Time and Speed - Train and Boat

PrepBank · পাতা ১৩ / ১৫ · ১,২০১–১,৩০০ / ১,৪৩৯

১,২০১.

A man covers half of his journey at 6 km/hr and the remaining half at 12 km/hr. His average speed is-

ক
ক) 7 km/hr
খ
খ) 9 km/hr
গ
গ) 10 km/hr
ঘ
ঘ) 8 km/hr

ব্যাখ্যা

Question: A man covers half of his journey at 6 km/hr and the remaining half at 12 km/hr. His average speed is-

Solution:
Let the total distance be 2x km.

Then,
Time taken = (x/6) + (x/12)
= (x + 2x)/12
= 3x/12
= x/4 hours

∴ Average speed= Total distance/Time taken
= 2x/(x/4) km/hr
=2x × (4/x) km/hr.
= 8 km/hr

১,২০২.

In a 1000 m race, A can beat B by 100 m. In a race of 400 m, B can beat C by 40 m. By how many metres will A beat C in a race of 500 m?

ক
ক) 85 m
খ
খ) 95 m
গ
গ) 105 m
ঘ
ঘ) 115 m

ব্যাখ্যা

A:B = 1000:900
B:C = 400:360 = 100:90 = 900:810
⇒ A:B:C = 1000:900:810
⇒ A:C = 1000:810
⇒ A:C = 500:405
⇒ In a 500 m race, A beats C by (500-405) m = 95 m

১,২০৩.

If Deep had walked 20 km/h faster he would have saved 1 hour in the distance of 600km. What is the usual speed of Deep?

ক
100
খ
120
গ
150
ঘ
None of these

ব্যাখ্যা

Question: If Deep had walked 20 km/h faster he would have saved 1 hour in the distance of 600km. What is the usual speed of Deep?

Solution:
Let the original speed be x km/h,
Then
600/x - 600/(x + 20) = 1
⇒ 600[{(x + 20) - x}/x(x + 20)] = 1
⇒ x² + 20x - 12000 = 0
⇒ x² + 120x - 100x - 12000 = 0
⇒ (x + 120)(x - 100) = 0
∴ x = - 120, 100 [- 120 not acceptable]

Therefore, original speed = 100 km/h

১,২০৪.

A train, 150 m long, passes a pole in 15 seconds and another train of the same length, travelling in the opposite direction in 10 seconds. What is the speed of the second train?

ক
56 km/h
খ
72 km/h
গ
78 km/h
ঘ
88 km/h
ঙ
None of these

ব্যাখ্যা

Question: A train, 150 m long, passes a pole in 15 seconds and another train of the same length, travelling in the opposite direction in 10 seconds. What is the speed of the second train?

Solution:
Given,
Length of the first train & second train = 150 m
Time to pass a pole = 15 seconds
Time taken by trains to cross each other = 10 sec

Speed of the first train = 150/15
= 10 m/s

And, the relative speed of two trains = (150 + 150)/10
= 30 m/s

Speed of the second train = (30 - 10) × (18/5)
= 20 × (18/5)
= 72 km/h

১,২০৫.

A motor boat takes 2 hours to travel a distance of 9 km downstream and it takes 6 hours to travel the same distance against the current. The speed of the boat in still water-

ক
ক) 3 km/h
খ
খ) 4 km/h
গ
গ) 4.5 km/h
ঘ
ঘ) 5 km/h

ব্যাখ্যা

Question: A motor boat takes 2 hours to travel a distance of 9 km downstream and it takes 6 hours to travel the same distance against the current. The speed of the boat in still water-

Solution:
Downstream speed,
= 9/2 kmph
= 4.5 kmph

Upstream speed,
= 9/6 kmph
= 1.5 kmph

Thus, speed of boat in still water,
= (4.5 + 1.5)/2
= 3 km/h

১,২০৬.

A train left Dhaka for Chittagong at 62 km per hour and at the same time another train left Chittagong for Dhaka at 48 km per hour on the same route. How far apart were the two trains one hour before they met?

ক
ক) 80
খ
খ) 92
গ
গ) 100
ঘ
ঘ) 110

ব্যাখ্যা

ঢাকা থেকে চট্টগ্রামগামী ট্রেনটি 1 ঘন্টায় 62 km. যায়।
চট্টগ্রাম থেকে ঢাকাগামী ট্রেনটি 1 ঘন্টায় 48 km. যায়।
∴ 1 ঘন্টায় দুটি বিপরীতমুখী ট্রেন মোট যায় (62 + 48)
= 110 km. দূরত্ব অতিক্রম করে।
অর্থাৎ মুখোমুখি হওয়ার 1 ঘন্টা আগে তাদের দুরত্ব = 110 km.

১,২০৭.

A jogger is running at a speed of 15 km/hr. In what time he will cross a track of length 400 meters?

ক
96 sec
খ
100 sec
গ
104 sec
ঘ
110 sec

ব্যাখ্যা

Question: A jogger is running at a speed of 15 km/hr. In what time he will cross a track of length 400 meters?

Solution:
Speed = 15 km/hr = 15 × (5/18) m/s = 75/18 m/s
Distance = 400 meters

∴ Required time = 400/(75/18) sec
= (400 × 18)/75 sec
= 96 sec

১,২০৮.

A train is moving at 120 km/hr. The length of the train is 150 meters. How long it will take to cross a platform of length 100 meters?

ক
10 seconds
খ
7.5 seconds
গ
20 seconds
ঘ
25 seconds
ঙ
None of these

ব্যাখ্যা

Question: A train is moving at 120 km/hr. The length of the train is 150 meters. How long it will take to cross a platform of length 100 meters?

Solution:
Speed of train in m/s = 120 × (5/18) m/s = 100/3 m/s
Distance covered to cross the platform is equal to the sum of length of the train and length of the platform.
So, distance= 150 + 100 = 250 meters

Time = distance/speed
= 250 × (3/100)
= 7.5 sec

১,২০৯.

If a boat goes 7 km upstream in 42 minutes and the speed of the stream is 3 kmph, then the speed of the boat in still water is -

ক
10 km/hr
খ
12 km/hr
গ
13 km/hr
ঘ
14 km/hr

ব্যাখ্যা

42 minutes = (42/60) hours = (7/10) hour
Rate upstream = 7 km / (7/10) hour = 10 km/hour
Speed of stream = 3 kmph.
Let speed in still water is x km/hr
Then, speed upstream = (x - 3) km/hr.
∴ x - 3 = 10 or x = 13 kmph

১,২১০.

A train with 20m/s velocity and 200m length is crossing a person with 5m/s velocity. If the person is coming towards the train, it will cross the person in -

ক
ক) 6 seconds
খ
খ) 8 seconds
গ
গ) 7 seconds
ঘ
ঘ) 10 seconds

ব্যাখ্যা

Question: A train with 20m/s velocity and 200m length is crossing a person with 5m/s velocity. If the person is coming towards the train, it will cross the person in -

Solution:
যেহেতু ট্রেন ও ব্যক্তি পরস্পর অভিমুখে আসে তাহলে তাদের আপেক্ষিক বেগ = (২০ + ৫) = ২৫ মি/সে

তাহলে, ব্যক্তিকে অতিক্রম করতে সময় লাগবে = ২০০/২৫ = ৮ সেকেন্ড

১,২১১.

Raju swims 26 km downstream in same time as 14 km upstream. What is his speed in still water if speed of stream is 3 km/hr?

ক
10 km/hr
খ
11 km/hr
গ
9 km/hr
ঘ
12 km/hr
ঙ
8 km/hr

ব্যাখ্যা

Question: Raju swims 26 km downstream in same time as 14 km upstream. What is his speed in still water if speed of stream is 3 km/hr?

Solution:
Let Raju's speed in still water = x km/hr
Speed of stream = 3 km/hr

∴ Downstream speed = x + 3 km/hr
∴ Upstream speed = x - 3 km/hr

Given that,
Time taken to swim 26 km downstream = Time taken to swim 14 km upstream
⇒ 26/(x + 3) = 14/(x − 3) ; [Time = Distance / Speed]
⇒ 26(x - 3) = 14(x + 3)
⇒ 26x - 78 = 14x + 42
⇒ 26x - 14x = 42 + 78
⇒ 12x = 120
⇒ x = 120/12
∴ x = 10

So Raju's speed in still water is 10 km/hr.

১,২১২.

A train takes 10 seconds to cross a pole and 20 seconds to cross a platform of length 200m. What is the length of the train?

ক
ক) 400m
খ
খ) 600m
গ
গ) 200m
ঘ
ঘ) 800m

ব্যাখ্যা

l/10 = (200 + l)/20
2l - l = 200
l = 200m

১,২১৩.

The ratio between the speeds of two trains, Train P and Train Q, is 7:9. If Train Q covers a distance of 270 km in 3 hours, find the speed of Train P in km/h.

ক
60 km/h
খ
70 km/h
গ
80 km/h
ঘ
90 km/h

ব্যাখ্যা

Question: The ratio between the speeds of two trains, Train P and Train Q, is 7:9. If Train Q covers a distance of 270 km in 3 hours, find the speed of Train P in km/h.

সমাধান:
ধরি,
ট্রেন P এবং ট্রেন Q এর গতিবেগ হলো যথাক্রমে 7x কিমি/ঘন্টা এবং 9x কিমি/ঘন্টা।

ট্রেন Q এর গতিবেগ = দূরত্ব / সময়
= 270 কিমি / 3 ঘন্টা
= 90 কিমি/ঘন্টা।

প্রশ্নমতে,
9x = 90
⇒ x = 90 / 9
⇒ x = 10

∴ ট্রেন P এর গতিবেগ = 7x
= (7 × 10) কিমি/ঘন্টা
= 70 কিমি/ঘন্টা।
∴ ট্রেন P এর গতিবেগ হলো 70 কিমি/ঘন্টা।

১,২১৪.

If speed is increased by 4 miles per hour, it takes 4 hour less to cover a distance of 32 km. Find the previous speed.

ক
ক) 8 kmph
খ
খ) 4 kmph
গ
গ) 12 kmph
ঘ
ঘ) 2 kmph

ব্যাখ্যা

ধরি
প্রকৃত বেগ = x kmph
নতুন বেগ = x + 4 kmph

এখানে
(32/x) - {32/(x + 4)} = 4
{32(x + 4) - 32x}/{x(x + 4)} = 4
(32x + 128 - 32x)/(x² + 4x) = 4
128/(x² + 4x) = 4
x² + 4x = 128/4
x² + 4x = 32
x² + 8x - 4x - 32 = 0
x(x + 8) - 4 (x + 8) = 0
(x + 8)(x - 4) = 0
হয়
x + 8 = 0
x = - 8 [গ্রহণযোগ্য নয় ]
অথবা
x - 4 = 0
x = 4

১,২১৫.

Walking 3/5 of his normal speed, Sami is 20 minutes late in reaching his coaching. The usual time taken by him to cover the distance between his home and coaching ?

ক
30 minutes
খ
25 minutes
গ
35 minutes
ঘ
20 minutes

ব্যাখ্যা

Question: Walking 3/5 of his normal speed, Sami is 20 minutes late in reaching his coaching. The usual time taken by him to cover the distance between his home and coaching ?

Solution:
Let,
Distance y meter and speed x meter/min.
usual time taken y/x min

According to the question,
(y/3x/5) - (y/x) = 20
⇒ (y/x) {(5/3) - 1} = 20
⇒ (y/x) (2/3) = 20
∴ (y/x) = 30

∴ The usual time taken by Sami to reach his coaching = 30 minutes

১,২১৬.

Two Motorists drove their cars at a speed of 45 km per hour and 50 km per hour respectively. One car took 10 minutes longer than the other to travel a distance. Find the distance travelled.

ক
60 km
খ
70 km
গ
65 km
ঘ
75 km

ব্যাখ্যা

Question: Two Motorists drove their cars at a speed of 45 km per hour and 50 km per hour respectively. One car took 10 minutes longer than the other to travel a distance. Find the distance travelled.

Solution:
let, the distance is x km

(x/45) - (x/50) = 10/60
⇒ (20x - 18x)/900 = 1/6
⇒ 2x = 150
∴ x = 75 km

১,২১৭.

Walking 6/7th of his usual speed, a man is 12 minutes too late. What is the usual time taken by him to cover that distance?

ক
1 hr 12 mins
খ
2 hr
গ
1 hr
ঘ
1 hr 42 mins
ঙ
None of these

ব্যাখ্যা

Question: Walking 6/7th of his usual speed, a man is 12 minutes too late. What is the usual time taken by him to cover that distance?

Solution:
New speed =6/7 of usual speed
Speed and time are inversely proportional.
Hence new time = 7/6 of usual time
Hence,
7/6 of usual time - usual time = 12 minutes
⇒ 1/6 of usual time = 12 minutes
∴ usual time = 12 × 6 = 72 minutes = 1 hour 12 minutes

১,২১৮.

In one hour, a boat goes 10 km/hr along the stream and 4 km/hr against the stream. The speed of the boat in still water (in km/hr) is-

ক
5 km/h
খ
8 km/h
গ
6 km/h
ঘ
7 km/h

ব্যাখ্যা

Question: In one hour, a boat goes 10 km/hr along the stream and 4 km/hr against the stream. The speed of the boat in still water (in km/hr) is-

Solution:
Speed in still water = (10 + 4)/2 kmph
= 14/2 kmph
= 7 kmph.

So, the speed of the boat in still water is 7 km/h.

১,২১৯.

A man walking at the rate of 5 km/hr. crosses a bridge in 15 minutes. The length of the bridge (in metres) is:

ক
680m
খ
850m
গ
1056m
ঘ
1250m

ব্যাখ্যা

Question: A man walking at the rate of 5 km/hr. crosses a bridge in 15 minutes. The length of the bridge (in metres) is:

Solution:
Speed of the man = 5km/hr
= 5 × (1000/60)m/min
= 250/3 m/min

Now,
Time taken to cross the bridge = 15 minutes
Length of the bridge = speed × time
= (250/3) × 15m
= 1250m

১,২২০.

A train 300 meters long passes a bridge 700 meters long in 40 seconds. How long will it take to pass a platform that is 500 meters long?

ক
48 seconds
খ
52 seconds
গ
36 seconds
ঘ
32 seconds

ব্যাখ্যা

Question: A train 300 meters long passes a bridge 700 meters long in 40 seconds. How long will it take to pass a platform that is 500 meters long?

Solution:
Total distance to pass bridge = 300 + 700 meters
= 1000 meters

∴ Train's speed = Distance/Time
= 1000/40 = 25 m/s

Total distance to pass the platform,
= Length of train + Length of platform
= 300 + 500
= 800 meters

∴ Required time to pass the platform = Distance/Speed
= 800/25
= 32 seconds

১,২২১.

A boat takes 6 hours to travel 30 km upstream and 3 hours to travel the same distance downstream. Find the distance travelled by the boat in 7 hours in still water.

ক
52.5 km
খ
42.6 km
গ
62.5 km
ঘ
45.5 km

ব্যাখ্যা

Question: A boat takes 6 hours to travel 30 km upstream and 3 hours to travel the same distance downstream. Find the distance travelled by the boat in 7 hours in still water.

Solution:
Let the speed of the boat in still water be b km/h
and the speed of the current (stream) be c km/h.
Then we get,
Upstream speed = b - c
Downstream speed = b + c

Now, upstream speed = 30/6 = 5 km/h
∴ b - c = 5 ……… (1)
And downstream speed = 30/3 = 10 km/h
∴ b + c = 10 ……… (2)

Add equations (1) and (2) then we get,
(b - c) + (b + c) = 5 + 10
⇒ 2b = 15
⇒ b = 15/2
∴ b = 7.5 km/h
So, the speed of the boat in still water is 7.5 km/h.

∴ Distance travel in still water in 7 hours = speed × time
= 7.5 × 7
= 52.5 km

So the boat will travel 52.5 km in 7 hours in still water.

১,২২২.

A train 'B' speeding with 100 kmph crosses another train C, running in the same direction, in 2 minutes. If the length of the train B and C be 150 metre and 250 metre respectively, what is the speed of the train C (in kmph)?

ক
90 kmph
খ
88 kmph
গ
80 kmph
ঘ
72 kmph

ব্যাখ্যা

Question: A train 'B' speeding with 100 kmph crosses another train C, running in the same direction, in 2 minutes. If the length of the train B and C be 150 metre and 250 metre respectively, what is the speed of the train C (in kmph)?

Solution:
Let,
the speed of train C be = x kmph.
Relative speed of B = (100 - x ) kmph.

Time taken in crossing = Length of both trains/Relative speed
2/60 = [{(150 + 250)/1000}/(100 - x)]
⇒ 1/30 = 2/{5(100 - x)}
⇒ 1/6 = 2/(100 - x)
⇒ 100 - x = 12
⇒ x = 100 – 12
∴ x = 88 kmph

১,২২৩.

A train 120 meters long takes 30 seconds to cross a 480-meter-long bridge. How much time will the train take to cross a 400-meter-long platform?

ক
24 seconds
খ
26 seconds
গ
30 seconds
ঘ
32 seconds

ব্যাখ্যা

Question: A train 120 meters long takes 30 seconds to cross a 480-meter-long bridge. How much time will the train take to cross a 400-meter-long platform?

Solution:
Length of train = 120 m
Length of bridge = 480 m
∴ Total distance to cross bridge = 120 + 480 = 600 m
Time taken = 30 seconds
∴ Speed of train = Total distance/Time
= 600/30 = 20 m/s

Length of platform = 400 m
∴ Total distance to cross platform = 120 + 400 = 520 m

∴ Time taken = Total distance/Speed
= 520/20 seconds
= 26 seconds

১,২২৪.

A boat covers a certain distance downstream in 1 hour, while it comes back in 7/3 hours. If the speed of the stream be 6 kmph, what is the speed of the boat in still water?

ক
ক) 9 kmph
খ
খ) 12 kmph
গ
গ) 15 kmph
ঘ
ঘ) 18 kmph

ব্যাখ্যা

Question: A boat covers a certain distance downstream in 1 hour, while it comes back in 7/3 hours. If the speed of the stream be 6 kmph, what is the speed of the boat in still water?

Solution:
Let the speed of the boat in still water be x kmph.
Then,
Speed downstream = (x + 6) kmph,
Speed upstream = (x - 6) kmph.

Now
(x + 6) x 1 =(x - 6)(7/3)
3(x + 6) = 7(x - 6)
3x + 18 = 7x - 42
7x - 3x = 42 + 18
4x = 60
x = 15 kmph

১,২২৫.

A boat running downstream covers a distance of 30 km in 3 hours, while it takes 5 hours to cover the same distance upstream. What is the speed of the boat in still water?

ক
9 km/h
খ
5 km/h
গ
8 km/h
ঘ
10 km/h

ব্যাখ্যা

Question: A boat running downstream covers a distance of 30 km in 3 hours, while it takes 5 hours to cover the same distance upstream. What is the speed of the boat in still water?

Solution:
Rate while running downstream = (30/3) km/h
= 10 km/h

Rate while running upstream = (30/5) km/h
= 6 km/h

∴ Speed of the boat in still water
= (10 + 6)/2 km/h
= 8 km/h

So, the speed of the boat in still water is 8 km/h.

১,২২৬.

A train covers a distance of 600 meters in 25 seconds, whereas a car covers a distance of 43.2 km in 36 minutes. What is the ratio of the speed of the train to the speed of the car?

ক
3 : 7
খ
6 : 5
গ
8 : 3
ঘ
7 : 4

ব্যাখ্যা

Question: A train covers a distance of 600 meters in 25 seconds, whereas a car covers a distance of 43.2 km in 36 minutes. What is the ratio of the speed of the train to the speed of the car?

Solution:
ট্রেনের গতিবেগ নির্ণয়:
দূরত্ব = 600 মিটার, সময় = 25 সেকেন্ড।
∴ ট্রেনের গতিবেগ = 600/25 মিটার/সেকেন্ড
= 24 মিটার/সেকেন্ড।

গাড়ির গতিবেগ নির্ণয়:
দূরত্ব = 43.2 কিমি = 43.2 × 1000 = 43200 মিটার।
সময় = 36 মিনিট = 36 × 60 = 2160 সেকেন্ড।
∴ গাড়ির গতিবেগ = 43200/2160 মিটার/সেকেন্ড
= 20 মিটার/সেকেন্ড।

গতিবেগের অনুপাত = ট্রেনের গতিবেগ : গাড়ির গতিবেগ
= 24 : 20
= 6 : 5
∴ তাদের গতিবেগের অনুপাত হলো 6 : 5

১,২২৭.

Two boys start running at the same time in the same direction at a speed of 10 km/hr and 12 km/hr respectively. In what time they will be 8 km apart?

ক
3 hours
খ
6 hours
গ
4 hours
ঘ
5 hours

ব্যাখ্যা

Question: Two boys start running at the same time in the same direction at a speed of 10 km/hr and 12 km/hr respectively. In what time they will be 8 km apart?

Solution:
Both the boys are in motion so we will consider relative speed to find the time.
Relative speed = 12 - 10 = 2 km/hr

Distance = 8 km

Time = 8/2 = 4 hours

১,২২৮.

If a man covers 15.2 km in 4 hours, the distance covered by him in 5 hours is:

ক
ক) 15 km
খ
খ) 17 km
গ
গ) 19 km
ঘ
ঘ) 20 km

ব্যাখ্যা

Question: If a man covers 15.2 km in 4 hours, the distance covered by him in 5 hours is:

Solution:
Speed = (15.2/4) km/hr
= 3.8 km/hr.

Distance covered in 5 hours = ( 3.8 × 5) km
= 19 km.

১,২২৯.

Two trains are 150 meters and 180 meters long. They are running on parallel tracks in the same direction. The faster train runs at 90 km/h and the slower at 54 km/h. How much time will the faster train take to completely pass the slower train?

ক
30 seconds
খ
33 seconds
গ
36 seconds
ঘ
45 seconds

ব্যাখ্যা

Question: Two trains are 150 meters and 180 meters long. They are running on parallel tracks in the same direction. The faster train runs at 90 km/h and the slower at 54 km/h. How much time will the faster train take to completely pass the slower train?

Solution:
Given,
Length of first train = 150 meters
Length of second train = 180 meters
Speed of faster train = 90 km/h
= (90 × 1000)/3600 = 25 m/s
Speed of slower train = 54 km/h
= (54 × 1000)/3600 = 15 m/s

Since they are moving in the same direction,
∴ Relative speed = 25 - 15 = 10 m/s

To pass completely, the faster train must cover the length of both trains.
∴ Total lenght = 150 + 180 = 330 meters

Now,
Time = Distance / Speed
= 330/10 = 33 seconds

১,২৩০.

A man swimming in a stream which flows (3/2) km/hr finds that in a given time he can swim twice as far with the stream as he can against it. At what rate does he swim?

ক
5.5 km/hr
খ
4.75 km/hr
গ
6.25 km/hr
ঘ
4.5 km/hr

ব্যাখ্যা

Question: A man swimming in a stream which flows (3/2) km/hr finds that in a given time he can swim twice as far with the stream as he can against it. At what rate does he swim?

Solution:
Let the rate of his swim x km/h
When he swim with the flow then speed =(x + 3/2) km/h
∴ S₁ = (x + 3/2) × t
When he swim against the flow stream then speed = (x - 3/2) km/h
∴ S₂ = (x - 3/2) × t

According to the question,
S₁ = 2 × S₂
⇒ (x + 3/2)t = 2(x - 3/2)t
⇒ (2x + 3)/2 = 2x - 3
⇒ 2x + 3 = 4x - 6
⇒ 9 = 2x
⇒ x = 9/2
∴ x = 4.5 km/hr

Therefore, the man swims at 9/2 km/h (or 4.5 km/h) in still water.

১,২৩১.

A train travels 180 km in 3 hours. What is its average speed?

ক
80 km/h
খ
60 km/h
গ
70 km/h
ঘ
50 km/h

ব্যাখ্যা

Question: A train travels 180 km in 3 hours. What is its average speed?

Solution:
ট্রেনটির অতিক্রান্ত দূরত্ব = 180 কি.মি.
মোট সময় = 3 ঘণ্টা

গড় গতিবেগ = 180/3 km/h
= 60 km/h

১,২৩২.

A train 220 m long is moving at 45km/h. The time taken by the train to cross a tunnel 260m long is-

ক
ক) 25 sec
খ
খ) 35 sec
গ
গ) 38 sec
ঘ
ঘ) 40 sec

ব্যাখ্যা

নির্ণেয় সময় = (220+260) / 45×(5/18)
= (480×18) / (45×5)
= 38.4 সেকেন্ড

সঠিক উত্তর হবে 38.4 সেকেন্ড কিন্তু অপশনে এই মান না থাকলেও কাছাকাছি মান আছে 38। তাই উত্তর হবে 38 সেকেন্ড।

১,২৩৩.

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:

ক
ক) 4
খ
খ) 5
গ
গ) 6
ঘ
ঘ) 7
ঙ
ঙ) 8

ব্যাখ্যা

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

30/(15 + x) +30/(15 - x) = 4(1/2)
900/(225 - X²) = 9/2
9X² = 225
X² = 25
X = 5 km/hr

১,২৩৪.

Three runners A, B and C run a race, with runner A finishing 12 metres ahead of runner B and 18 metres ahead of runner C, in another race of same type runner B finished 8 metres ahead of runner C. Each runner travels the entire distance at a constant speed. The length of the race is:

ক
ক) 24 meters
খ
খ) 48 meters
গ
গ) 72 meters
ঘ
ঘ) 96 meters

ব্যাখ্যা

Question: Three runners A, B and C run a race, with runner A finishing 12 metres ahead of runner B and 18 metres ahead of runner C, in another race of same type runner B finished 8 metres ahead of runner C. Each runner travels the entire distance at a constant speed. The length of the race is:

Solution:
Let A finish the race of x metres
B finish the race of = x - 12 meters.
C finish the race of = x - 18 - - - - - - - - (1)
In another race of B & C
B finish race of = x meters.
C finish race of = x - 8 - - - - - - - - (2)
Ratio of speeds of B & C
(x - 12)/(x - 18) = x/(x - 8)
⇒ (x - 12)(x - 8) = x(x -18)
⇒ x² - 12x - 8x + 96 = x² - 18x
⇒ x^{2 -}20x + 96 = x² - 18x
⇒ 96 = 2x
∴ x = 48 metres.

∴ The length of the race is 48 meters.

১,২৩৫.

A train 300 metres long is running at a speed of 25 metre per second. It will cross a bridge of 200 metres long in:

ক
ক) 20 sec
খ
খ) 15 sec
গ
গ) 10 sec
ঘ
ঘ) 25 sec

ব্যাখ্যা

Question: A train 300 metres long is running at a speed of 25 metre per second. It will cross a bridge of 200 metres long in:

Solution:
Speed of train = 25 m/s.
Total distance covered while passing bridge = 300 + 200 = 500 m

∴ Time = 500/25
= 20 seconds.

১,২৩৬.

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 :

ক
ক) 4 hours 30 minutes
খ
খ) 4 hours 45 minutes
গ
গ) 5 hours
ঘ
ঘ) 6 hours

ব্যাখ্যা

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

১,২৩৭.

Fahim rows a boat at 8 km/h in still water. He finds that it takes him 2 hours longer to row upstream than downstream for a certain distance. If the speed of the stream is s km/h, and the distance is 30 km, find s.

ক
2 km/hr
খ
3 km/hr
গ
4.2 km/hr
ঘ
6 km/hr

ব্যাখ্যা

Question: Fahim rows a boat at 8 km/h in still water. He finds that it takes him 2 hours longer to row upstream than downstream for a certain distance. If the speed of the stream is s km/h, and the distance is 30 km, find s.

সমাধান:
স্থির পানিতে নৌকার গতিবেগ = 8 কিমি/ঘন্টা
স্রোতের গতিবেগ = s কিমি/ঘন্টা
মোট দূরত্ব = 30 কিমি

স্রোতের অনুকূলে গতিবেগ (Downstream) = (8 + s) কিমি/ঘন্টা
স্রোতের প্রতিকূলে গতিবেগ (Upstream) = (8 - s) কিমি/ঘন্টা

স্রোতের অনুকূলে সময় = দূরত্ব/গতিবেগ = 30/(8 + s) ঘন্টা
স্রোতের প্রতিকূলে সময় = দূরত্ব/গতিবেগ = 30 / (8 - s) ঘন্টা

স্রোতের প্রতিকূলে যেতে 2 ঘন্টা বেশি সময় লাগে।
প্রশ্নমতে,
{30/(8 - s)} - {30/(8 + s)} = 2
⇒ 30[{(1/(8 - s)} - {1/(8 + s)}] = 2
⇒ {1/(8 - s)} - {1/(8 + s)} = 2/30
⇒{(8 + s) - (8 - s)}/{(8 - s)(8 + s)} = 1/15
⇒ 2s /(64 - s²) = 1/15
⇒ 30s = 64 - s²
⇒ s² + 30s - 64 = 0
⇒ s² + 32s - 2s - 64 = 0
⇒ s(s + 32) - 2(s + 32) = 0
⇒ (s - 2)(s + 32) = 0

সুতরাং, s = 2 অথবা s = - 32
যেহেতু স্রোতের গতিবেগ ঋণাত্মক হতে পারে না, তাই গ্রহণযোগ্য মান হলো s = 2 কিমি/ঘন্টা।

সুতরাং, স্রোতের গতিবেগ হলো 2 কিমি/ঘন্টা।

১,২৩৮.

Two ships, Alpha and Beta, start towards each other from two ports, 160 km apart. The speeds of Ship Alpha and Ship Beta in still water are 16 km/h and 24 km/h respectively. If Ship Alpha proceeds downstream and Ship Beta proceeds upstream, they will meet after how many hours?

ক
4 hours
খ
5 hours
গ
6 hours
ঘ
6.5 hours

ব্যাখ্যা

Question: Two ships, Alpha and Beta, start towards each other from two ports, 160 km apart. The speeds of Ship Alpha and Ship Beta in still water are 16 km/h and 24 km/h respectively. If Ship Alpha proceeds downstream and Ship Beta proceeds upstream, they will meet after how many hours?

সমাধান:
ধরি, স্রোতের গতিবেগ হলো x কিমি/ঘন্টা।
ধরি, জাহাজ দুটি t ঘন্টা পর মিলিত হবে।

জাহাজ Alpha (স্রোতের অনুকূলে) এর গতিবেগ = (16 + x) কিমি/ঘন্টা।
জাহাজ Beta (স্রোতের প্রতিকূলে) এর গতিবেগ = (24 - x) কিমি/ঘন্টা।

তারা একে অপরের দিকে আসছে, তাই তাদের আপেক্ষিক গতিবেগ হলো তাদের গতির যোগফল।
আপেক্ষিক গতিবেগ = (16 + x) + (24 - x) কিমি/ঘন্টা
= (16 + 24 + x - x) কিমি/ঘন্টা
= 40 কিমি/ঘন্টা।

প্রশ্নমতে,
দূরত্ব = আপেক্ষিক গতিবেগ × সময়।
160 = 40 × t
⇒ t = 160/40
⇒ t = 4 ঘন্টা।
∴ জাহাজ দুটি 4 ঘন্টা পর মিলিত হবে।

১,২৩৯.

A train 300 meters long passes a pole in 20 seconds. How long will it take to pass a platform that is 450 meters long?

ক
30 seconds
খ
45 seconds
গ
50 seconds
ঘ
60 seconds

ব্যাখ্যা

Question: A train 300 meters long passes a pole in 20 seconds. How long will it take to pass a platform that is 450 meters long?

Solution:
Train's speed = Distance/Time
= 300/20
= 15 m/s

Total distance to pass the platform = Length of train + Length of platform
= 300 m + 450 m
= 750 m

∴ Required time = Distance/Speed
= 750/15
= 50 seconds

∴ The train will take 50 seconds to pass the platform.

১,২৪০.

A car covers the first 30km in 15 minutes and the second 30km in 10 minutes. Between the two actions, the car stopped for 5 minutes for fuel. What is the average speed of the car?

ক
100 km/hr
খ
120 km/hr
গ
140 km/hr
ঘ
90 km/hr

ব্যাখ্যা

Question: A car covers the first 30km in 15 minutes and the second 30km in 10 minutes. Between the two actions, the car stopped for 5 minutes for fuel. What is the average speed of the car?

Solution:
total distance = 30 + 30 = 60km
total time = 30 minutes = 1/2 hour

average = 60/(1/2) = 120km/hr.

১,২৪১.

A train, 150 m long, passes a pole in 15 seconds and another train of the same length travelling in the opposite direction in 12 seconds. The speed of the second train is-

ক
54 km/hr
খ
45 km/hr
গ
48 km/hr
ঘ
52 km/hr

ব্যাখ্যা

Question: A train, 150 m long, passes a pole in 15 seconds and another train of the same length travelling in the opposite direction in 12 seconds. The speed of the second train is-

Solution:
Given that,
Length of both trains = 150 m
First train passes a pole in 15 s
First train passes the second train (opposite direction) in 12 s

Now,
Speed of the first train,
= 150/15
= 10 m/s
Time taken by trains to cross each other = 12 sec
And, relative speed of two trains :
= (150 + 150)/12
= 25 m/s

∴ Speed of the second train is
= (25 - 10) × 18/5
= 15 × 18/5
= 54 km/hr

So the speed of the second train is 54 km/hr.

১,২৪২.

A boat covers a certain distance downstream in 1 hour, while it comes back in 3/2 hours. If the speed of the stream be 3 km/hr, what is the speed of the boat in still water?

ক
ক) 16km/hr
খ
খ) 14km/hr
গ
গ) 15km/hr
ঘ
ঘ) 12km/hr

ব্যাখ্যা

Question: A boat covers a certain distance downstream in 1 hour, while it comes back in 3/2 hours. If the speed of the stream be 3 km/hr, what is the speed of the boat in still water?

Solution:
Let the speed of the boat in still water be x km/hr
Then
Speed downstream = (x + 3) km/hr
Speed upstream = (x - 3) km/hr

ATQ,
(x + 3) × 1 = (x - 3) × (3/2)
⇒ 2x + 6 = 3x - 9
∴ x = 15km/hr

১,২৪৩.

How many hours will it take a man to travel M miles at the rate of H miles per hour?

ক
ক) HM
খ
খ) H/M
গ
গ) M/H
ঘ
ঘ) H/2M

ব্যাখ্যা

question: How many hours will it take a man to travel M miles at the rate of H miles per hour?

solution:
given,
Distance = M miles
Speed = H miles per hour
Time = ?

we know,
Distance = Speed × Time
Time = Distance/Speed
= M/H

১,২৪৪.

Pavel travels 96 km at a speed of 16 km/hr using a bike, 124 km at 31 km/h by car and another 105 km at 7 km/h in horse cart. Find his average speed for the entire distance travelled.

ক
20 km/h
খ
12 km/h
গ
13 km/h
ঘ
16 km/h

ব্যাখ্যা

Question: Pavel travels 96 km at a speed of 16 km/hr using a bike, 124 km at 31 km/h by car and another 105 km at 7 km/h in horse cart. Find his average speed for the entire distance travelled.

Solution:
We know,
Average speed = Total distance/Total time

Total distance = 96 + 124 + 105 = 325 km

Now,
Bike: 96 km at 16 km/h ∴ Time = 96/16 = 6 hours
Car: 124 km at 31 km/h ∴ Time = 124/31 = 4 hours
Horse cart: 105 km at 7 km/h ∴ Time = 105/7 = 15 hours

∴ Total time = 6 + 4 + 15 = 25 hours

∴ Average speed = Total distance/Total time
= 325/25
= 13 km/h

So Pavel's average speed for the entire journey is 13 km/h.

১,২৪৫.

Find the speed of the boat in still water, if a boat covers a certain distance upstream in 2 hours, while it comes back in 3/2 hours. If the speed of the stream be 3 kmph.

ক
12 Kmph
খ
18 Kmph
গ
21 Kmph
ঘ
24 Kmph

ব্যাখ্যা

Question: Find the speed of the boat in still water, if a boat covers a certain distance upstream in 2 hours, while it comes back in 3/2 hours. If the speed of the stream be 3 kmph.

Solution:
Let the speed of the boat in still water be x kmph.
Then,
Speed downstream = (x + 3) kmph,
Speed upstream = (x - 3)kmph.

(x - 3) × 2 = (x + 3) × (3/2)
⇒ 4x - 12 = 3x + 9
∴ x = 21 kmph.

১,২৪৬.

A man is running at a speed of 3 km/hr in the direction of the train whose length is 500 meters. If the train is moving at a speed of 63 km/hr then how many seconds will this train take to cross the man?

ক
ক) 38 sec
খ
খ) 35 sec
গ
গ) 34 sec
ঘ
ঘ) 30 sec

ব্যাখ্যা

Question: A man is running at a speed of 3 km/hr in the direction of the train whose length is 500 meters. If the train is moving at a speed of 63 km/hr then how many seconds will this train take to cross the man?

Solution:
ট্রেন ও মানুষের আপেক্ষিক গতি = (63 - 3) km/hr
= 60 km/hr
= (60 x 1000)/3600 m/sec
= 50/3 m/sec

মানুষটি পার হতে সময় লেগেছে = 500 x (3/50) sec
= 30 sec

১,২৪৭.

Speed of a boat along and against the current are 14 kms/hr and 8 kms/hr respectively. The speed of the current is?

ক
4 km/hr
খ
3 km/hr
গ
2.5 km/hr
ঘ
2 km/hr

ব্যাখ্যা

Question: Speed of a boat along and against the current are 14 kms/hr and 8 kms/hr respectively. The speed of the current is?

Solution:
Let the speed of the boat is S
the speed of the current is W

S + W = 14 . . . . . . (i)
S - W = 8 . . . . . . . .(ii)
_________________
from equation (i) & (ii)
S = 11 km/hr, W = 3 km/hr
Speed of current = 3 km/hr

১,২৪৮.

A boat can travel with a speed of 12 km/hr in still water. If the speed of the stream is 5 km/hr, find the time taken by the boat to go 136 km downstream.

ক
ক) 10 hours
খ
খ) 8 hours
গ
গ) 6 hours
ঘ
ঘ) 4 hours

ব্যাখ্যা

Question: A boat can travel with a speed of 12 km/hr in still water. If the speed of the stream is 5 km/hr, find the time taken by the boat to go 136 km downstream.

Solution:
Speed downstream = (12 + 5) km/hr = 17 km/hr

Time taken to travel 136 km downstream = 136/17 hr
= 8 hr

১,২৪৯.

A motorcyclist covers the first 40 km in 20 minutes and the second 60 km in 30 minutes. Between these two segments, the motorcyclist stopped for 10 minutes for a rest. What is the average speed of the motorcycle in km/h?

ক
80 km/h
খ
90 km/h
গ
100 km/h
ঘ
120 km/h

ব্যাখ্যা

Question: A motorcyclist covers the first 40 km in 20 minutes and the second 60 km in 30 minutes. Between these two segments, the motorcyclist stopped for 10 minutes for a rest. What is the average speed of the motorcycle in km/h?

Solution:
মোট দূরত্ব = 40 কিমি + 60 কিমি = 100 কিমি।

প্রথম অংশের সময় = 20 মিনিট।
দ্বিতীয় অংশের সময় = 30 মিনিট।
বিশ্রামের জন্য বিরতি = 10 মিনিট।
মোট সময় = 20 + 30 + 10 মিনিট = 60 মিনিট = 1 ঘন্টা

গড় গতিবেগ = 100 কিমি/1 ঘন্টা
= 100 কিমি/ঘন্টা
∴ মোটরসাইকেলটির গড় গতিবেগ হলো 100 কিমি/ঘন্টা।

১,২৫০.

A man completes a journey in 8 hours. He travels the first half of the journey at the rate of 40 km/hr and the second half at the rate of 60 km/hr. Find the total distance of the journey in km.

ক
290 km.
খ
304 km.
গ
354 km.
ঘ
384 km.

ব্যাখ্যা

Question: A man completes a journey in 8 hours. He travels the first half of the journey at the rate of 40 km/hr and the second half at the rate of 60 km/hr. Find the total distance of the journey in km.

Solution:
Let the total distance of the journey be d km.
Then, the first half of the journey = d/2 km and the second half = d/2 km.

Time taken for the first half,
= (d/2) / 40 hours
= d/80 hours

And,
Time taken for the second half,
= (d/2) / 60 hours
= d/120 hours

According to the question,
(d/80) + (d/120) = 8
⇒ (3d + 2d)/240 = 8
⇒ 5d/240 = 8
⇒ 5d = 8 × 240
⇒ 5d = 1920
⇒ d = 1920/5
⇒ d = 384 km

∴ The total distance of the journey is 384 km.

১,২৫১.

A 180-meter-long train takes 18 seconds to pass a pole. How long will it take to pass a 330-meter-long platform?

ক
30 seconds
খ
33 seconds
গ
42 seconds
ঘ
51 seconds

ব্যাখ্যা

Question: A 180-meter-long train takes 18 seconds to pass a pole. How long will it take to pass a 330-meter-long platform?

Solution:
আমরা জানি,
কোনো ট্রেন একটি খুঁটিকে অতিক্রম করলে ট্রেনটি তার নিজের দৈর্ঘ্যকে অতিক্রম করে।
আবার,
কোনো প্ল্যাটফর্মকে অতিক্রম করতে হলে ট্রেনকে নিজের দৈর্ঘ্য এবং প্ল্যাটফর্মের দৈর্ঘ্যের সমান দূরত্ব অতিক্রম করতে হয়।

∴ প্ল্যাটফর্ম অতিক্রম করলে ট্রেনের অতিক্রান্ত দূরত্ব = (ট্রেনের দৈর্ঘ্য + প্ল্যাটফর্মের দৈর্ঘ্য) = (180 + 330) মিটার = 510 মিটার

এখন,
180 মিটার অতিক্রম করে = 18 সেকেন্ডে
∴ 1 মিটার অতিক্রম করে = 18/180 সেকেন্ডে
∴ 510 মিটার অতিক্রম করে = (18 × 510)/180 = 51 সেকেন্ডে

১,২৫২.

How long does a train 110 metres long running at the speed of 72 km/hr take to cross a bridge 132 metres in length?

ক
9.8 sec
খ
12.1 sec
গ
12.42 sec
ঘ
14.3 sec

ব্যাখ্যা

Question: How long does a train 110 metres long running at the speed of 72 km/hr take to cross a bridge 132 metres in length?

Solution:
Speed = 72 km/hr
= (72 × 1000)/3600 m/s
= 20 m/s

Train has to be treavelled = (110 + 132) metres = 242 metres

Required time = 242/20 seconds
= 12.1 seconds

১,২৫৩.

The speed of P and Q are in the ratio 5 : 8. Q takes 24 minutes less than P to reach a destination. Time in which Q reaches the destination?

ক
32 minutes
খ
40 minutes
গ
48 minutes
ঘ
36 minutes

ব্যাখ্যা

Question: The speed of P and Q are in the ratio 5 : 8. Q takes 24 minutes less than P to reach a destination. Time in which Q reaches the destination?

Solution:
Given, speed of P and Q = 5 : 8

So, ratio of time taken = 8 : 5 [Time ∝ 1/Speed]

Let time taken by P and Q be 8x and 5x minutes respectively.

According to the question,
8x - 5x = 24
⇒ 3x = 24
⇒ x = 8

Hence, time taken by Q = 5 × 8 = 40 minutes

১,২৫৪.

A person takes 20 minutes more to cover a certain distance by decreasing his speed by 20%. What is the time taken to cover the distance at his original speed?

ক
1 hr
খ
1 hr 20 min
গ
1 hr 10 min
ঘ
50 min

ব্যাখ্যা

Question: A person takes 20 minutes more to cover a certain distance by decreasing his speed by 20%. What is the time taken to cover the distance at his original speed?

Solution:
Let the distance and original speed be 'd' km and 'k' kmph respectively.
d/0.8k - d/k = 20/60
⇒ 5d/4k - d/k = 1/3
⇒ (5d - 4d)/4k = 1/3
⇒ d = (4/3)k

Time taken to cover the distance at original speed
= d/k = 4/3 hours = 1 hour 20 minutes

১,২৫৫.

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-

ক
48 km/hr
খ
54 km/hr
গ
82 km/hr
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.

১,২৫৬.

Two trains 140 m and 160 m long run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other, is:

ক
11.6 sec
খ
12.5 sec
গ
10.8 sec
ঘ
12.8 sec
ঙ
None

ব্যাখ্যা

Question: Two trains 140 m and 160 m long run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other, is:

Solution:
Relative speed = (60 + 40) km/hr = 100 km/hr = 250/9 m/sec
Distance covered in crossing each other = (140 + 160) m = 300 m.

Required time = 300 × (9/250) sec
= 10.8 sec

১,২৫৭.

A monkey climbs a 40m high pole. In the first minute, he climbs 3m and slips down 1m in the next minute. How much time is required by it to reach the top?

ক
38 minutes.
খ
39 minutes.
গ
40 minutes.
ঘ
37 minutes.

ব্যাখ্যা

Question: A monkey climbs a 40m high pole. In the first minute, he climbs 3m and slips down 1m in the next minute. How much time is required by it to reach the top?

Solution:
in two minutes,
total distance covered by the monkey is = 3 - 1 = 2m
in 1 minute = 2/2 = 1m

in 36 minutes it will cover = 36m
in the 37th minute, it will go up to = 36 +3 = 39m
in the 38th minute, it will fall 1m ad thus covers = 39 - 1 = 38m

the rest of the 2m of total 40m will be climbed in 1 minute.
thus total time to reach the peak is = 38 + 1 = 39 minutes.

১,২৫৮.

ক ঘণ্টায় ৫৪ কি.মি. বেগে এবং খ প্রতি সেকেন্ডে ২০ মিটার যায়। উভয়ের গতিবেগের পার্থক্য কত মিটার/সেকেন্ড?

ক
ক) ২ মিটার/সেকেন্ড
খ
খ) ৪ মিটার/সেকেন্ড
গ
গ) ৫ মিটার/সেকেন্ড
ঘ
ঘ) ১০ মিটার/সেকেন্ড

ব্যাখ্যা

ক এর গতিবেগ = ৫৪ কি.মি./ঘণ্টা
= (৫৪ × ১০০০)/৩৬০০
= ১৫ মিটার/সেকেন্ড
খ এর গতিবেগ = ২০ মিটার/সেকেন্ড

উভয়ের গতিবেগের পার্থক্য = (২০ - ১৫) মিটার/সেকেন্ড
= ৫ মিটার/সেকেন্ড

১,২৫৯.

Two men P and Q start from a place walking at 5 km/hr and 6.5 km/hr, respectively. What is the time they will take to be 92 km apart, if they walk in opposite directions?

ক
ক) 2 hr
খ
খ) 4 hr
গ
গ) 6 hr
ঘ
ঘ) ৪ hr

ব্যাখ্যা

Distance = 92 km,
As they walk in opposite direction, their relative Speed = 5 + 6.5 = 11.5 km/h
So, Required time = Distance/Relative speed = 92/11.5 = 8 h

১,২৬০.

Two trains 130 m and 170 m long move towards each other on parallel tracks. If they cross each other in 5 seconds and the speed of the first train is 54 km/h, find the speed of the second train.

ক
72 km/h
খ
162 km/h
গ
120 km/h
ঘ
92 km/h

ব্যাখ্যা

Question: Two trains 130 m and 170 m long move towards each other on parallel tracks. If they cross each other in 5 seconds and the speed of the first train is 54 km/h, find the speed of the second train.

Solution:
দেওয়া আছে,
প্রথম ট্রেনের দৈর্ঘ্য, L₁= 130 মিটার
দ্বিতীয় ট্রেনের দৈর্ঘ্য, L₂ = 170 মিটার
সময়, t = 5 সেকেন্ড
প্রথম ট্রেনের গতি, S₁= 54 কিমি/ঘণ্টা
দ্বিতীয় ট্রেনের গতি, S₂ =?

এখন,
প্রথম ট্রেনের গতি, S₁ = 54 কিমি/ঘণ্টা = 54 × (1000/3600) = 15 মি./সে.

মোট দৈর্ঘ্য, L = L₁ + L₂ = 130 + 170 = 300 মিটার
যেহেতু দুটি ট্রেন একে অপরের দিকে আসছে, তাই আপেক্ষিক গতি = মোট দূরত্ব/সময় = ৩০০/৫ = ৬০ মি./সে.

∴ দ্বিতীয় ট্রেনের গতি S₂ = 60 - 15 = 45 মি./সে. = 45 × (3600/1000) = 162 কিমি/ঘণ্টা

১,২৬১.

Walking 3/4 of his normal speed, Rakib is 20 minutes late in reaching his office. The usual time taken by him to cover the distance between his home and office-

ক
ক) 48 min
খ
খ) 55 min
গ
গ) 60 min
ঘ
ঘ) 65 min

ব্যাখ্যা

Questin: Walking 3/4 of his normal speed, Rakib is 20 minutes late in reaching his office. The usual time taken by him to cover the distance between his home and office-

Solution:
let, distance y meter and speed x meter/min.
usual time taken y/x min

ATQ,
(y/3x/4) - (y/x) = 20
⇒ (y/x) {(4/3) - 1} = 20
⇒ (y/x) (1/3) = 20
∴ y/x = 60 min

১,২৬২.

Sarah is planning a bike ride to reach point B. If she cycles at 8 kmph, she will reach at 4 P.M., but if she cycles at 12 kmph, she will reach at 2 P.M. At what speed should she cycle to reach B at 3 P.M.?

ক
8.4 kmph.
খ
9.6 kmph
গ
7.8 kmph.
ঘ
8 kmph.

ব্যাখ্যা

Question: Sarah is planning a bike ride to reach point B. If she cycles at 8 kmph, she will reach at 4 P.M., but if she cycles at 12 kmph, she will reach at 2 P.M. At what speed should she cycle to reach B at 3 P.M.?

Solution:
Let the distance traveled be x km.
Then,
(x/8) - (x/12) = 2
⇒ 3x - 2x = 48
∴ x = 48 km.
Time taken to travel 48 km at 8 km/hr = 48/8 hrs = 6 hrs.
So, Sarah started 6 hours before 4 P.M. i.e., at 10 A.M.
∴ Required speed = 48/5 kmph = 9.6 kmph.

১,২৬৩.

A car traveling with 5/7 of its actual speed covers 42 km in 1 hr 40 min 48 sec. What is the actual speed of the car?

ক
40 km/hr
খ
25 km/hr
গ
30 km/hr
ঘ
45 km/hr
ঙ
35 km/hr

ব্যাখ্যা

time = 1 hr 40 min 48 sec
= 1 hr + 40/60 hr + 48/3600 hr
= 1 + 2/3 + 1/75 = 126/75 hr
distance = 42 km
speed = distance/time = 42/( 126/75 ) = (42 × 75)/126 = 25 km/hr
⇒ 5/7 of the actual speed = 25
⇒ Actual speed = 25 × 7/5 = 35 km/hr

১,২৬৪.

Sara covered a distance of 450 miles between city X and city Y in a total of 7 hours. If part of the distance was covered at 50 miles per hour speed and the remaining distance at 75 miles per hour speed, how many hours did she travel at 50 miles per hour?

ক
2 hours
খ
2 hours 20 minute
গ
3 hours
ঘ
3 hours 30 minute

ব্যাখ্যা

Question: Sara covered a distance of 450 miles between city X and city Y in a total of 7 hours. If part of the distance was covered at 50 miles per hour speed and the remaining distance at 75 miles per hour speed, how many hours did she travel at 50 miles per hour?

Solution:
2 hours were spent traveling at 50 miles per hour,
(7 - x) hours were spent traveling at 75 miles per hour.

ATQ,
50x + 75(7 - x) = 450
⇒ 50x + 525 - 75x = 450
⇒ - 25x + 525 = 450
⇒ - 25x = - 75
∴ x = 3

∴ Sara traveled 3 hours at 50 miles per hour.

১,২৬৫.

A boat sailing against a stream of river takes 6 hours to travel 36 km while sailing with the stream it takes 3 hours to travel the same distance. What is the speed of the stream?

ক
3 km/h
খ
3.5 km/h
গ
4 km/h
ঘ
5 km/h

ব্যাখ্যা

Question: A boat sailing against a stream of river takes 6 hours to travel 36 km while sailing with the stream it takes 3 hours to travel the same distance. What is the speed of the stream?

Solution:
Let,
Speed of boat = x km/h and,
Speed of stream = y km/h

ATQ,
x - y = 36/6 = 6 ......... (i)
x + y = 36/3 = 12 ....... (ii)

From,
(ii) - (i)
x + y - x + y = 12 - 6
⇒ 2y = 6
∴ y = 3

So, Speed of the stream = 3 km/h

১,২৬৬.

A man takes 20 minutes to row 12 km upstream which is one-third more than the time he takes on his way downstream. What is his speed in still water?

ক
ক) 72 km/hr
খ
খ) 42 km/hr
গ
গ) 36 km/hr
ঘ
ঘ) 48 km/hr

ব্যাখ্যা

Question: A man takes 20 minutes to row 12 km upstream which is one-third more than the time he takes on his way downstream. What is his speed in still water?

Solution:
Let,
His speed in still water is x km/hr
The speed of stream is y km/hr
Time taken downstream is t min
ATQ,
t + (t/3) = 20 min
⇒ (3t + t)/3 = 20
⇒ 4t = 20 × 3
⇒ t = 60/4
∴ t = 15 min

In 20 minutes he rows in upstream 12 km
∴ In 60 minutes he rows in upstream (12 × 60)/20 km
= 36 km

∴ x - y = 36 ..............(1)

In 15 minutes he rows In downstream 12 km
∴ In 60 minutes he rows In downstream (12 × 60)/15 km
= (720 × 3)/40 km
= 48 km

∴ x + y = 48 ................(2)

From (2) + (1) we get,
x + y + x - y = 48 + 36
⇒ 2x = 84
∴ x = 42

∴ His speed in still water is 42 km/hr

১,২৬৭.

A train moving at speed of 108 km/hr crosses a pole in 12 seconds. Find the length of the train.

ক
250 meters
খ
280 meters
গ
300 meters
ঘ
360 meters

ব্যাখ্যা

Question: A train moving at speed of 108 km/hr crosses a pole in 12 seconds. Find the length of the train.

Solution:
Length of the train is equal to the distance covered by train to cross the pole.
So, we will find the distance travelled by the train in 12 seconds.

Now,
Speed is given in Km/hr so we will convert it into m/s
Speed = 108 × (5/18) = 30 m/s
Time = 12 seconds

We know,
Distance = Speed × Time
∴ Length of train = 30 × 12 = 360 meters

১,২৬৮.

An inspector notices a thief from a distance of 200 meters after this thief starts running and the inspector chases him. The inspector and the thief run at the speed of 11 km/hr and 10 km/hr respectively. The distance between them after 6 minutes is?

ক
120 m
খ
110 m
গ
100 m
ঘ
90 m

ব্যাখ্যা

Question: An inspector notices a thief from a distance of 200 meters after this thief starts running and the inspector chases him. The inspector and the thief run at the speed of 11 km/hr and 10 km/hr respectively. The distance between them after 6 minutes is?

Solution:
চোর ও ইন্সপেক্টরের আপেক্ষিক গতি = (11 - 10) km/hr
= 1 km/hr
6 মিনিটে অতিক্রম করে = {(1/60) × 6}km
= 1/10 km
= 100 m

∴ চোর ও ইন্সপেক্টরের মধ্যবর্তী দূরত্ব = (200 - 100) m
= 100 m

১,২৬৯.

A wheel of an engine of 450 cm in circumference makes 20 revolutions in 6 seconds. What is the speed of the wheel in km/h?

ক
48 km/h
খ
54 km/h
গ
36 km/h
ঘ
60 km/h

ব্যাখ্যা

Question: A wheel of an engine of 450 cm in circumference makes 20 revolutions in 6 seconds. What is the speed of the wheel in km/h?

Solution:
Total distance = (450 × 20) cm
= 9000 cm
= 9000/100 m
= 90 m

We know,
Speed = (Total distance ÷ Time)
= (90 ÷ 6) m/sec
= 15 m/sec
= (15 × 18/5) km/h
= 54 km/h

১,২৭০.

If a runner takes as much time in running 20 metres as the car takes in covering 50 metres. The distance covered by the runner during the time the car covers 1 km is -

ক
ক) 400 metres
খ
খ) 40 metres
গ
গ) 440 metres
ঘ
ঘ) None of these

ব্যাখ্যা

According to the question,
∴ 50 m = 20 m
∴ 1m = 20/50 m
∴ 1000 m = (20/50)×1000 m
= 400 metres

১,২৭১.

A cyclist covers 15 rounds of a circular track of 400 meters every day. The time taken by the cyclist for three consecutive days are 75, 85, and 80 minutes respectively. On an average, what is the speed of the cyclist in meters/minute?

ক
70 meters/minute
খ
75 meters/minute
গ
80 meters/minute
ঘ
90 meters/minute

ব্যাখ্যা

Question: A cyclist covers 15 rounds of a circular track of 400 meters every day. The time taken by the cyclist for three consecutive days are 75, 85, and 80 minutes respectively. On an average, what is the speed of the cyclist in meters/minute?

সমাধান:
মোট অতিক্রান্ত দূরত্ব (৩ দিনে) = (দিনের সংখ্যা × প্রতি দিনের রাউন্ড × ট্র্যাকের দৈর্ঘ্য)
= (3 × 15 × 400) মিটার
= 18000 মিটার।

মোট সময় লেগেছে = (75 + 85 + 80) মিনিট
= 240 মিনিট।

গড় গতিবেগ = মোট দূরত্ব / মোট সময়
= 18000 / 240 মিটার/মিনিট
= 75 মিটার/মিনিট।

∴ সাইকেল আরোহীর গড় গতিবেগ হলো 75 মিটার/মিনিট।

১,২৭২.

If a man swims 5 meters upstream at 1 mph and back downstream to the same point at 5 mph, what is his average speed?

ক
1.07 mph
খ
1.37 mph
গ
1.67 mph
ঘ
1.97 mph

ব্যাখ্যা

1 mph এ 5m যায় 5 ঘণ্টায়
5 mph এ 5m যায় 1 ঘণ্টায়
∴6 ঘণ্টায় যায় 10m
∴1 ঘণ্টায় যায় (10/6)m=1.67 m

So,the average speed is 1.67 mph

১,২৭৩.

A person crosses a 600m long street in 5 minutes. What is his speed in km per hour?

ক
ক) 7.2
খ
খ) 3.6
গ
গ) 8.4
ঘ
ঘ) 10

ব্যাখ্যা

Question: A person crosses a 600m long street in 5 minutes. What is his speed in km per hour?

Solution:
Speed = 600 meters/5 minutes
= (600 × 60)/(5 × 1000) km/hr
= 7.2 km/hr

১,২৭৪.

What is the speed of a train if it overtakes two persons who are walking in the same direction at the rate of a m/s and (a + 1) m/s and passes them completely in b seconds and (b + 1) seconds respectively?

ক
ক) (a + b) m/s
খ
খ) (a + b + 1) m/s
গ
গ) (2a + 1) m/s
ঘ
ঘ) (2a + 1)/2 m/s

ব্যাখ্যা

Let the length of the train be x metres and its speed be y m/s.
Then,
x/(y - a) = b and x/{y - (a + 1) = (b + 1)
⇒ x = b (y - a) and x = (b + 1)(y - a - 1)
⇒ b (y - a) = (b + 1) (y - a - 1)
⇒ by - ba = by - ba - b + y - a - 1
⇒ y = (a + b + 1).

১,২৭৫.

A biker was riding at 60 km/h, but if he had gone at 80 km/h, he could’ve gone 100 km more in the same time. How far did he actually travel?

ক
250 km
খ
300 km
গ
350 km
ঘ
480 km

ব্যাখ্যা

Question: A biker was riding at 60 km/h, but if he had gone at 80 km/h, he could’ve gone 100 km more in the same time. How far did he actually travel?

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

Then,
x/60 = (x + 100)/80
⇒ x/6 = (x + 100)/8
⇒ 6(x + 100) = 8x
⇒ 6x + 600 = 8x
⇒ 8x - 6x = 600
⇒ 2x = 600
⇒ x = 600/2
⇒ x = 300 km

১,২৭৬.

A moving train, 66 metres long, overtakes another train of 88 metres long, moving in the same direction in 0.168 minutes. If the second train is moving at 30 km/hr, at what speed is the first train moving?

ক
55 kmph
খ
60 kmph
গ
74 kmph
ঘ
85 kmph

ব্যাখ্যা

Question: A moving train, 66 metres long, overtakes another train of 88 metres long, moving in the same direction in 0.168 minutes. If the second train is moving at 30 km/hr, at what speed is the first train moving?

Solution:
Suppose, the speed of first train be = x kmph
Speed of second train = 30 kmph = (30 × 1000)/60
= 500 m per min.

ATQ,
Total Distance/Relativespeed
⇒ (66 + 88)/(x - 500) = 0.168
⇒ 154/(x - 500) = 0.168
⇒ 0.168x - 84 = 154
⇒ 0.168x = 238
⇒ x = 238/0.168

Now,
{(238 × 1000)/168} m per min
= {(238 × 1000)/168} × (3/50) kmph
= 85 kmph

১,২৭৭.

A 300-meter-long train passes a person in 15 seconds. The person was running at 6 km/hr in the opposite direction of the train. What is the speed of the train in km/hr?

ক
50 km/hr
খ
66 km/hr
গ
75 km/hr
ঘ
86 km/hr
ঙ
none of these

ব্যাখ্যা

Question: A 300-meter-long train passes a person in 15 seconds. The person was running at 6 km/hr in the opposite direction of the train. What is the speed of the train in km/hr?

Solution:
দেওয়া আছে,
ট্রেনের দৈর্ঘ্য = 300 মিটার
অতিক্রম করার সময় = 15 সেকেন্ড
ব্যক্তির গতিবেগ = 6 কিমি/ঘন্টা

আমরা জানি, আপেক্ষিক গতিবেগ = দূরত্ব/সময়
আপেক্ষিক গতিবেগ = 300/15 মিটার/সেকেন্ড
= 20 মিটার/সেকেন্ড
= 20 × (18/5) কিমি/ঘন্টা
= 72 কিমি/ঘন্টা

যেহেতু ট্রেন এবং ব্যক্তি বিপরীত অভিমুখে গতিশীল,
তাই আপেক্ষিক গতিবেগ = ট্রেনের গতিবেগ + ব্যক্তির গতিবেগ।

মনে করি, ট্রেনের গতিবেগ = v কিমি/ঘন্টা

প্রশ্নমতে,
v + 6 = 72
⇒ v = 72 - 6
∴ v = 66

সুতরাং, ট্রেনের গতিবেগ 66 কিমি/ঘন্টা।

১,২৭৮.

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 _____

ক
28 km/hr
খ
62 km/hr
গ
86 km/hr
ঘ
82 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.

১,২৭৯.

A man covers half of his journey at 9 km/hr and the remaining half at 3 km/hr. His average speed is-

ক
ক) 4 kmph
খ
খ) 4.5 kmph
গ
গ) 5 kmph
ঘ
ঘ) 6 kmph

ব্যাখ্যা

Question: A man covers half of his journey at 9 km/h and the remaining half at 3 km/h. His average speed is-

Solution:
ধরি, মোট দূরত্ব 2x কিলোমিটার

x কিলোমিটার অতিক্রম করতে সময় লাগে = x/9 ঘণ্টা
বাকি x কিলোমিটার অতিক্রম করতে সময় লাগে = x/3 ঘণ্টা

গড় বেগ = মোট দূরত্ব/মোট সময়
= 2x/(x/9 + x/3)
= 2/4/9
= 9/2
= 4.5 কিমি/ঘণ্টা

১,২৮০.

Two stations A and B are on a straight line. One train starts from A at 7 a.m. and travels towards B at 20 Kmph. Another train starts from B at 8 a.m. and travel towards A at a speed of 25 kmph. If they meet at 10 a.m., what is the distance between two stations A and B?

ক
90 km
খ
100 km
গ
103 km
ঘ
110 km

ব্যাখ্যা

Question: Two stations A and B are on a straight line. One train starts from A at 7 a.m. and travels towards B at 20 Kmph. Another train starts from B at 8 a.m. and travel towards A at a speed of 25 kmph. If they meet at 10 a.m., what is the distance between two stations A and B?

Solution:
train started at 7 a.m. traveled for (10 - 7) = 3 hour
so in 3 hour at 20 kmph, the train travelled (20 × 3) = 60 km

train started at 8 a.m. traveled for (10 - 8) = 2 hour
so in 2 hour at 25 kmph, the train travelled (25 × 2) = 50 km

∴ the distance between two station A and B is = (60 + 50) km
= 110 km

১,২৮১.

A train 800 meters long is running at a speed 78 kmph. If it crosses a tunnel in 1 min, then the length of the tunnel is-

ক
ক) 500 m
খ
খ) 700 m
গ
গ) 800 m
ঘ
ঘ) 1300 m

ব্যাখ্যা

Question: A train 800 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 - ৮০০ = ৫০০ মি

১,২৮২.

A man covers half of his journey at 15 km/h and the remaining half at 5 km/h. His average speed is-

ক
3.25 km/h
খ
5 km/h
গ
7.5 km/h
ঘ
10 km/h
ঙ
15 km/h

ব্যাখ্যা

Question: A man covers half of his journey at 15 km/h and the remaining half at 5 km/h. His average speed is-

Solution:
Here, x = 15 km/h and y = 5 km/h

We know,
Average speed = 2xy/(x + y)
= (2 × 15 × 5)/(15 + 5)
= 150/20
= 7.5 km/h

১,২৮৩.

একটি দৌড় প্রতিযোগিতায় A এবং B এর গতিবেগের অনুপাত 3 : 4। গন্তব্যে পৌঁছাতে A এর চেয়ে B এর সময় 30 মিনিট কম লাগে। তাহলে A গন্তব্যে পৌঁছাতে কত ঘণ্টা সময় নেয়?

ক
1
খ
2
গ
1.5
ঘ
3
ঙ
কোনোটিই নয়

ব্যাখ্যা

প্রশ্ন: একটি দৌড় প্রতিযোগিতায় A এবং B এর গতিবেগের অনুপাত 3 : 4। গন্তব্যে পৌঁছাতে A এর চেয়ে B এর সময় 30 মিনিট কম লাগে। তাহলে A গন্তব্যে পৌঁছাতে কত ঘণ্টা সময় নেয়?

সমাধান:
ধরি,
A ও B এর গতিবেগ যথাক্রমে 3x ও 4x কিমি/ঘণ্টা
এবং উভয়ের জন্য দূরত্ব = D কিমি

আমরা জানি,
সময় = দূরত্ব/গতি

∴ A এর সময় = D/3x
∴ B এর সময় = D/4x

প্রশ্নমতে,
D/3x - D/4x = 30 মিনিট
⇒ D/3x - D/4x = 30/60 ঘণ্টা
⇒ (4D - 3D)/12x = 1/2
⇒ D/12x = 1/2
⇒ D = 6x

তাহলে, A এর সময় = D/3x = 6x/3x = 2 ঘণ্টা

১,২৮৪.

The ratio of speeds of A and B is 2 : 3 and therefore A takes 20 minutes more time than B. What is the ratio of time taken by A and B?

ক
3 : 2
খ
4 : 1
গ
1 : 2
ঘ
2 : 5
ঙ
None of the above

ব্যাখ্যা

Question: The ratio of speeds of A and B is 2 : 3 and therefore A takes 20 minutes more time than B. What is the ratio of time taken by A and B?

Solution:
When distance is constant then speed is inversely proportional to time.
ST = D

When distance is constant,
S ∝ 1/T

So, the ratio of time taken by A and B = 3 : 2

১,২৮৫.

A man can row at 6 kmph in still water. If the velocity of current is 2 kmph and it takes his 3 hours to row to a place and come back, how far is the place?

ক
ক) 6 km
খ
খ) 8 km
গ
গ) 10 km
ঘ
ঘ) 12 km

ব্যাখ্যা

Question: A man can row at 6 kmph in still water. If the velocity of current is 2 kmph and it takes his 3 hours to row to a place and come back, how far is the place?

Solution:
Speed of downstream = (6 + 2) kmph = 8 kmph
Speed of upstream = (6 - 2) kmph = 4 kmph

Let the required distance be x km

ATQ,
(x/8) + (x/4) = 3
⇒ (3x + 6x)/24 = 3
⇒ 9x = 3 × 24
⇒ 9x = 72
⇒ x = 72/9
∴ x = 8

Hence, the place is 8 km away

১,২৮৬.

The ratio of the speed of a boat in still water to the speed of the stream is 5 : 3. The boat covers 16 km upstream in 4 hours and x km downstream in 2 hours. Find the value of x.

ক
32 km
খ
36 km
গ
40 km
ঘ
48 km
ঙ
None of these

ব্যাখ্যা

Question: The ratio of the speed of a boat in still water to the speed of the stream is 5 : 3. The boat covers 16 km upstream in 4 hours and x km downstream in 2 hours. Find the value of x.

Solution:
ধরি, স্থির পানিতে নৌকার গতিবেগ = 5k কিমি/ঘণ্টা এবং স্রোতের গতিবেগ = 3k কিমি/ঘণ্টা।
স্রোতের প্রতিকূলে (Upstream) নৌকার গতিবেগ = (5k - 3k) = 2k কিমি/ঘণ্টা।
স্রোতের অনুকূলে (Downstream) নৌকার গতিবেগ = (5k + 3k) = 8k কিমি/ঘণ্টা।

প্রশ্ন অনুযায়ী,
স্রোতের প্রতিকূলে 16 কিমি যেতে সময় লাগে 4 ঘণ্টা।
∴ স্রোতের প্রতিকূলে গতিবেগ = 16 / 4 = 4 কিমি/ঘণ্টা।

শর্তমতে,
2k = 4
⇒ k = 4/2
⇒ k = 2

এখন, স্রোতের অনুকূলে নৌকার গতিবেগ = 8k = 8 × 2 = 16 কিমি/ঘণ্টা।
∴ স্রোতের অনুকূলে 2 ঘণ্টায় অতিক্রান্ত দূরত্ব, x = 16 কিমি/ঘণ্টা × 2 ঘণ্টা
= 32 কিমি

∴ x = 32 কিমি

১,২৮৭.

A man can row at 10 kmph in still water. If the speed of the current is 2 kmph and it takes him 1.5 hours to row to a place and come back, how far is the place?

ক
7.2 km away
খ
7 km away
গ
8.2 km away
ঘ
6 km away

ব্যাখ্যা

Question: A man can row at 10 kmph in still water. If the speed of the current is 2 kmph and it takes him 1.5 hours to row to a place and come back, how far is the place?

Solution:
Speed downstream = 10 + 2 = 12 kmph
Speed upstream = 10 - 2 = 8 kmph

Let the required distance = x km

ATQ,
(x/12) + (x/8) = 1.5
Or, (2x + 3x)/24 = 1.5
Or, 5x/24 = 1.5
Or, 5x = 36
∴ x = 36/5
∴ x = 7.2 km

∴ The place is 7.2 km away.

১,২৮৮.

Train A travels at a speed of 90 km/h and takes 9 seconds to pass a Man. It meets Train B, which is moving in the opposite direction at 72 km/h and length of the train B is x m. If Train A crosses Train B completely in 9 seconds. Find the value of x?

ক
220 m
খ
180 m
গ
160 m
ঘ
190 m

ব্যাখ্যা

Question: Train A travels at a speed of 90 km/h and takes 9 seconds to pass a Man. It meets Train B, which is moving in the opposite direction at 72 km/h and length of the train B is x m. If Train A crosses Train B completely in 9 seconds. Find the value of x?

Solution:
Given that,
Speed of Train A = 90 km/h
Speed of Train B = 72 km/h
Train A crosses a man in 9 sec
Train A crosses Train B in 9 sec

Now,
Train A speed = 90 × (5/18) = 25 m/s
∴ Length of Train A = 25 × 9 = 225 m

Relative speed = (90 + 72) × (5/18) = 162 × (5/18) = 45 m/s

∴ Total length = 45 × 9 = 405 m

∴ Length of Train B is, x = 405 - 225 = 180 m
∴ x = 180 m

১,২৮৯.

Nasim starts climbing an 11 m high wall at 4 pm. In each minute he climbs up 1 m but slips down 50 cm. At what time will he climb the wall?

ক
ক) 4:18 pm
খ
খ) 4:20 pm
গ
গ) 4:21 pm
ঘ
ঘ) 4:22 pm

ব্যাখ্যা

Question: Nasim starts climbing an 11 m high wall at 4 pm. In each minute he climbs up 1 m but slips down 50 cm. At what time will he climb the wall?

Solution:
প্রতি মিনিটে উঠে ১ মিটার বা ১০০ সেমি, নামে ৫০ সেমি
প্রতি মিনিটে উঠে ১০০ - ৫০ সেমি = ৫০ সেমি বা ১/২ মিটার

১/২ মিটার উঠতে সময় লাগে ১ মিনিট
১ মিটার উঠতে সময় লাগে ২ মিনিট
১০ মিটার উঠতে সময় লাগে ২০ মিনিট

পরের ১ মিটার ১ মিনিটে উঠে যায়।

সময় লাগে = ২০ + ১ মিনিট
= ২১ মিনিট

∴ ৪ টা ২১ মিনিটে দেয়ালের উপরে উঠে যাবে।

১,২৯০.

Two runners start running together for a certain distance, one at 5 km/h and another at 8 km/h. The former arrives one and a half an hour before the latter. The distance in Km is -

ক
10 km
খ
15 km
গ
20 km
ঘ
25 km

ব্যাখ্যা

Question: Two runners start running together for a certain distance, one at 5 km/h and another at 8 km/h. The former arrives one and a half an hour before the latter. The distance in Km is -

Solution:
Let the distance be = P
time of the first runner = P/5 hour
second runner = P/8

ATQ,
P/5 - P/8 = 3/2
3P/40 = 3/2
3P = 60
P = 20 km

১,২৯১.

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?

ক
9 sec
খ
11 sec
গ
15 sec
ঘ
18 sec

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.

১,২৯২.

A 70 metres long train running at the speed of 120 kmph crosses another train running in the opposite direction at the speed of 80 kmph in 9 seconds. What is the length of the other train?

ক
230 m
খ
240 m
গ
260 m
ঘ
320 m

ব্যাখ্যা

Relative speed = (120 + 80) km/hr
(200 times; 5/18) m/s
(500/9) m/s

Let the length of other train x meter
Then, (x +270)/9 = 500/9
⇒ x + 270 = 500
⇒ x = 230

Answer : 230 m

১,২৯৩.

If a biker rides at 70 km/hr instead of 50 km/hr, he would have ride 100 km more. The actual distance travelled by biker is:

ক
250 km
খ
225 km
গ
200 km
ঘ
175 km

ব্যাখ্যা

Question: If a biker rides at 70 km/hr instead of 50 km/hr, he would have ride 100 km more. The actual distance travelled by biker is:

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

Then,
x/50 = (x + 100)/70
⇒ x/5 = (x + 100)/7
⇒ 7x = 5x + 500
⇒ 7x - 5x = 500
⇒ 2x = 500
⇒ x = 500/2
∴ x = 250 km

১,২৯৪.

A thief is stopped by a policeman from a distance of 400 metres. When the policeman starts the chase, the thief also starts running. Assuming the speed of the thief as 5 km/h and that of policeman as 9 km/h, how far the thief would have run, before he is over taken by the policeman?

ক
400 m
খ
450 m
গ
500 m
ঘ
535 m

ব্যাখ্যা

Question: A thief is stopped by a policeman from a distance of 400 metres. When the policeman starts the chase, the thief also starts running. Assuming the speed of the thief as 5 km/h and that of policeman as 9 km/h, how far the thief would have run, before he is over taken by the policeman?

Solution:
Distance between thief and policeman = 400 metre
Relative speed of policeman with respect to thief
= (9 - 5) kmph = 4 kmph
= (4 × 5)/18 m./sec.
= 10/9 m./sec.

Now, Time taken in overtaking the thief = 400/(10/9) second
= (400 × 9)/10 second
= 360 s

∴ Distance covered by thief = Speed × Time
= {5 × 5 × (360/18)} m
= 500 m

১,২৯৫.

Two trains running in opposite directions cross a man standing on the platform in 37 seconds and 27 seconds respectively and they cross each other in 33 seconds. The ratio of their speeds is:

ক
ক) 3 : 2
খ
খ) 3 : 1
গ
গ) 4 : 3
ঘ
ঘ) 5 : 4

ব্যাখ্যা

Question: Two trains running in opposite directions cross a man standing on the platform in 37 seconds and 27 seconds respectively and they cross each other in 33 seconds. The ratio of their speeds is:

Solution:
Let,
The speed of the 1st train be x m/sec
The speed of the 2nd train be y m/sec

∴ Length of the 1st train = 37x metres
∴ Length of the 2nd train = 27y metres

ATQ,
(37x + 27y)/(x + y) = 33 [ time = distance/speed]
⇒ 37x + 27y = 33x + 33y
⇒ 37x - 33x = 33y - 27y
⇒ 4x = 6y
⇒ 2x = 3y
⇒ x/y = 3/2
∴ x : y = 3 : 2

১,২৯৬.

A train takes 18 seconds to pass completely through a station 162m long and 15 seconds through another station 120m long. The length of the train is-

ক
100 m
খ
90 m
গ
80 m
ঘ
70 m

ব্যাখ্যা

Question: A train takes 18 seconds to pass completely through a station 162m long and 15 seconds through another station 120m long. The length of the train is-

Solution:
ট্রেনটি 18 সেকেন্ডে অতিক্রম করে 162 মিটার + ট্রেনের দৈর্ঘ্য
ট্রেনটি 15 সেকেন্ডে অতিক্রম করে 120 মিটার + ট্রেনের দৈর্ঘ্য
ট্রেনটি 3 সেকেন্ডে অতিক্রম করে = 42 মিটার
ট্রেনটি 1 সেকেন্ডে অতিক্রম করে = 42/3 মিটার
ট্রেনটি 15 সেকেন্ডে অতিক্রম করে = (42 × 15)/3 মিটার
= 210 মিটার

∴ ট্রেনটির দৈর্ঘ্য = (210 - 120) মিটার = 90 মিটার।

১,২৯৭.

The current of a stream runs at the rate of 2 km per hr. A motorboat goes 10 km upstream and back again to the starting point in 55 min. Find the speed of the motorboat -

ক
ক) 16 km/hr
খ
খ) 20 km/hr
গ
গ) 12 km/hr
ঘ
ঘ) 22 km/hr

ব্যাখ্যা

Let the speed of the boat in still water=x km/hr
Speed of the current = 2 km/hr
Then, speed downstream = (x + 2) km/hr
speed upstream = (x - 2) km/hr
Total time taken to travel 10 km upstream and back = 55 minutes
= (55/60) hr
= 11/12 hr.
According to question,
10/(x - 2) + 10/(x + 2) = 11/12
⇒ 120(x+2)+120(x−2) = 11(x² - 4)
⇒ 240x = 11x² - 44
⇒ 11x² - 240x - 44 = 0
⇒ 11x(x -22) + 2(x - 22) = 0
⇒ (x -22) (11x + 2) = 0
Since x cannot be negative.
So, x = 22 km/hr.
Hence, the Speed of the motorboat is 22 km/hr.

১,২৯৮.

With an average speed of 40 km/hr, a train reaches its destination in time. If it goes with an average speed of 35 km/hr, it is late by 15 minutes. Find the length of the total journey?

ক
70 kms
খ
60 kms
গ
45 kms
ঘ
30 kms

ব্যাখ্যা

Question: With an average speed of 40 km/hr, a train reaches its destination in time. If it goes with an average speed of 35 km/hr, it is late by 15 minutes. Find the length of the total journey?

Solution:
Let the time taken by train be 't' hrs.
Then,
40t = 35(t + 15/60)
⇒ 40t = 35t + 35/4
∴ t = 7/4 hrs

Therefore, Required length of the total journey d = v × t
= 40 × 7/4
= 70 kms.

১,২৯৯.

A cyclist moving on a circular track of radius 100 meters completes one revolution in 2 minutes. What is the average speed of a cyclist (approx.)?

ক
ক) 314 m/min
খ
খ) 200 m/min
গ
গ) 300 m/min
ঘ
ঘ) 900 m/min
ঙ
ঙ) 500 m/min

১,৩০০.

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:

ক
ক) 3 : 2
খ
খ) 3 : 4
গ
গ) 3 : 1
ঘ
ঘ) 2 : 1

ব্যাখ্যা

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