যা
Question: A man rows downstream at 20 km/hr and rows upstream at 15 km/hr. At what speed he can row in still water?

Solution:
Speed downstream = 20 km/hr
Speed upstream = 15 km/hr

∴ Required speed = (20 + 15)/2 km/hr
= 35/2 = 17.5 km/hr
৪৯৯.
Rahman is a boatman. He can row a boat at the speed of 5 km/hr upstream and 15 km/hr downstream. Find the speed of the stream.
  1. ক
    5 km/hr
  2. খ
    8 km/hr
  3. গ
    6 km/hr
  4. ঘ
    4 km/hr
ব্যাখ্যা

Question: Rahman is a boatman. He can row a boat at the speed of 5 km/hr upstream and 15 km/hr downstream. Find the speed of the stream.

Solution:
Let’s denote:
B as Speed of the boat in still water (km/h)
S as Speed of the stream (km/h)

Speed of the boat upstream is the speed of the boat in still water minus the speed of the stream:
B - S = 5 km/hr -------- (1)

Speed of the boat downstream is the speed of the boat in still water plus the speed of the stream:
B + S = 15 km/hr -------- (2)

(1) + (2)
B - S = 5
B + S = 15
2B = 20
∴ B = 10 km/hr

Putting the value of B in (2)
∴ S = (15 - 10) km/hr = 5 km/hr

∴ The speed of the stream is 5 km/hr

৫০০.
A boatman goes 4 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 7 km in stationary water?
  1. ক
    1 h 12 min
  2. খ
    1 h 24 min
  3. গ
    2 h 12 min
  4. ঘ
    2 h 24 min
  5. ঙ
    None of this
ব্যাখ্যা

Question: A boatman goes 4 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 7 km in stationary water?

Solution: 
Speed along current = 60/10 km/h
= 6 km/h
Speed against current  = 4 km/h

∴ Speed in still water = (Speed against current + Speed along current)/2
= (4 + 6)/2
= 10/2
= 5

∴ Required time = 7/5 h
= [(7/5) × 60] min
= 84 min
= 60 min + 24 min 
= 1 h 24 min