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Time and Speed - Train and Boat

মোট প্রশ্ন১,৪৩৯এই পাতা১০০প্রতি পাতা১০০
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উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Time and Speed - Train and Boat

PrepBank · পাতা ৭ / ১৫ · ৬০১–৭০০ / ১,৪৩৯

৬০১.
A man is walking at a speed of 10 kmph. After every km, he takes rest for 5 minutes. How much time will he take to cover a distance of 5 km?
  1. ক
    ক) 60 min
  2. খ
    খ) 50 min
  3. গ
    গ) 55 min
  4. ঘ
    ঘ) 70 min
ব্যাখ্যা

Time taken by man if he did not stop
= 5 km/10 kmph
= 1/2 hr
= 30 min 
∵Man takes rest for 5 minutes on each km
Total rest time= 5×4= 20 min
Total travelling time:
= 30 min+20 min
= 50 min

৬০২.
Working 10 hours a day, Polas can Complete a work in 4 days. Working 3 hours a day, Mahin can complete the same work in 20 days. Working 4 hours a day, they can jointly complete the work in
  1. ক
    7 days
  2. খ
    5 days
  3. গ
    3 days
  4. ঘ
    4 days
ব্যাখ্যা
Questin: Working 10 hours a day, Polas can Complete a work in 4 days. Working 3 hours a day, Mahin can complete the same work in 20 days. Working 8 hours a day, they can jointly complete the work in

Solution: 
Given,
Working 10 hours a day, Polash can complete the work in 4 days
Polash's total working hour = (10 × 4) hours 
= 40 hours

∴ Polas's 1 hour's work = 1/40 part

Working 3 hours a day, Mahin can complete the work in 20 days
Mahin's total working hour = (3 × 20) hours 
= 60 hours

∴ Mahin's 1 hour's work = 1/60 part

( Polas + Mahin )'s 1 hour's work = (1/40 + 1/60) part
= ( 3 + 2)/120 part
= 5/120
= 1/24 part

So (Polas + Mahin) complete whole part = 24 hours

Hence, they work 8 hours a day
i.e. they require 3 days to complete the work.
৬০৩.
Two trains of equal length, running in opposite directions, pass a pole in 24 and 12 seconds. The train will cross each other in -
  1. ক
    ক) 16 sec
  2. খ
    খ) 18 sec
  3. গ
    গ) 14 sec
  4. ঘ
    ঘ) 20 sec
ব্যাখ্যা
Question: Two trains of equal length, running in opposite directions, pass a pole in 24 and 12 seconds. The train will cross each other in -

Solution:
Let the length of both train be x meters
Speed of the first train = x/24 m/s
and speed of the second train = x/12 m/s

When running in opposite directions, relative speed = x/24 + x/12
= 3x/24
= x/8

To cross each other, distance to be covered = x + x = 2x meters

Taken taken = (2x)/(x/8) = 16 sec
৬০৪.
Two trains start from the same starting points and move towards the same destination. The first one starts half an hour earlier than the second one. The first one runs at a speed of 90 km/hr while the second one runs at 30 km/hr faster. At what distance from the starting point will the two trains meet?
  1. ক
    ক) 150 km
  2. খ
    খ) 180 km
  3. গ
    গ) 360 km
  4. ঘ
    ঘ) 450 km
ব্যাখ্যা

We know,
Distance(D) = Speed(S) × Time(T)
⇒ D = S × T
∴ S = D/T; T = D/S

Since the second train is 30 km/hr faster, it is moving at 120 km/hr
Now, let train A travel T hours before meeting train B.
Time for which train B travels = (T - 1/2) hrs.
This is because it starts half an hour late
Distance travelled is same.

∴ D = D
∴ 90 km/hr × T hrs = 120 km/hr × {T - (1/2} hrs
∴ 90T = 120T - 60
∴ T = 2 hours

Distance travelled by train A = 90 km/hr × 2 hours = 180km.
Thus they meet 180 km from starting point.

৬০৫.
A man's speed with the current is 18 km/h and the speed of the current is 3 km/h. The man's speed against the current is-
  1. ক
    10 km/h
  2. খ
    12 km/h
  3. গ
    14 km/h
  4. ঘ
    15 km/h
ব্যাখ্যা
Question: A man's speed with the current is 18 km/h and the speed of the current is 3 km/h. The man's speed against the current is-

Solution:
Given,
man's speed with the current is 18 km/h
the speed of the current is 3 km/h

∴ Man's rate in still water = (Man's speed with the current - The speed of the current)
= (18 - 3) km/h
= 15 km/h

∴ Man's rate against the current = (Man's rate in still water - The speed of the current)
= (15 - 3) km/h
= 12 km/h
৬০৬.
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
    ক) 400 meters
  2. খ
    খ) 300 meters
  3. গ
    গ) 200 meters
  4. ঘ
    ঘ) 100 meters
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 length of the train be x m and speed of the train is s kmph.
Speed, s = (x + 800)/100 . . . . . (i)
Speed, s = (x + 400)/60. . . . . (ii) 

Equating equation (i) and (ii),
we get,
(x + 800)/100 = (x + 400)/60
Or, (x + 800)/5 = (x + 400)/3
Or, 5x + 2000 = 3x + 2400
Or, 2x = 400
∴ x = 200m

∴ The length of the train is 200 meters.
৬০৭.
A train takes 15 seconds to pass a stationary point and covers 20 km in 25 minutes. Find the length of the train.
  1. ক
    300 meters
  2. খ
    150 meters
  3. গ
    200 meters
  4. ঘ
    180 meters
  5. ঙ
    320 meters
ব্যাখ্যা
Question: A train takes 15 seconds to pass a stationary point and covers 20 km in 25 minutes. Find the length of the train.

Solution:
Speed = 20km/25min
= (20 × 1000)m/(25 × 60)sec
= (40/3)m/s

∴ Length = Speed × Time to pass point
= (40/3) × 15
= 200 meters
৬০৮.
The ratio of speed of a motor-boat to that of the current of water is 5 : 1. The boat goes along with the current in 3 hours. It will come back in-
  1. ক
    5 hours
  2. খ
    3 hours
  3. গ
    5.8 hours
  4. ঘ
    2.5 hours
  5. ঙ
    None of these
ব্যাখ্যা
Question: The ratio of speed of a motor-boat to that of the current of water is 5 : 1. The boat goes along with the current in 3 hours. It will come back in-

Solution:
Since the ratio 5 : 1 is given.
Let the speed of boat in still water = 5 km/hr and speed of stream = 1 km/hr
Now,
Downstream speed = 5 + 1 = 6 km/hr
Upstream speed = 5 - 1 = 4 km/hr

∴ Distance = Downstream speed × downstream time = 6 × 3 = 18km

∴ Upstream time = Distance/upstream speed = 18/4 = 4.5 hours
৬০৯.
A farmer travelled a distance of 61 km in 9 hours. He travelled partly on foot 4 km/hr and partly on bicycle at 9 km/hr. The distance travelled on foot is:
  1. ক
    ক) 8 km
  2. খ
    খ) 14 km
  3. গ
    গ) 16 km
  4. ঘ
    ঘ) 20 km
ব্যাখ্যা
Question: A farmer travelled a distance of 61 km in 9 hours. He travelled partly on foot 4 km/hr and partly on bicycle at 9 km/hr. The distance travelled on foot is:

Solution:
Let the distance travelled on foot be x km
Then,
distance travelled on bicycle = (61 - x) km

So,
x/4 + (61 - x)/9 = 9
⇒ 9x + 4(61 - x) = 9
⇒ 9x + 244 - 4x = 9 × 36
⇒ 5x = 324 - 244
⇒ 5x = 80
⇒ x = 80/5
∴ x = 16 km.

∴ The distance travelled on foot is 16 km.
৬১০.
দুইটি ট্রেন দুটি স্টেশন থেকে একসাথে যাত্রা শুরু করে পরস্পরের দিকে রওনা দেয়। স্টেশন দুটি ২০০ কিমি দূরে অবস্থিত। তারা একে অপরকে একটি স্টেশন থেকে ১১০ কিমি দূরত্বে অতিক্রম করে। ট্রেন দুইটির গতিবেগের অনুপাত কত?
  1. ক
    ১১ : ৯
  2. খ
    ১২ : ৫
  3. গ
    ১৩ : ১১
  4. ঘ
    ৭ : ৩
  5. ঙ
    কোনোটিই নয়
ব্যাখ্যা

প্রশ্ন: দুইটি ট্রেন দুটি স্টেশন থেকে একসাথে যাত্রা শুরু করে পরস্পরের দিকে রওনা দেয়। স্টেশন দুটি ২০০ কিমি দূরে অবস্থিত। তারা একে অপরকে একটি স্টেশন থেকে ১১০ কিমি দূরত্বে অতিক্রম করে। ট্রেন দুইটির গতিবেগের অনুপাত কত?

সমাধান:
ধরি,
প্রথম ট্রেনের গতিবেগ = ক কিমি/ঘণ্টা
দ্বিতীয় ট্রেনের গতিবেগ = খ কিমি/ঘণ্টা

প্রথম ট্রেন অতিক্রম করে = ১১০ কিমি
দ্বিতীয় ট্রেন অতিক্রম করে = ২০০ - ১১০ = ৯০ কিমি

যেহেতু উভয় ট্রেন একসাথে শুরু করে এবং একই সময়ে একে অপরকে অতিক্রম করে, অতএব তাদের সময় সমান।

সময় = দূরত্ব/গতি

তাহলে,
১১০/ক = ৯০/খ
⇒ ক/খ = ১১০/৯০
⇒ ক : খ = ১১০ : ৯০
⇒ ক : খ = ১১ : ৯

∴ ট্রেন দুইটির গতির অনুপাত ১১ : ৯

৬১১.
A boy rides his bicycle 10 km at an average speed of 12 km/hr and travel 12 km at an average speed of 10 km/hr. His average speed for the entire trip is approximately :
  1. ক
    ক) 10.4 km/hr
  2. খ
    খ) 10.8 km/hr
  3. গ
    গ) 11.0 km/hr
  4. ঘ
    ঘ) 12.2 km/hr
ব্যাখ্যা

Average speed= Total distance/Total time
Average speed= (10+12)/(10/12)+(12/10)
Average speed= 10.8 km/hr

৬১২.
A train 360 metre long runs with a speed of 45 km/hr. What time will it take to pass a platform 140 metre long?
  1. ক
    ক) 25 seconds
  2. খ
    খ) 30 seconds
  3. গ
    গ) 48 seconds
  4. ঘ
    ঘ) 40 seconds
ব্যাখ্যা
Speed = 45 km/hr = 45 × 5/18
= 25/2 m/s
Distance travelled = Length of the train + Length of the platform
= 360 + 140
= 500 metre.
Time taken to cross the platform = 500/(25/2)
= 40 seconds
৬১৩.
A man can row at 5 kmph in still water. If the velocity of current is 1 km/hr and it takes him 1 hour to row to place and come back, how is the place? 
  1. ক
    ক) 3.4 km
  2. খ
    খ) 2.4 km
  3. গ
    গ) 1.4 km
  4. ঘ
    ঘ) 3.0 km
ব্যাখ্যা
Speed downstream =(5 + 1)kmph=6kmph
Speed upstream =(5 - 1)kmph=4kmph
Let the required distance be x km.

x/6 + x/4 = 1
2x+ 3x = 12
5x = 12 
x = 12/5 
x = 2.4
৬১৪.
A boat's speed with the current is 15 km/hr and the speed of the current is 2.5 km/hr. The boat's speed against the current is-
  1. ক
    8.5 km/hr
  2. খ
    9 km/hr
  3. গ
    10 km/hr
  4. ঘ
    12.5 km/hr
ব্যাখ্যা
Question: A boat's speed with the current is 15 km/hr and the speed of the current is 2.5 km/hr. The boat's speed against the current is-

Solution:
ধরি,
নৌকার বেগ x km/hr
স্রোতের বেগ y = 2.5 km/hr

স্রোতের অনুকূলে বেগ x + y = 15 km/hr

∴ x + y = 15
⇒ x = 15 - 2.5
∴ x = 12.5

∴ স্রোতের প্রতিকূলে বেগ x - y = 12.5 - 2.5 = 10 km/hr
৬১৫.
The length of the bridge, which a train 150 meters long and travelling at 45 km/h can cross in 30 seconds, is-
  1. ক
    220 meters
  2. খ
    225 meters
  3. গ
    235 meters
  4. ঘ
    250 meters
ব্যাখ্যা
Question: The length of the bridge, which a train 150 metres long and travelling at 45 km/h can cross in 30 seconds, is-

Solution:
Given,
Speed = 45 km/hr
= 45 × (5/18) m/sec
=25/2 m/sec
Time = 30 sec

Let, the length of bridge = x metres

Now
(150 + x)/30 = 25/2
⇒ 300 + 2x = 750
⇒ 2x = 750 - 300
⇒ 2x = 450
⇒ x = 450/2
∴ x = 225
৬১৬.
A boat can travel 36 km downstream in 45 minutes. The ratio of the speed of the boat in still water to the speed of the stream is 5 : 1. How much time will the boat take to cover 24 km upstream?
  1. ক
    45 min
  2. খ
    36 min
  3. গ
    25 min
  4. ঘ
    20 min
ব্যাখ্যা
Question: A boat can travel 36 km downstream in 45 minutes. The ratio of the speed of the boat in still water to the speed of the stream is 5 : 1. How much time will the boat take to cover 24 km upstream?

Solution:
Downstream speed = (36/45) × 60 km/h
= 48 km/hr.

Let,
speed of water and stream is 5x and x
Downstream speed = 5x + x = 6x
and upstream speed = 5x - x = 4x

ATQ, 
6x = 48
∴ x = 8

So, upstream speed = 4 × 8 = 32 km/hr.
Time taken to go 24 km upstream = (24 × 60)/32 = 45 min.

∴ The boat will take 45 minutes to cover 24 km upstream.
৬১৭.
A boat travels 120 km upstream in 6 hours and the same distance downstream in 4 hours. What is the speed (in km/h) of the stream?
  1. ক
    5 km/h
  2. খ
    10 km/h
  3. গ
    3 km/h
  4. ঘ
    6 km/h
ব্যাখ্যা

Question: A boat travels 120 km upstream in 6 hours and the same distance downstream in 4 hours. What is the speed (in km/h) of the stream?

সমাধান:
ধরি, 
​স্থির জলে নৌকাটির গতিবেগ = x কিমি/ঘন্টা
এবং স্রোতের গতিবেগ = y কিমি/ঘন্টা

স্রোতের প্রতিকূলে গতিবেগ = (x - y) কিমি/ঘন্টা
স্রোতের অনুকূলে গতিবেগ = (x + y) কিমি/ঘন্টা

প্রশ্নমতে,
120/(x - y) = 6
⇒ x - y = 120/6
∴ x - y = 20 …………(1)

আবার,
120/(x + y) = 4
⇒ x + y = 120/4
∴ x + y = 30 …………(2)

(2) নং থেকে (1) নং সমীকরণ বিয়োগ করে পাই,
(x + y) - (x - y) = 30 - 20
⇒ x + y - x + y = 10
⇒ 2y = 10
⇒ y = 10/2
∴ y = 5

সুতরাং, স্রোতের গতিবেগ 5 কিমি/ঘন্টা।

৬১৮.
A train passes two bridges of the lengths 500 m and 250 m in 100 seconds and 60 seconds respectively. The length of the train is -
  1. ক
    152 m
  2. খ
    125 m
  3. গ
    250 m
  4. ঘ
    120 m
ব্যাখ্যা

Length of train = x metre (let)
Speed of train
5x + 1250 = 3x + 1500.
⇒ 5x – 3x = 1500 – 1250.
⇒ 2x = 250.
∴ x= 125 metres.

৬১৯.
A train 150 m long passes a man, running at 6 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is-
  1. ক
    45 km/hr.
  2. খ
    54 km/hr.
  3. গ
    60 km/hr.
  4. ঘ
    48 km/hr.
ব্যাখ্যা
Question: A train 150 m long passes a man, running at 6 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is-

Solution:
Speed of the train relative to man = (150/10)m/sec
= 15 m/sec
= 15 × (18/5) km/hr
= 54 km/hr

Let the speed of the train be x km/hr. Then, relative speed = (x - 6) km/hr.

ATQ,
⇒ x - 6 = 54
∴ x = 60 km/hr.
৬২০.
A boat can travel with a speed of 13 km/h in still water. If the speed of the stream is 4 km/h, find the time taken by the boat to go 68 km downstream.
  1. ক
    3 hours
  2. খ
    4 hours
  3. গ
    5 hours
  4. ঘ
    6 hours
ব্যাখ্যা
Speed in downstream = (13 + 4) km/h = 17 km/h
Boat goes 17 km in 1 hour
Therefore, boat goes 68 km in (68/17) hour or 4 hours [ time = distance/speed ]
৬২১.
How many seconds will a 450 metre long train take to cross a man walking with a speed of 4 km/hr in the direction of the moving train if the speed of the train is 64 km/hr?
  1. ক
    ক) 24 sec
  2. খ
    খ) 25 sec
  3. গ
    গ) 27 sec
  4. ঘ
    ঘ) 30 sec
ব্যাখ্যা
Speed of the train relative to man
= (64 - 4) km/hr
= 60 km/hr
= 60× (5/18) sec
= 50/3 sec

Time taken to pass the man
=450 ×(3/50)
= 27 sec
৬২২.
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?
  1. ক
    4.5 km/hr
  2. খ
    5.5 km/hr
  3. গ
    6.5 km/hr
  4. ঘ
    7.5 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, speed upstream = x km/hr
speed downstream = 2x km/hr
Speed of stream = (2x - x)/2 = x/2 km/hr.

ATQ,
x/2 = 3/2
∴ x = 3

speed upstream = 3 km/hr
Speed downstream = 2 × 3 = 6 km/hr

Hence, rate of swimming = (3 + 6)/2 = 4.5 km/hr
৬২৩.
A boat travels at 12 km/h in still water, and the stream's speed is 3 km/h. If a boatman rows 90 km to a destination and comes back, how much time does he take in total?
  1. ক
    9 hours
  2. খ
    15 hours
  3. গ
    16 hours
  4. ঘ
    18 hours
ব্যাখ্যা
Question: A boat travels at 12 km/h in still water, and the stream's speed is 3 km/h. If a boatman rows 90 km to a destination and comes back, how much time does he take in total?

Solution:
Given,
Speed of the boat in still water = 12 km/h
Speed of the current = 3 km/h

Speed of the boat downstream = (Speed in still water + Speed of current) = (12 + 3) km/h = 15 km/h
Speed of the boat upstream = (Speed in still water − Speed of current) = (12 − 3) km/h = 9 km/h

Time taken to go downstream = Distance / Speed = (90/15) hours = 6 hours
Time taken to go upstream = Distance / Speed = (90/9) hours = 10 hours

Total time for the round trip = (6 + 10) hours = 16 hours
৬২৪.
A boat takes 6 hours to cover a distance while travelling upstream, whereas while travelling downstream it takes 4 hours. If the speed of the current is 6 kmph, what is the speed of the boat in still water? 
  1. ক
    ক) 18 kmph 
  2. খ
    খ) 24 kmph 
  3. গ
    গ) 30 kmph 
  4. ঘ
    ঘ) 38 kmph 
ব্যাখ্যা
Let the speed of the boat in still water be x kmph 
Then Speed downstream = (x + 6) kmph 
Speed up stream = (x - 6) kmph 

Now 
(x + 6) × 4 = (x - 6) × 6 
4x + 24 = 6x - 36
6x - 4x = 36 + 24 
2x = 60
x = 30 kmph
৬২৫.
A boat goes 13 km upstream in 39 minutes. The speed of stream is 3 km/h. What is the speed of the boat in still water?
  1. ক
    6 km/h
  2. খ
    16 km/h
  3. গ
    23 km/h
  4. ঘ
    26 km/h
ব্যাখ্যা
Question: A boat goes 13 km upstream in 39 minutes. The speed of stream is 3 km/h. What is the speed of the boat in still water?

Solution:
Against the stream,
Speed of the boat = Distance/Time
= 13/(39/60) km/h
= (13 × 60)/39 km/h
= 20 km/h

We know,
Speed of the boat upstream = Speed in still water − Speed of the stream
⇒ Speed in still water = Speed of the boat upstream + Speed of the stream
⇒ Speed in still water = (20 + 3) km/h = 23 km/h

∴ Speed of the boat in still water = 23 km/h.
৬২৬.
A pair of runners begin a race at the same time. One runs at 5 km/h and the other at 8 km/h. The faster runner finishes 1.5 hours sooner. Find the total distance.
  1. ক
    18 km
  2. খ
    20 km
  3. গ
    25 km
  4. ঘ
    30 km
  5. ঙ
    None of the above
ব্যাখ্যা
Question: A pair of runners begin a race at the same time. One runs at 5 km/h and the other at 8 km/h. The faster runner finishes 1.5 hours sooner. Find the total distance.
(দুইজন দৌড়বিদ সমান সময়ে একটি দৌড় প্রতিযোগিতা শুরু করেন। প্রথম ব্যক্তি ঘণ্টায় ৫ কিমি এবং দ্বিতীয় ব্যক্তি ঘণ্টায় ৮ কিমি গতিতে এগোন। দ্রুততম ব্যক্তি ১.৫ ঘণ্টা আগে সমাপ্ত করেন। মোট দূরত্ব নির্ণয় করুন।)

Solution: 
ধরা যাক, দূরত্ব = Z
প্রথম দৌড়বিদের সময় = Z/5 hour
দ্বিতীয় দৌড়বিদের সময় = Z/8

প্রশ্নমতে,
Z/5 - Z/8 = 3/2
⇒ 3Z/40 = 3/2
⇒ 3Z = 60
⇒ Z = 20 km
৬২৭.
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?
  1. ক
    300 km
  2. খ
    250 km
  3. গ
    380 km
  4. ঘ
    290 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 a km.

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

So the actual distance travelled = 300 km

৬২৮.
A train speeds past a pole in 20 seconds and speeds past a platform 100 meters in length in 30 seconds. What is the length of the train?
  1. ক
    ক) 100 meters
  2. খ
    খ) 150 meters
  3. গ
    গ) 180 meters
  4. ঘ
    ঘ) 200 meters
ব্যাখ্যা

Let the length of the train be x meters and its speed be y m/sec.
Then, xy = 20
⇒ y = x/20
∴ (x+100)/30 = x/20
⇒ 30x = 20x + 2000
⇒ 10x = 2000
⇒ x = 200 meters

৬২৯.
A man can row upstream at 12km/hr and downstream at 18 km/hr. The man rowing speed in still water is?
  1. ক
    5 km/h
  2. খ
    10 km/h
  3. গ
    15 km/h
  4. ঘ
    20 km/h
ব্যাখ্যা
Speed of boat in still water = (18 + 12)/2
                                            = 15 km/h
৬৩০.
Rakib completes two rounds of a circular field of radius 50 metres in 5 minutes. What is his speed in m/s?
  1. ক
    ক) 2.09 m/s
  2. খ
    খ) 3.5 m/s
  3. গ
    গ) 4 m/s
  4. ঘ
    ঘ) 2.5 m/s
ব্যাখ্যা
Question: Rakib completes two rounds of a circular field of radius 50 metres in 5 minutes. What is his speed in m/s?

Solution: 
বৃত্তাকার মাঠের ব্যাসার্ধ, r = 50 metres
সময়, t = 5 minutes = 300 seconds
দুইবার সম্পূর্ণ ঘুরলে মোট অতিক্রান্ত দূরত্ব = 2 × 2πr
= (4πr)
= 4 × 3.1416 × 50
= 628.32 metres

speed = মোট দূরত্ব / মোট সময়
= ৬২৮.৩২/৩০০
= ২.০৯ মি/সে
৬৩১.
If A travels to his school from his house at the speed of 3 km/h, then he reaches the school 7 minutes late. If he travels at the speed of 4 km/h, he reaches the school 8 minutes earlier than school time. The distance of his school from his house is:
  1. ক
    ক) 2 km
  2. খ
    খ) 3 km
  3. গ
    গ) 4 km
  4. ঘ
    ঘ) 5 km
ব্যাখ্যা
Question: If A travels to his school from his house at the speed of 3 km/h, then he reaches the school 7 minutes late. If he travels at the speed of 4 km/h, he reaches the school 8 minutes earlier than school time. The distance of his school from his house is:

Solution: 
Let the distance between school and home be x km.
The difference of time when A goes school to school with these two different speeds is = 8 - (- 7) = 15 min
=15/60 hour

Now
(x/3) - (x/4) = 15/60
(4x - 3x)/12 = 1/4
x/12 = 1/4
x = 12/4
x = 3
৬৩২.
A boat covers a certain distance downstream in 1 hour, while it comes back in 7/2 hours. If the speed of the stream be 5 kmph, what is the speed of the boat in still water?
  1. ক
    ক) 13 kmph
  2. খ
    খ) 11 kmph
  3. গ
    গ) 9 kmph
  4. ঘ
    ঘ) 7 kmph
ব্যাখ্যা
Let the speed of the boat in still water be x kmph.
Then,
Speed downstream = (x + 5) kmph,
Speed upstream = (x - 5) kmph.

Now 
(x + 5) x 1 =(x - 5)(7/2)
2(x + 5) = 7(x - 5)
2x + 10 = 7x - 35 
7x - 2x = 35 +10 
5x = 45
x = 9 kmph
৬৩৩.
A train, 800metre long is running with a speed of 78 km/hr. It crosses a tunnel in 1 minute. What is the length of the tunnel (in metres)?
  1. ক
    ক) 440 metre
  2. খ
    খ) 260 metre
  3. গ
    গ) 500 metre
  4. ঘ
    ঘ) 430 metre
ব্যাখ্যা

Let the length of the tunnel = x metre
Then, distance = (800 + x) metre.
Time = 1 minute = 60 seconds.

∴ Speed = 78 km/hr
= 78 × (5/18)
= (65/3) m/s

According to the question,
800 + x = {60 × (65/3)}
⇒ 800 + x = 1300
⇒ x = 500 metre.
Hence, The length of the tunnel is 500 metre.

৬৩৪.
A thief is noticed by a policeman from a distance of 300 meters. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km/hr and 12 km/hr respectively. What is the distance between them after 6 minutes?
  1. ক
    50 meters
  2. খ
    100 meters
  3. গ
    150 meters
  4. ঘ
    200 meters
ব্যাখ্যা

Question: A thief is noticed by a policeman from a distance of 300 meters. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km/hr and 12 km/hr respectively. What is the distance between them after 6 minutes?

সমাধান:
যেহেতু পুলিশ ও চোর একই দিকে দৌড়াচ্ছে, তাই তাদের আপেক্ষিক গতিবেগ (relative speed) হবে তাদের গতিবেগের পার্থক্য।
∴ আপেক্ষিক গতিবেগ = (12 - 10) কিমি/ঘন্টা
= 2 কিমি/ঘন্টা।
= (2 × 1000)/60 মিটার/মিনিট
= 200/6 মিটার/মিনিট

6 মিনিটে পুলিশ চোরের থেকে যে দূরত্ব কমিয়ে আনবে, তা হলো:
অতিক্রান্ত দূরত্ব = আপেক্ষিক গতিবেগ × সময়
= (200/6)× 6 
= 200 মিটার

সুতরাং, 6 মিনিট পরে তাদের মধ্যে অবশিষ্ট দূরত্ব = প্রাথমিক দূরত্ব - অতিক্রান্ত দূরত্ব
= 300 মিটার - 200 মিটার
= 100 মিটার

৬৩৫.
A bus is running with a uniform speed of 37 km/h. What distance will be covered by bus at 8 h?
  1. ক
    ক) 246 km
  2. খ
    খ) 289 km
  3. গ
    গ) 296 km
  4. ঘ
    ঘ) 276 km
  5. ঙ
    ঙ) 250 km
ব্যাখ্যা

We know that, speed = Distance/Time
Therefore, Required distance = Speed × Time
=37 × 8
= 296km

৬৩৬.
Train A, 600 metres long is running at 80 kmph will take how much time to cross a man sitting in another train which is 400 metres long, running at 64 kmph in the opposite direction?
  1. ক
    ক) 5 seconds
  2. খ
    খ) 10 seconds
  3. গ
    গ) 15 seconds
  4. ঘ
    ঘ) 20 seconds
  5. ঙ
    ঙ) 25 seconds
ব্যাখ্যা

Distance = 600 metres
Total Speed = 64 + 80 = 144 kmph (added because they are travelling in opposite directions)
In m/sec, speed = 144×(5/18) = 40 m/sec
Distance = Speed×Time
600 = 40×Time
Therefore, Time = 15 seconds

৬৩৭.
A man rows to a place 48 km distant and back in 14 hours. He finds that he can row 4 km with the stream at the same time as 3 km against the stream. The rate of the stream is:
  1. ক
    ক) 1 km/hr
  2. খ
    খ) 1.5 km/hr
  3. গ
    গ) 1.8 km/hr
  4. ঘ
    ঘ) 3.5 km/hr
ব্যাখ্যা

Suppose he moves 4 km downstream in x hours.
Then, Speed downstream = (4/x) km/hr
Speed upstream = (3/x) km/hr

∴ 48/(4/x) + 48/(3/x) = 14
⇒ 48x/4 + 48x/3 = 14
⇒ 48{(x/4) + (x/3)} = 14
⇒ 7x/12 = 14/48
⇒ 7x/12 = 7/24
⇒ 7x = (7/24) × 12
⇒ 7x = 7/2
⇒ 14x = 7
⇒ x = 7/14
⇒ x = 1/2

So, speed downstream = 7 km/hr
Rate of the stream = 1/2(8 - 6) km/hr = 1 km/hr.

৬৩৮.
A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is-
  1. ক
    2mph
  2. খ
    3mph
  3. গ
    4mph
  4. ঘ
    5mph
ব্যাখ্যা
Question: A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is-

Solution:
Let,
the speed of the stream = b mph

Then,
Speed downstream = (10 + b) mph
Speed upstream = (10 - b) mph

ATQ,
{36/(10 - b) - 36/(10 + b)} = 90/60
⇒ (72b × 60) = 90(100 - b2)
⇒ b2 + 48b - 100 = 0
⇒ (b + 50)(b - 2) = 0
⇒ b = 2mph
৬৩৯.
The speed ratio of car, train and bus are 5 : 9 : 4. If the average speed of the 3 vehicles is 72 km per hour, what is the average speed of the car and the bus?
  1. ক
    ক) 76 km/hr.
  2. খ
    খ) 54 km/hr.
  3. গ
    গ) 67 km/hr.
  4. ঘ
    ঘ) 86 km/hr.
ব্যাখ্যা
Question: The speed ratio of car, train and bus are 5 : 9 : 4. If the average speed of the 3 vehicles is 72 km per hour, what is the average speed of the car and the bus?

Solution: 
ধরি 
গাড়ির গতিবেগ = ৫ক কি.মি./ ঘণ্টা
ট্রেনের গতিবেগ = ৯ক কি.মি./ ঘণ্টা
বাসের গতিবেগ = ৪ক কি.মি./ ঘণ্টা

প্রশ্নমতে,
(৫ক + ৯ক + ৪ক)/৩ = ৭২
বা, ১৮ক/৩ = ৭২
বা, ৬ক = ৭২
     ক = ১২

গাড়ির গতিবেগ = ৫ × ১২ = ৬০ কি.মি./ ঘণ্টা
বাসের গতিবেগ = ৪ × ১২ = ৪৮ কি.মি./ ঘণ্টা

গাড়ি ও বাসের গড় গতিবেগ = (৬০ + ৪৮)/২ = ৫৪ কি.মি./ঘণ্টা
৬৪০.
A boat covers a certain distance in half an hour downstream with the speed of 40km/hr in still water and the speed of the current is 8km/hr. Then the distance travelled by the boat is:
  1. ক
    ক) 6km
  2. খ
    খ) 7km
  3. গ
    গ) 12km
  4. ঘ
    ঘ) 24km
ব্যাখ্যা
Given that,
The speed of a boat in still water is 40 km/hr
The speed of the stream is 8 km/hr.
Then,
The speed downstream = (40 + 8) km/hr
= 48 km/hr
The distance travelled in half an hour = 48 x 1/2
= 24 km
Hence the distance travelled by the boat is 24 km.
৬৪১.
A train with a length of 350 m long, takes 40 seconds to cross a 450m long bridge. How much time will the train take to cross a 550 m long platform?
  1. ক
    35 sec
  2. খ
    45 sec
  3. গ
    50 sec
  4. ঘ
    55 sec
  5. ঙ
    None
ব্যাখ্যা
Question: A train with a length of 350 m long, takes 40 seconds to cross a 450m long bridge. How much time will the train take to cross a 550 m long platform?

Solution:
Length of the train = 350 m
Length of the bridge = 450 m
∴ Total length = (450 + 350) m
= 800m

∴ Speed of the train = 800/40 m/sec
= 20 m/sec

Now total length of the train and new bridge = (550 + 350) m
= 900 m

∴ Time taken = (900/20) sec
= 45 sec
৬৪২.
A man traveled from the village to the post-office at the rate of 25 kmph and walked back at the rate of 4 kmph. If the whole journey took 5 hours 48 minutes, find the distance of the post-office from the village?
  1. ক
    40 km
  2. খ
    30 km
  3. গ
    20 km
  4. ঘ
    10 km
ব্যাখ্যা
Question: A man traveled from the village to the post-office at the rate of 25 kmph and walked back at the rate of 4 kmph. If the whole journey took 5 hours 48 minutes, find the distance of the post-office from the village?

Solution:
Average speed = (2ab)/(a + b) here a = 25, b = 4
= (2 × 25 × 4)/(25 + 4)
= 200/29 km/hr.

Distance covered in 5 hours 48 minutes
= Speed × time = (200/29) × (29/5)
= 40

Distance covered in 5 hours 48 minutes = 40 kms.
∴ Distance of the post office from the village = (40/2) = 20 km.
৬৪৩.
A man rows a boat a certain distance downstream in 9 hours, while it takes 18 hours to row the same distance upstream. How many hours will it take him to row three-fifth of the same distance in still water?
  1. ক
    9.5 hours
  2. খ
    10 hours
  3. গ
    12 hours
  4. ঘ
    7.2 hours
ব্যাখ্যা

Question: A man rows a boat a certain distance downstream in 9 hours, while it takes 18 hours to row the same distance upstream. How many hours will it take him to row three-fifth of the same distance in still water?

Solution: 
Let distance = d
So downstream speed = d/9​
And upstream speed = d/18​

We know, 
Speed in still water = (Downstream + Upstream)/2
= (d/9 + d/18)/2 
= {(2d + d)/18}/2
= (d/6)/2
= d/12
So, still water speed = d/12
Distance to travel = 3d/5

∴ Time = (3d/5)/(d/12) = (3 × 12)/5 = 36/5 = 7.2 hours

So it will take the man 7.2 hours to row three-fifth of the distance in still water. 

৬৪৪.
A person is walking at a speed of 5 km/hr along a railway track. If he is 200 m ahead of the train which is 100 m long and runs at a speed of 60 km/hr in same direction, then what is the time required to pass the person?
  1. ক
    ক) 19.64 sec
  2. খ
    খ) 15.19 sec
  3. গ
    গ) 3.2 min
  4. ঘ
    ঘ) 5 min
ব্যাখ্যা

Given: Speed of the person = 5 km/hr, length of train = 100 m, speed of train = 60 km/hr
Speed of train relative to walking person = (60–5) = 55 km/hr
Convert km/hr into m/s
55 km/hr = 55 x(5/18) = 15.27 m/s
Distance to be covered by the train = 200 + 100 = 300 m
Therefore, time taken by the train to cross the person
= Distance over speed =300/15.27 = 19.64 sec

৬৪৫.
A car travels a distance of 500 km in 10 hours. What is its speed in m/s?
  1. ক
    50 m/s
  2. খ
    16.67 m/s
  3. গ
    13.89 m/s
  4. ঘ
    180 m/s
ব্যাখ্যা
Question: A car travels a distance of 500 km in 10 hours. What is its speed in m/s?

Solution: 
Speed = Distance/Time 
= 500/10 km/hr.
= 50 km/hr.
= (50 × 1000)/3600 m/s
= 13.89 m/s
৬৪৬.
A motorist travels to a place 150 km away at an average speed of 50 km and returns at 30 km per hour. His average speed for the whole journey in km per hour is-
  1. ক
    ক) 35
  2. খ
    খ) 37
  3. গ
    গ) 37.5
  4. ঘ
    ঘ) 40
ব্যাখ্যা

Average speed
= (2xy / x+y) km/hr
= (2×50×30 / 50+30) km/hr
= 37.5 km/hr

৬৪৭.
An Aeroplan covers a certain distance at a speed of 300 kmph in 5 hours. To cover the same distance in 5/3 hours, it must travel at a speed of-
  1. ক
    900 km/h
  2. খ
    850 km/h
  3. গ
    720 km/h
  4. ঘ
    690 km/h
ব্যাখ্যা
Question: An Aeroplan covers a certain distance at a speed of 300 kmph in 5 hours. To cover the same distance in 5/3 hours, it must travel at a speed of-

Solution:
Given,
Distance = (300 × 5)
= 1500 km

We know,
Speed = (Distance ÷ Time)

∴ Required speed = (1500 ÷ 5/3) km/h
= (1500 × 3/5) km/h
= 900 km/h
৬৪৮.
A bus travels a distance of 315 km at a uniform speed. If the speed of the bus were 90% of the original speed, it would have taken 2.5 hours more to cover the same distance. The original speed was-
  1. ক
    ক) 14 Km/hour
  2. খ
    খ) 13 Km/hour
  3. গ
    গ) 12.6 Km/hour
  4. ঘ
    ঘ) 11.6 Km/hour
  5. ঙ
    ঙ) None
ব্যাখ্যা
Question: A bus travels a distance of 315 km at a uniform speed. If the speed of the bus were 90% of the original speed, it would have taken 2.5 hours more to cover the same distance. The original speed was-

Solution: 
Let the original speed of car is x kmph
Then time taken in moving 315 km = 315/x kmph

Now 
 (315/90% of x) - (315/x) = 2.5 
 {315/(90x/100)} - (315/x) = 2.5 
{(315 × 10)/9x} - (315/x) = 2.5 
(3150 - 2835)/9x = 2.5 
315/9x = 2.5
9x × 2.5 = 315
x = 315/(9 × 2.5 )
x = 14
৬৪৯.
A boatman rows his boat 48 km upstream and same distance downstream. The boat takes 12 hrs and 8 hrs to go upstream and downstream respectively. Find the speed of the boat in stagnant water and the speed of the stream respectively.
  1. ক
    4 km/hr ; 3 km/hr
  2. খ
    5 km/hr ; 1 km/hr
  3. গ
    6 km/hr ; 4 km/hr
  4. ঘ
    12 km/hr ; 6 km/hr
ব্যাখ্যা

We know,
Man's/Boat's Speed = X
Stream/Current/River Speed = Y

∴ Downstream speed = X + Y
Upstream speed = X - Y

Downstream speed = Distance covered/Time taken
= 48/8 = 6 km/hr
Upstream Speed = Distance covered/Time taken
= 48/12 = 4 km/hr

X+Y = 6 km/hr and X-Y = 4 km/hr
Adding them we get,
X + Y + X - Y = 10 km/hr

∴ X=5 km/hr = Speed of Boat
Y=6 - 5 = 1 km/hr = Speed of stream

৬৫০.
A train passes a station platform in 36 seconds and a man standing on the platform in 15 seconds. If the speed of the train is 90 km/h, what is the length of the platform?
  1. ক
    525 m
  2. খ
    440 m
  3. গ
    650 m
  4. ঘ
    475 m
ব্যাখ্যা
Question: A train passes a station platform in 36 seconds and a man standing on the platform in 15 seconds. If the speed of the train is 90 km/h, what is the length of the platform?

Solution:
Given,
Speed of the train = 90 km/h
= (90 × 5/18) m/sec
= 25 m/sec

Length of the train = (25 x 15) m
= 375 m

Let,
The length of the platform = x metres.

ATQ,
(x + 375)/36 = 25
⇒ x + 375 = 900
⇒ x = 900 - 375
∴ x = 525 m
৬৫১.
After travelling 50km, a train is meeting with an accident and travels at (3/4)th of the usual Speed and reaches 45 min late. Had the accident happened 10km further on it would have reached 35 min late. Find the usual Speed?
  1. ক
    ক) 20 km/hr
  2. খ
    খ) 25 km/hr
  3. গ
    গ) 30 km/hr
  4. ঘ
    ঘ) 32 km/hr
ব্যাখ্যা

Here there are 2 cases
Case 1: accident happens at 50 km
Case 2: accident happens at 60 km
Difference between two cases is only for the 10 kms between 50 and 60.
Time difference of 10 minutes is only due to these 10 kms.
In case 1, 10 kms between 50 and 60 is covered at (3/4)th Speed.
In case 2, 10 kms between 50 and 60 is covered at usual Speed.
So the usual Time 't' taken to cover 10 kms, can be found out as below.
(4/3)t – t = 10 mins [s = vt]
⇒ t = 30 mins, d = 10 kms
so usual
Speed = Distance/Time
= 10/30min
= 10/(1/2) km/hr.
= 20 km/hr.

৬৫২.
If a person walks at 18 km/h instead of 12 km/h, he would have walked 24 km more. The actual distance travelled by him is-
  1. ক
    48 km
  2. খ
    50 km
  3. গ
    52 km
  4. ঘ
    53 km
ব্যাখ্যা
Question: If a person walks at 18 km/h instead of 12 km/h, he would have walked 24 km more. The actual distance travelled by him is-

Solution:
Let
the actual distance travelled be x km.

Then,
x/12 = (x + 24)/18
⇒ 18x = 12x + 288
⇒ 6x = 288
∴ x = 48 km
৬৫৩.
A boat covers 143 km upstream in 13 hours and the same distance downstream in 11 hours. What is the speed (in km/h) of the boat in still (without stream) water?
  1. ক
    8 km/h
  2. খ
    10 km/h
  3. গ
    15 km/h
  4. ঘ
    12 km/h
ব্যাখ্যা

Question: A boat covers 143 km upstream in 13 hours and the same distance downstream in 11 hours. What is the speed (in km/h) of the boat in still (without stream) water?

Solution:
Let, the speed of the boat in still water = x km/h
and the speed of the stream = y km/h

According to the question,
143/(x - y) = 13
⇒ 13(x - y) = 143
⇒ x - y = 11 ............(1)

Again,
143/(x + y) = 11
⇒ 11(x + y) = 143
⇒ x + y = 13 ............(2)

Adding (1) and (2):
x - y + x + y = 11 + 13
2x = 24
⇒ x = 12 km/h

৬৫৪.
If a train of length 300 meters takes 81 seconds to cover a 600 meters wall then which of the following will be its speed in km/hr?
  1. ক
    38
  2. খ
    42
  3. গ
    35
  4. ঘ
    40
ব্যাখ্যা

We know,
Time taken by a train of length l metres to pass a stationary object of length b metres is
the time taken by the train to cover (l + b) metres.
Given that,
Time taken by a train of length 300 meters to pass a wall of length 600 meters is
the time taken by the train to covers (300+600) meters.
That is, 81 seconds.
We know that,
distance/time = speed
Therefore, required speed = (300+600)/81 m/sec
= 900/81 m/sec
Converting the above to km/hr
= (900/81) x (18/5) km/hr [m/s to km/hr conversion: a m/s = a x 18/5 km/hr.]
= 40 km/hr
Hence the answer is 40 km/hr.

৬৫৫.
A man walks a at a rate of 10 mph. After every ten miles, he rests for 6 minutes. How much time does he take to walk 50 miles?
  1. ক
    300
  2. খ
    318
  3. গ
    322
  4. ঘ
    324
ব্যাখ্যা

Question: A man walks a at a rate of 10 mph. After every ten miles, he rests for 6 minutes. How much time does he take to walk 50 miles?

Solution:
The man needs time = 50/10 = 5 hours
= (5 × 60)
= 300 minutes

He will also rest 4 times after 10, 20, 30 and 40 miles
Total resting time = 4 × 6 = 24 minutes.

Total time = 300 + 24 = 324 minutes

৬৫৬.
A train 300 m long is running at a speed of 90 km/hr. If it passes through a tunnel in 40 seconds, then the length of the tunnel is-
  1. ক
    600 meters
  2. খ
    700 meters
  3. গ
    900 meters
  4. ঘ
    1000 meters
ব্যাখ্যা

Question: A train 300 m long is running at a speed of 90 km/hr. If it passes through a tunnel in 40 seconds, then the length of the tunnel is-

Solution:
ট্রেনের গতিবেগ = 90 কিমি/ঘন্টা
= (90 × 5/18) মিটার/সেকেন্ড
= 25 মিটার/সেকেন্ড

একটি সুড়ঙ্গ অতিক্রম করার সময় ট্রেনটি তার নিজের দৈর্ঘ্যের সাথে সুড়ঙ্গের দৈর্ঘ্যও অতিক্রম করে।
সুতরাং, মোট অতিক্রান্ত দূরত্ব = ট্রেনের গতিবেগ × সময়
= 25 মিটার/সেকেন্ড × 40 সেকেন্ড
= 1000 মিটার

ধরা যাক, সুড়ঙ্গটির দৈর্ঘ্য = L মিটার
মোট অতিক্রান্ত দূরত্ব = ট্রেনের দৈর্ঘ্য + সুড়ঙ্গের দৈর্ঘ্য
⇒ 1000 = 300 + L
⇒ L = 1000 - 300
⇒ L = 700 মিটার

সুতরাং, সুড়ঙ্গটির দৈর্ঘ্য হলো 700 মিটার।

৬৫৭.
A monkey climbing up a pole ascends 6 meters and slips 3 meters in alternative minutes. If the pole is 60 meters high, how long will it take the monkey to reach the top?
  1. ক
    ক) 31 miles
  2. খ
    খ) 33 miles
  3. গ
    গ) 35 miles
  4. ঘ
    ঘ) 37 miles
ব্যাখ্যা

Net height ascended in 2 min = (6 - 3) m = 3 m.
Net height ascended in 36 min = (3/2 × 36) = 54m.
In the 37th min,
the monkey ascends 6m and reaches the top.
Hence,
Total time taken = 37 minutes.

৬৫৮.
A steamer goes downstream from one part to another in 4 hours. It covers the same distance upstream in 5 hours. If the speed of stream is 2 km/hr, the distance between the two ports is:
  1. ক
    60 km
  2. খ
    72 km
  3. গ
    80 km
  4. ঘ
    96 km
ব্যাখ্যা
Question: A steamer goes downstream from one part to another in 4 hours. It covers the same distance upstream in 5 hours. If the speed of stream is 2 km/hr, the distance between the two ports is:

Solution:
Let, the distance between the two parts = 'x' km
and the speed of steamer in still water = 'y' km/hr

ATQ,
x/(y + 2) = 4
⇒ x = 4y + 8 …. (1)

And, x/(y - 2) = 5
⇒ x = 5y - 10 ….. (2)

From (1) and (2)⇒
4y + 8 = 5y - 10
⇒ y = 18

∴ From (1)
x = 4 × 18 + 8
⇒ x = 80 km.
৬৫৯.
A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?
  1. ক
    220 m
  2. খ
    195 m
  3. গ
    180 m
  4. ঘ
    100 m
ব্যাখ্যা

Question: A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?

Solution:
Relative speed of the thief and policeman  =  (11 - 10) km/hr = 1 km/hr 
Distance covered in 6 minutes  = (1/60) × 6 km   = 1/10 km = 100 m
Therefore, Distance between the thief and policeman = (200 - 100) m = 100 m.

৬৬০.
A boat takes 12 hours to cover a certain distance upstream and 8 hours to cover the same distance downstream. What is the ratio between the speed of the boat and the speed of the water current, respectively?
  1. ক
    7 : 1 
  2. খ
    4 : 1 
  3. গ
    5 : 1 
  4. ঘ
    3 : 1 
ব্যাখ্যা

Question: A boat takes 12 hours to cover a certain distance upstream and 8 hours to cover the same distance downstream. What is the ratio between the speed of the boat and the speed of the water current, respectively?

Solution:
Let,
Upstream speed = x kmph
Downstream speed = y kmph

Since distance is same,
x × 12 = y × 8
⇒ 12x = 8y
⇒ y = 3x/2

Now,
Speed of boat = (y + x)/2
Speed of current = (y - x)/2

∴ Required ratio
= (y + x) : (y - x)
= [(3x/2) + x] : [(3x/2) - x]
= (5x/2) : (x/2)
= 5x : x
= 5 : 1

∴ The ratio of the speed of the boat to the speed of the water current is 5 : 1.

৬৬১.
How long will it take to row 20 km upstream if one can row 10 km in 10 minutes in still water and the same distance in 8 minutes with the stream?
  1. ক
    ক) 13.33 min
  2. খ
    খ) 24 min
  3. গ
    গ) 25.55 min
  4. ঘ
    ঘ) 26.67 min
ব্যাখ্যা
Question: How long will it take to row 20 km upstream if one can row 10 km in 10 minutes in still water and the same distance in 8 minutes with the stream?

Solution:
Let,
x be the speed of man in still water 
y be the speed of stream.
∴ Speed of man x = 10/10 km/min
= 1 km/min
= 60 km/hr

In downstream in 8 min he can row 10 km
∴ In downstream in 60 min he can row (10 × 60)/8 km
= 75 km

∴ x + y = 75
⇒ y = 75 - 60
∴ y = 15
∴ Speed of stream = 15 km/hr.

Hence upstream speed = 60 – 15 = 45 km/hr.

Time taken to cover 45 km = 60 min
∴ Time taken to cover 20 km = (60 × 20)/45 min  
= 26.67 min
৬৬২.
A passenger train of 200 m runs at a speed of 55 km/hr. A person traveling in it observes that the goods train moving in opposite direction takes 10 seconds to cross him. Find the speed of the goods train, if it is 250 m long.
  1. ক
    ক) 30.23 m/s
  2. খ
    খ) 29.73 m/s
  3. গ
    গ) 42.11 m/s
  4. ঘ
    ঘ) 55 m/s
ব্যাখ্যা

Given: Speed of passenger train = 55 km/hr, length of goods train (P) = 250, length of passenger train (Q)= 200m
Hint:
Time = (P+Q)/(V1+V2) sec
Goods train and the passenger train move in opposite direction. Hence, the relative speed is the addition of two speeds.
Convert 55 km/hr into m/s
55 x (5/18) = 15.277 m/s
Therefore,
10 = (250+200)/(15.27+V2)
V2 = 29.73 m/s 

৬৬৩.
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 :
  1. ক
    ক) 4 km/hr
  2. খ
    খ) 5 km/hr
  3. গ
    গ) 6 km/hr
  4. ঘ
    ঘ) 10 km/hr
ব্যাখ্যা

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

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

৬৬৪.
A delivery cart went from Candle ford to Lark Rise and back at an average speed of 2/3 miles per hour. If the distance from Candle ford to Lark Rise is 1 mile, and the trip back took half as much time as the trip there, what was the average speed of the delivery cart on the way to Lark Rise?
  1. ক
    ক) 1/3
  2. খ
    খ) 3/4
  3. গ
    গ) 1/2
  4. ঘ
    ঘ) 2/3
ব্যাখ্যা

Total distance = Average speed × total time
So, 2 = 2/3×T
⇒ T = 3 hours
So, the whole journey took 3 hours
Since trip back took half as much time as the trip there, it took 2 hours to reach there and 1 hour to come back.
So, speed of the delivery cart while going to Lark Rise = distance/time = 1/2 mph

৬৬৫.
A train 110 m long is running at the speed of 60 km/hr. In what time will it pass a man who is running at the speed of 6 km/hr in the opposite direction in which the train is moving?
  1. ক
    ক) 3 sec 
  2. খ
    খ) 5 sec 
  3. গ
    গ) 6 sec 
  4. ঘ
    ঘ) 9 sec 
ব্যাখ্যা
Question: A train 110 m long is running at the speed of 60 km/hr. In what time will it pass a man who is running at the speed of 6 km/hr in the opposite direction in which the train is moving?

Solution: 
Relative speed = 60 km/hr + 6 km/hr
= 66 km/hr
= 66 ×1000/3600 m/sec 
= 66 × 5/18 m/sec
= 55/3 m/sec

time needed = 110 /55/3 sec
= 110 × 3/55 sec 
= 6 sec 
৬৬৬.
A train 150 meters long passes a signal post in 15 seconds. How long will it take to pass a bridge that is 450 meters long?
  1. ক
    1 minute
  2. খ
    3 minute
  3. গ
    2 minute
  4. ঘ
    1.5 minute
ব্যাখ্যা

Question: A train 150 meters long passes a signal post in 15 seconds. How long will it take to pass a bridge that is 450 meters long?

Solution:
Train's speed = Distance/Time
= 150/15 = 10 m/s

Total distance to pass the bridge,
= Length of train + Length of bridge
= 150 m + 450 m
= 600 m

∴ Required time = Distance/Speed
= 600/10
= 60 seconds
​= 1 minute

∴ The train will take 60 seconds or 1 minute to pass platform.

৬৬৭.
A man covers half of his journey at 12 km/h and the remaining half at 6 km/h. His average speed is-
  1. ক
    10 km/h
  2. খ
    8 km/h
  3. গ
    7 km/h
  4. ঘ
    9 km/h
ব্যাখ্যা
Question: A man covers half of his journey at 12 km/h and the remaining half at 6 km/h. His average speed is-

Solution:
Given,
Man covers half of his journey at 12 km/h
The remaining half at 6 km/h

His average speed is = (2 × 12 × 6)/(12 + 6)
= 144/18
= 8 km/h
৬৬৮.
A man travels from his home to the office at 4 km/hr and reaches his office 5 min late. If the speed had been 5 km/hr he would have reached 10 min early. Find the distance from his home to the office?
  1. ক
    3 km
  2. খ
    5 km
  3. গ
    6 km
  4. ঘ
    10 km
ব্যাখ্যা

Question: A man travels from his home to the office at 4 km/hr and reaches his office 5 min late. If the speed had been 5 km/hr he would have reached 10 min early. Find the distance from his home to the office?

Solution:
ধরি, বাড়ি থেকে অফিসের দূরত্ব = d কিমি।
4 কিমি/ঘণ্টা বেগে সময় লাগে = d/4 ঘণ্টা
5 কিমি/ঘণ্টা বেগে সময় লাগে = d/5 ঘণ্টা

সময়ের পার্থক্য = 5 মিনিট (দেরি) + 10 মিনিট (আগে) = 15 মিনিট
= 15/60 = 1/4 ঘণ্টা

প্রশ্নমতে,
(d/4) - (d/5) = 1/4
⇒ (5d - 4d) / 20 = 1/4
⇒ d / 20 = 1/4
⇒ d = 20 / 4
∴ d = 5

সুতরাং, বাড়ি থেকে অফিসের দূরত্ব 5 কিমি।

৬৬৯.
A man on tour travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. The average speed for the first 320 km of the tour is-
  1. ক
    35.55 km/hr
  2. খ
    36 km/hr
  3. গ
    71.11 km/hr
  4. ঘ
    71 km/hr
ব্যাখ্যা
Question: A man on tour travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. The average speed for the first 320 km of the tour is-

Solution:
Total time taken = (160/64 + 160/80)hrs
= 9/2 hrs.

Average speed = 320 × (2/9) km.hr
= 71.11 km/hr.
৬৭০.
A boat goes 30 km downstream and comes back in a total of 4 hours 30 minutes. Its speed is 15 km/hr in still water. What is the speed of the stream (in km/hr)?
  1. ক
    4 km/h
  2. খ
    5 km/h
  3. গ
    6 km/h
  4. ঘ
    18 km/h
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) = 4.5
⇒ x2 = 25
⇒ x = 5km/hr
৬৭১.
A man completes a certain journey by a car. If he covered 30% of the distance at the speed of 20kmph. 60% of the distance at 40km/h and the remaining of the distance at 10 kmph, his average speed is:
  1. ক
    25 km/h
  2. খ
    45 km/h
  3. গ
    35 km/h
  4. ঘ
    27 km/h
ব্যাখ্যা
Suppose, total distance = 100 km

He covered 30% of the distance at the speed of 20kmph
His time taken = 30/20 hours

He covered 60% of the distance at 40km/h
His time taken = 60/40 hours

He covered the remaining of the distance at 10 kmph
The remaining of the distance = 100% - (30 + 60)% = 10%
His time taken = 10/10 hour

Average speed = total distance/time
                          = 100/(30/20 + 60/40 + 10/10)
                           = 25 km/h
৬৭২.
A dog starts chasing to a cat 2 hours later. It takes 2 hours to dog to catch the cat. If the speed of the dog is 30 km/h, what is the speed of cat?
  1. ক
    15 km/hr
  2. খ
    18 km/hr
  3. গ
    20 km/hr
  4. ঘ
    25 km/hr
  5. ঙ
    None of the above
ব্যাখ্যা
Question: A dog starts chasing to a cat 2 hours later. It takes 2 hours to dog to catch the cat. If the speed of the dog is 30 km/h, what is the speed of cat?

Solution:
Let the speed of the cat be = x km/hr
The speed of the dog be = 30 km/hr

Distance covered by cat in 2 hrs = (x × 2) = 2x km

Relative speed = (30 - x) km/hr

t = s/v
⇒ 2 = 2x/(30 - x) [It takes 2 hours to dog to catch the cat, so t = 2]
⇒ x = 30 - x
⇒ 2x = 30
∴ x = 15 km/hr
৬৭৩.
A man walk at a speed of 10 km/h. After every 3 kilometers, he takes a rest of 5 minutes. How much time will he take to cover a distance of 15 km?
  1. ক
    1 hour
  2. খ
    1 hour 30 minutes
  3. গ
    40 minutes
  4. ঘ
    1 hour 50 minutes
ব্যাখ্যা

Question: A man walk at a speed of 10 km/h. After every 3 kilometers, he takes a rest of 5 minutes. How much time will he take to cover a distance of 15 km?

সমাধান:
দূরত্ব = 15 কিমি
গতিবেগ = 10 কিমি/ঘন্টা

∴ সময় = দূরত্ব/গতিবেগ
= 15 কিমি/10 কিমি/ঘন্টা
= 1.5 ঘন্টা
= 90 মিনিট

সে প্রতি 3 কিলোমিটারের পরে 5 মিনিট বিশ্রাম নেয়।
15 কিমি দূরত্বে সে (15/3) - 1 = 4 বার বিশ্রাম নেবে (3 কিমি, 6 কিমি, 9 কিমি এবং 12 কিমি অতিক্রম করার পর)।
শেষ কিলোমিটারের পরে তার আর বিশ্রাম নেওয়ার প্রয়োজন নেই।

মোট বিশ্রামের সময় = 4 × 5 মিনিট = 20 মিনিট।

∴ মোট সময় = হাঁটার সময় + বিশ্রামের সময়
= 90 মিনিট + 20 মিনিট
= 110 মিনিট
= 1 ঘন্টা 50 মিনিট

সুতরাং, লোকটি 15 কিমি দূরত্ব অতিক্রম করতে মোট 110 মিনিট বা 1 ঘন্টা 50 মিনিট সময় নেবে।

৬৭৪.
The current of a stream runs at the rate of 2 km per hr. A motor boat goes 10 km upstream and back again to the starting point in 55 min. Find the speed of the motor boat -
  1. ক
    16 km/hr
  2. খ
    20 km/hr
  3. গ
    12 km/hr
  4. 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.

৬৭৫.
A thief committed a crime and escaped from the spot at a speed of 12 m/s. A Security guard started chasing him 20 minutes after the thief started running and caught him in the next 20 minutes. What is the speed (in m/s) of the Security guard?
  1. ক
    32 m/s
  2. খ
    24 m/s
  3. গ
    18 m/s
  4. ঘ
    36 m/s
ব্যাখ্যা

Question: A thief committed a crime and escaped from the spot at a speed of 12 m/s. A Security guard started chasing him 20 minutes after the thief started running and caught him in the next 20 minutes. What is the speed (in m/s) of the Security guard?

Solution: 
Given that,
Thief's speed = 12 m/s
Security guard starts 20 minutes or 1200 s later and catches thief in next 20 minutes or 1200 s

Now,
Distance covered by thief before guard starts,
d1 = speed × time = 12 × 1200 = 14400 m ; [20 minutes = 1200 s]
So, thief has a 14400 m head start.

And
Guard catches thief in 20 minutes = 1200 s
d2 = 12 × 1200 s = 14400 m 
∴ Guard covers total distance = head start + distance thief runs during chase = 14400 + 14400 = 28800 m
and Time taken by guard = 1200 s

∴ Speed of guard = Distance/Time = 28800/1200 = 24 m

৬৭৬.
The speed of the boat in still water is 5 times that of current, it takes 1.1 hour to row to point B from point A downstream. The distance between point A and point B is 13.2 km. How much distance (in km) will it cover in 312 minutes upstream?
  1. ক
    ক) 43.2 km
  2. খ
    খ) 48 km
  3. গ
    গ) 41.6 km
  4. ঘ
    ঘ) 44.8 km
ব্যাখ্যা

Let the speed of the current be x kmph
Then speed of the boat in still water = 5x

∴ Downstream speed
= (5x + x) = 6x km/hr.
Upstream speed
= (5x - x)
= 4x km/hr

Now,
According to the question,
1.1 × 6x = 13.2 km
6.6x = 13.2
x = 13.2/6.6
x = 2 km/hr.

Upstream speed
= 4x = 4 × 2 = 8 km/hr

∴ 312 minutes
= 312/60 hours
= 5(1/5) hours

∴ Required Distance travelled upstream
= Speed × Time
= 8 × (26/5)
= 41.6 km

৬৭৭.
A man goes to his office from his house at a speed of 3 km/hr and returns at a speed of 2 km/hr. If he takes 5 hours in going and coming, what is the distance between his house and office?
  1. ক
    3 km
  2. খ
    5 km
  3. গ
    6 km
  4. ঘ
    4 km
ব্যাখ্যা

Average speed = (2 × 3 × 2)/(2 + 3)
= 12/5 km/h
Total time taken = 5 hours
Distance travelled = (12/5) × 5
= 12 km
Therefore, distance between his house and office
= 12/2
= 6 km.

৬৭৮.
A man riding his bicycle covers 150 meters in 25 seconds. What is his speed in km per hour?
  1. ক
    ক) 20
  2. খ
    খ) 23
  3. গ
    গ) 21.6
  4. ঘ
    ঘ) 25
ব্যাখ্যা
Question: A man riding his bicycle covers 150 meters in 25 seconds. What is his speed in km per hour?

Solution:
Given,
Distance = 150 meters.
TIme = 25 seconds

∴ Speed = Distance/Time
= 150/25 m/s
= (150 × 3600)/(25 × 1000) km/hr.
= 21.6 km/hr.
৬৭৯.
Excluding stoppages, the speed of a bus is 80 kmph and including stoppages, it is 60 kmph. For how many minutes does the bus stop per hour?
  1. ক
    10 minutes
  2. খ
    12 minutes
  3. গ
    15 minutes
  4. ঘ
    18 minutes
ব্যাখ্যা

Question: Excluding stoppages, the speed of a bus is 80 kmph and including stoppages, it is 60 kmph. For how many minutes does the bus stop per hour?

Solution:
স্টপেজ ছাড়া বাসের গতিবেগ = 80 কিমি/ঘন্টা
স্টপেজ সহ বাসের গতিবেগ = 60 কিমি/ঘন্টা

প্রতি ঘন্টায় বাসটি যে দূরত্ব কম অতিক্রম করে = (80 - 60) কিমি = 20 কিমি।

এই 20 কিমি দূরত্ব অতিক্রম করতে বাসটির যে সময় লাগতো, সেই সময়টুকুই বাসটি থেমে থাকে।

গতিবেগ 80 কিমি/ঘন্টা হলে 20 কিমি যেতে সময় লাগে:
সময় = দূরত্ব/গতিবেগ
= 20 কিমি/80 কিমি/ঘন্টা
= 1/4 ঘন্টা
= (1/4 × 60) মিনিট
= 15 মিনিট

সুতরাং, প্রতি ঘন্টায় বাসটি 15 মিনিট থামে।

৬৮০.
Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is-
  1. ক
    50 m
  2. খ
    72 m
  3. গ
    80 m
  4. ঘ
    82 m
  5. ঙ
    None of these
ব্যাখ্যা
Question: Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is-

Solution:
Let the length of each train be x metres.
Then, distance covered = 2x metres.
Relative speed = (46 - 36) km/hr
= 10(5/18) m/sec
= 25/9  m/sec

ATQ,
2x/36 = 25/9
⇒ 2x =100
∴ x = 50 m
৬৮১.
A man can row 6 km/hr in still water. If the speed of the current is 2 km/hr, it takes 3 hrs more in upstream than in the downstream for the same distance. The distance is-
  1. ক
    ক) 30km
  2. খ
    খ) 24km
  3. গ
    গ) 20km
  4. ঘ
    ঘ) 32km
ব্যাখ্যা
Question: A man can row 6 km/hr in still water. If the speed of the current is 2 km/hr, it takes 3 hrs more in upstream than in the downstream for the same distance. The distance is-

Solution: 
Let distance = x
Speed of man in still water, = 6 km/h
Speed of current, = 2 km/h

Now
{x/(6 - 2)} - {x/(6 + 2)} = 3
⇒ (x/4) - (x/8) = 3
⇒ (2x - x)/8 = 3
⇒ x/8 = 3
x = 24 

Distance = 24 km
৬৮২.
A boat can travel 20 km downstream in 24 min. The ratio of the speed of the boat in still water to the speed of the stream is 4 : 1. How much time will the boat take to cover 15 km upstream?
  1. ক
    25 min
  2. খ
    30 min
  3. গ
    45 min
  4. ঘ
    50 min
ব্যাখ্যা
Question: A boat can travel 20 km downstream in 24 min. The ratio of the speed of the boat in still water to the speed of the stream is 4 : 1. How much time will the boat take to cover 15 km upstream?

Solution:
Downstream speed = (20/24) × 60 km/hr
= 50 km/hr.

Let, speed of water and stream is 4x and x
Downstream speed = 4x + x = 5x and upstream speed = 4x - x = 3x

ATQ, 
5x = 50
∴ x = 10

So, upstream speed = 3 × 10 = 30 km/hr.
Time taken to go 15 km upstream = (15 × 60)/30 = 30 min.
৬৮৩.
A train is driven at the speed of 100 kmph and stops for 10 minutes at the end of every 150 km. To cover a distance of 1000 km, it will be-
  1. ক
    ক) 9 hr
  2. খ
    খ) 10 hr
  3. গ
    গ) 11 hr
  4. ঘ
    ঘ) 12 hr
ব্যাখ্যা
Question: A train is driven at the speed of 100 kmph and stops for 10 minutes at the end of every 150 km. To cover a distance of 1000 km, it will be-

Solution:
Here, time taken to 150 km at 100 kmph  = 1 hour 30 min + 10 min
= 1 hr 40 min
= 1 hr + 2/3 hr
= 5/3 hr

time taken to cover (150 × 6) = 900 km in (5/3 × 900) = 10 hrs
Remaining 100 km is covered in 1 hr

So, total time = 10 + 1 = 11 hr
৬৮৪.
A man can row 30 km upstream in 6 hours. If the speed of the man in still water is 6 km/hr, find how much he can row downstream in 10 hours.
  1. ক
    ক) 70 km
  2. খ
    খ) 140 km
  3. গ
    গ) 95 km
  4. ঘ
    ঘ) 50 km
ব্যাখ্যা
Question: A man can row 30 km upstream in 6 hours. If the speed of the man in still water is 6 km/hr, find how much he can row downstream in 10 hours.

Solution:
Speed of upstream = 30/6 = 5 km/hr.
Speed of man in still water = 6 km/hr.

∴ Speed of current = 6 - 5 = 1 km / hr.
So speed of downstream = 6 + 1 = 7 km / hr.

∴ Distance traveled in 10 hrs = 10 × 7 = 70 km.
৬৮৫.
A train covered first 160 km at a speed of 20 km/h and then covered the remaining 240 km at a speed of 30 km/h. Find its average speed.
  1. ক
    35 km/h
  2. খ
    30 km/h
  3. গ
    27 km/h
  4. ঘ
    25 km/h
ব্যাখ্যা
Question: A train covered first 160 km at a speed of 20 km/h and then covered the remaining 240 km at a speed of 30 km/h. Find its average speed.

Solution:
Total distance = 160 + 240 = 400 km.

Time taken for the first 160 km = 160/20 = 8 hrs.
Time taken for the next 240 km = 240/30 = 8 hrs.
Total time taken = 8 + 8 = 16 hrs.

Average Speed =Total distance travelled/Total time taken
= 400/16
= 25 km/h
৬৮৬.
A man travelled a distance of 61km 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
    18 km
  2. খ
    14 km
  3. গ
    12 km
  4. ঘ
    16 km
  5. ঙ
    20 km
ব্যাখ্যা

Let the time in which he travelled on foot = x hr
Then the time in which he travelled on bicycle = (9 − x ) hr
distance = speed × time
⇒ 4 x + 9 (9 − x) = 61
⇒ 4 x + 81 − 9 x = 61
⇒ 5 x = 20
⇒ x = 4
Distance travelled on foot = 4x
= 4 × 4
= 16 km

৬৮৭.
In a 500 m race, A gives B a start of 50 m and still beats him by 25 m. What is the ratio of their speeds A : B ?
  1. ক
    21 : 20
  2. খ
    20 : 17
  3. গ
    18 : 16
  4. ঘ
    22 : 19
  5. ঙ
    None of the above
ব্যাখ্যা
Question: In a 500 m race, A gives B a start of 50 m and still beats him by 25 m. What is the ratio of their speeds A : B ?

Solution:
একটি ৫০০ মিটার রেসে B , A এর ৫০ মিটার আগে থেকে দৌড় শুরু করে। 
তাহলে A কে দৌড়াত হবে ৫০০ মিটার কিন্তু B কে দৌড়াতে হবে ৪৫০ মিটার

আবার
রেসে A, B কে ২৫ মিটার দূরত্বে হারায়। 
এখানে A যখন B কে অতিক্রম করে জিতে যায় তখন B অতিক্রম করে = ৪৫০ - ২৫ = ৪২৫ মিটার
এখন যেহেতু সময় একই তাই তাদের গতির অনুপাত হবে = A : B
= ৫০০ : ৪২৫
= ২০ : ১৭
৬৮৮.
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
    ক) 480 km.
  2. খ
    খ) 520 km.
  3. গ
    গ) 620 km.
  4. ঘ
    ঘ) 420 km.
ব্যাখ্যা
Question: 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?

Solution:
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
∴ x = 60 km/hr

Required distance = 8 × 60
= 480 km.
৬৮৯.
The time taken by a man to travel 36 miles downstream is 90 min less than to go the same distance upstream. The speed of the man in still water is 10 mph. Find the speed of the stream?
  1. ক
    ক) 2 mph
  2. খ
    খ) 2.5 mph
  3. গ
    গ) 3.5 mph
  4. ঘ
    ঘ) 5 mph
ব্যাখ্যা
ধরি,
স্রোতের বেগ x mph 

স্রোতের অনুকূলে লোকটির বেগ =(10 + x) mph 
স্রোতের প্রতিকূলে লোকটির বেগ = (10 - x) mph

প্রশ্নমতে, 
           36/(10 - x) - 36 /(10 + x) = 90/60
          ⇒ (360 + 36x - 360 + 36x)/ (10 - x) (10 + x) = 90/60 
          ⇒ 72x/ (100 -x2) = 3/2 
          ⇒ 24x/(100 - x2 )= 1/2
          ⇒ 48x = 100 - x2
          ⇒ x2 + 48x -100 = 0 
          ⇒ x2 +50x - 2x - 100=0
          ⇒ x(x +50) - 2(x +50)=0  
             ∴  (x +50) (x - 2) = 0
 এখানে,                            
          x - 2 = 0                     x +50 = 0 
           x =2                           x ≠ - 50 


স্রোতের বেগ 2 mph
৬৯০.
The speed of a boat in still water is 15 km/hr and the rate of current is 3 km/hr. The distance traveled downstream in 12 minutes is
  1. ক
    ক) 1.2 km
  2. খ
    খ) 3.6 km
  3. গ
    গ) 1.8 km
  4. ঘ
    ঘ) 2.4 km
ব্যাখ্যা

এখানে, 12 মিনিট = 12/60 = 1/5 ঘন্টা
স্রোতের অনুকূলে নৌকার বেগ = 15 + 3 = 18 kmph
অর্থাৎ নৌকাটি 1 ঘন্টা বা 60 মিনিটে যায় = 18 km
নৌকাটি 12 মিনিটে যায় = (18 × 12)/60
= 3.6 km

৬৯১.
The speed of a boat is 15km/h and the rate of current is 3km/h. How much distance downstream can be travelled in 15 minutes?
  1. ক
    ক) 2.5km
  2. খ
    খ) 3km
  3. গ
    গ) 4km
  4. ঘ
    ঘ) 4.5km
ব্যাখ্যা
Question: The speed of a boat is 15km/h and the rate of current is 3km/h. How much distance downstream can be travelled in 15 minutes?

Solution: 
here, 
the speed of the boat is = 15km/h
rate of current is = 3km/h

in downstream total speed, S = 15 + 3 = 18km/h
time, T = 15 minutes = 15/60 = 1/4 hours

the distance, D = S × T 
= 18 × 1/4
= 4.5 km
৬৯২.
A man covers a total distance of 100 km on bicycle. For the first 2 hours, the speed was 20 km/hr and for the rest of the journey, it came down to 10 km/hr. The average speed will be -
  1. ক
    ক) 12.5 km/hr
  2. খ
    খ) 13 km/hr
  3. গ
    গ) 14.5 Km/hr
  4. ঘ
    ঘ) 20 Km/hr
ব্যাখ্যা

Distance covered by man in first 2 hrs = 2 × 20 = 40 km
Remaining distance = (100 - 40) = 60 km
Time to cover the remaining distance = (60 /10) kmh = 6 hrs
∴ Required average speed = 100 / (2 + 6) = 12.5 kmh

৬৯৩.
A 180 m long train takes 18 seconds to pass a pole. How long will it take to pass a 330 m long platform?
  1. ক
    55 seconds
  2. খ
    60 seconds
  3. গ
    50 seconds
  4. ঘ
    51 seconds
ব্যাখ্যা

Question: A 180 m long train takes 18 seconds to pass a pole. How long will it take to pass a 330 m 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

৬৯৪.
A motorboat, whose speed is 12 km/hr in still water goes 24 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/h) is :
  1. ক
    9 km/h
  2. খ
    7 km/h
  3. গ
    4 km/h
  4. ঘ
    10 km/h
ব্যাখ্যা
Question: A motorboat, whose speed is 12 km/hr in still water goes 24 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/h) is :

Solution:
Let the speed of the stream be x km/h
Then, Speed downstream = (12 + x) km/h
Speed upstream = (12 - x) km/h

ATQ,
24/(12 + x) + 24/(12 - x) = 4(1/2)
⇒ 576/(144 - x2) = 9/2
⇒ 9(144 - x2) = 1152
⇒ 1296 - 9x2 = 1152
⇒ 1296 - 1152 = 9x2
⇒ 9x2 = 144
⇒ x2 = 16
∴ x = 4
৬৯৫.
In a race of 200 meters, B can give a start of 10 meters to A, and C can give a start of 20 meters to B. The starts that C can give to A, in the same race is:
  1. ক
    33 meters
  2. খ
    31 meters
  3. গ
    29 meters
  4. ঘ
    27 meters
  5. ঙ
    None of the above
ব্যাখ্যা
Question: In a race of 200 meters, B can give a start of 10 meters to A, and C can give a start of 20 meters to B. The starts that C can give to A, in the same race is:

Solution:
According to the question,
When B runs 200 m meters, A runs 190 meters;
Hence, when B runs 180 meters,

A runs = 190 × (180/200) meters
= 171 meters

When C runs 200m, B runs 180 meters.

Hence,
C will give a start to A by = (200 - 171) meters
= 29 meters
৬৯৬.
A distance is covered in 3 hrs 30 minutes at 5 km/hr. How much time will be taken to cover it at 21 km/hr?
  1. ক
    ক) 50 min
  2. খ
    খ) 60 min
  3. গ
    গ) 45 min
  4. ঘ
    ঘ) 55 min
ব্যাখ্যা
Question: A distance is covered in 3 hrs 30 minutes at 5 km/hr. How much time will be taken to cover it at 21 km/hr?

Solution: 
Distance = [5 × 3 (1/2)]
= 5 × (7/2)
= 17.5 km.

Time taken to cover it at 21 km/hr = [(17.5/21) × 60] min
= 50 min.
৬৯৭.
A man can row a boat 120 km with stream in 5 hours. If speed of the boat is double the speed of the stream, then the speed of stream is
  1. ক
    12 km/h
  2. খ
    10 km/h
  3. গ
    9 km/h
  4. ঘ
    14 km/h
  5. ঙ
    None
ব্যাখ্যা
Question: A man can row a boat 120 km with stream in 5 hours. If speed of the boat is double the speed of the stream, then the speed of stream is

Solution:
Speed of the boat downstream = 120/5 = 24 km/h
Ratio of speeds of boat and stream = 2 : 1
∴ Speed of the stream = (1/3) × 24
= 8 km/h
৬৯৮.
In a 500 m race, the speeds of two runners, A and B are in the ratio 5 : 6. If A is given a start of 100m, by how many meters does A win the race?
  1. ক
    20 meters
  2. খ
    40 meters
  3. গ
    25 meters
  4. ঘ
    60 meters
ব্যাখ্যা

Question: In a 500 m race, the speeds of two runners, A and B are in the ratio 5 : 6. If A is given a start of 100m, by how many meters does A win the race?

Solution:
Total race length = 500 meters.
A is given a start of 100 meters, so A runs 500 - 100 = 400 meters.

Speed ratio A : B = 5 : 6.

Let, B runs = X meter

Therefore,
400/X = 5/6
⇒ X = (6 × 400)/5
∴ X = 480m

Remaining distance for B = 500 - 480 = 20 meters.
Therefore, A wins by 20 meters.

৬৯৯.
A man travels from A to B at a speed of 36 km/hr in 74 minutes and he travels a distance from B to C with a speed of 45 km/hr in 111 minutes. Find the average speed of whole journey.
  1. ক
    39.8 km/h
  2. খ
    43.3 km/h
  3. গ
    40.5 km/h
  4. ঘ
    41.4 km/h
ব্যাখ্যা

Question: A man travels from A to B at a speed of 36 km/hr in 74 minutes and he travels a distance from B to C with a speed of 45 km/hr in 111 minutes. Find the average speed of whole journey.

Solution:
Given that,
A man travels from A to B at a speed of 36 km/hr in 74 minutes and he travels a distance from B to C with a speed of 45 km/hr in 111 minutes. 

We know,
Average speed = Total distance/Total time taken

Now,
Time taken = 74 min : 111 min   [given]
Ratio of Time taken = 2 : 3

∴ Average Speed = {(36 × 2) + (45 × 3)}/(2 + 3) = 207/5
 = 41.4 km/hr

∴ So the average speed of whole journey is 41.4 km/h

৭০০.
Traveling at a speed of 50 kmph, how long is it going to take to travel 60 km?
  1. ক
    1 hour and 12 minutes.
  2. খ
    1 hour and 15 minutes.
  3. গ
    1 hour and 18 minutes.
  4. ঘ
    1 hour and 21 minutes.
ব্যাখ্যা
Question: Traveling at a speed of 50 kmph, how long is it going to take to travel 60 km?

Solution:
Time = Distance ÷ Speed 
= 60/50
= 1.2 hours
= 1 hour and 12 minutes.