PrepBank · বিষয়ভিত্তিক প্রশ্ন
PrepBank · পাতা ৫ / ১৫ · ৪০১–৫০০ / ১,৪৩৯
Let distance travelled upstream be x km
x/(4 − 2) +( x − 4)/( 4 + 2) = 6
⇒ x/2 + (x − 4)/6 = 6
⇒ 3 x + x − 4 = 36
⇒ 4 x = 40
⇒ x = 10
Since, the train travels at 60 km/h, it's speed per minute is 1 km per minute. Hence, if it's speed with stoppage is 40 km/h, it will travel 40 minutes per hour i.e. train stops 20 min per hour.
Time taken to travel 300 km with stoppage,
= 300/40 = 7.5 hours
Time taken for stoppage as it stops for 20 min per hour
= (7 × 20 + 10)
= 140 + 10
= 150 minutes.
= 150/60 hours
= 2.5 hours.
Relative speed = 40 - 20
= 20 km/hr
= 20 × (5/18)
= 50/9 m/s
Time = 5 s
Distance = (50/9) × 5
= 250/9
= 27(7/9).
Question: Rahim ran half the distance at 5 km/h and the remaining half at 10 km/h. What was his average speed for the entire run?
Solution:
Let the total distance = 2x km, so each half = x km.
Time for first half = x / 5 hours
Time for second half = x / 10 hours
∴ Total time = (x/5) + (x/10) = (2x + x)/10 = 3x/10 hours
∴Average speed = Total distance / Total time
= 2x ÷ (3x/10)
= 2x × (10/3x)
= 20/3
= 6.67 km/h
Speed of the man in still water = 12 km/hr.
Speed of the stream = 3 km/hr.
Speed downstream = 12 + 3
= 15 km/hr.
Speed upstream = (12 - 3)
= 9 km/hr.
Average speed = (Speed downstream × Speed upstream)/Speed in still water
= (15 × 9)/12
= (15 × 3)/4
= 45/4
= 11(1/4) km/hr.
Distance covered = 120 + 120 = 240 m
Time = 12s
Let the speed of each train = v.
Then relative speed = v + v = 2v
2v = distance/time
= 240/12
= 20 m/s
Speed of each train = v = 20/2
= 10 m/s
= 10 × 36/10 km/hr
= 36 km/hr
Let the time in which he traveled on foot = x hour
Time for travelling on bicycle = (9-x) hr
Distance = Speed×Time, and Total distance = 61 km
So,
4x + 9(9-x) = 61
=> 5x = 20
=> x = 4
So distance traveled on foot = 4x4 = 16 km
Question: A train travelling at the speed of x km/h crossed a 300 m long platform in 30 seconds, and overtook a man walking in the same direction at 6 km/h in 20 seconds. What is the value of x?
Solution:
Train speed = x km/h
Length of train = L
Length of platform = 300m
Man's speed = 6 km/h
∴ (x - 6) × (5/18) = L/20
⇒ (5x - 30)/18 = L/20
⇒ 100x - 600 =18L . . . . . . (i)
And, x × (5/18) = (L+ 300)/30
⇒ 150x = 18L + 5400
⇒ v150x - 5400 = 18L . . . . . . (ii)
From equations (i) & (ii)
⇒ 100x - 600 = 150x - 5400
⇒ 50x = 4800
∴ x = 96
Question: A train 150 metres long is travelling at 72 km/h. How much time will it take to completely cross a railway platform that is 250 metres long?
Solution:
Here,
Speed of the running train = 72 km/hr
= {72 × (5/18)} m/sec
= 20 m/sec
And length of the train is = 150 metres
Length of platform = 250 m
So, the time will taken by the train = (Length of train + Length of platform)/Speed
= (150 + 250)/30
= 400/20
= 20 sec
B is 60% more efficient than A, which means that the rate of B is 1.6 times as greater as that of A.
So, if A can complete the job in 12 days,
then B completes the same job in 12/1.6 = 12/(8/5)
= 15/2
= 7.5 days.
Alternative method:
Ratio of times taken by
A and B = 160 : 100 = 8 : 5
Suppose B alone takes x days to do the job.
Then, 8 : 5 :: 12 : x
or, 8x = 5 × 12
or, x = 7.5 days
Question: A train takes 12 seconds to cross a telegraph pole. It takes 42 seconds to cross a tunnel. What is the ratio of the length of the tunnel to that of the train?
Solution:
Let the speed of the train be x m/s
While crossing the telegraph pole,
The train travels 12 × x meters = 12x meters, which is the length of the train.
While crossing the tunnel,
The train travels 42 × x meters = 42x meters
∴ Length of the tunnel = 42x - 12x = 30x meters
∴ Length of the tunnel : Length of the train = 30x : 12x
= 30 : 12
= 5 : 2
Let, initial speed = x
ATQ,
⇒ 60/x - 60/(x+2) = 1
⇒ {60(x+2) - 60x} / x(x+2) = 1
⇒ 60x + 120 - 60x = x2 + 2x
⇒ x2 + 2x - 120 = 0
⇒ x2 + 12x - 10x - 120 = 0
⇒ x(x + 12) - 10(x + 12) = 0
⇒ (x + 12)(x - 10) = 0
Either, x = -12 [Not acceptable] or, x = 10
Question: In a race of 1km, A can beat B by 100m. In a 400m, B beats C by 40m. In a race of 500m. A will beat C by-
Solution:
We know,
1km = 1000m
∴ While A covers 1000 B covers 900
∴ while A covers 500 B covers 450m
∴ While B covers 400, C covers 360m
∴ While B covers 450, C covers (360 × 450)/400 = 405m
∴ in a 500m race A will beat C by = (500 - 405) = 95m
That means when A runs 500 meter then B can run 450m then C runs 405m.
Speed of the train relative to man
= (125/10) m/sec
= (25/2) m/sec
= (25/2)×(18/5) km/hr
= 45 km/hr
Let the speed of the train be x km/hr.
Then, relative speed = (x−5) km/hr
∴ x−5 = 45
⇒ x = 50km/hr
Let the speed in still water be x km/hr.
Then,
Speed downstream = (x+ 4) km/hr,
speed upstream = (x-4) km/hr.
6/(x+4) + 6 /(x-4) = 2
=> 1/(x+4) +1/(x-4)=2/6 = 1/3
=> (x+4)+(x-4)/x2-16= 1/3
=> x2-16= 6x
=> x2 -6x-16= 0
=> (x-8) (x+2) = 0
=> x = 8.
∴ Speed of boat in still water = 8 km/hr.
Let AB be the lighthouse and the two boats be at C and D
AB = 125 m
tan30° = BC/AB
= x/125
= 1/v3
x = 72.17 m
tan45° = BD/AB
= y/125
= 1
y = 125 m
Therefore,
the distance between the two boats is = x + y
= 72.17 + 125
= 197.17 m
Question: A boat can travel 48 km upstream in 6 hours. If the speed of the stream is 2 km/hr, how much time will the boat take to cover a distance of 120 km downstream?
Solution:
Distance covered by a boat in 6 hours = 48 km
Rate upstream of boat = 48/6
= 8 km/hr
Now,
Speed of stream = 2 km/hr
∴ Speed of boat in still water = (8 + 2)
= 10 km/hr
∴ Rate downstream of boat = (10 + 2) km/hr
= 12 km/hr
∴ Time taken in covering 120 km distance = 120/12
= 10 hours
Let PQ = Qr = x km
Let speed downstream = a km/hr.
and speed upstream = b km/hr.
Then,
x/a + x/b = 10
x = 10ab/(a + b) .........(i)
And,
2x/a = 4
x = 4a/2
x = 2a .............(ii)
From (i) and (ii) we have:
2a = 10ab/(a + b)
5b = a + b
a = 4b
Required ratio = Speed in the water/Speed of river
= {1/2(a + b)}/{(1/2) (a - b)}
= (a + b)/(a - b)
= (4b + b)/(4b - b)
= 5b/3b
= 5/3
Question: A train passes a man at 110 kmph running towards the train at the speed of 10 kmph. If it took 3 seconds to cross the man, what would be the length of the train?
Solution:
As both of them are facing towards each other.
total speed will be = 110 + 10 kmph
= 120 kmph
length of the train = 120 × (3/3600) km
= 0.1 km
= 100 m
Given that,
time is taken to travel upstream = 2 × times taken to travel downstream
When the distance is constant, speed is inversely proportional to the time
Hence, 2 × speed upstream = speed downstream
Let speed upstream = x
Then speed downstream = 2x
we have,
1/2(x + 2x) = speed in still water
⇒ 1/2(3x)=7.5
⇒ 3x = 15
⇒ x = 5
∴ speed upstream = 5 km/hr
∴ Rate of stream = 1/2(2x - x)
= x/2
= 5/2
= 2.5 km/hr.
Question: A train passes two bridges of length 800 m and 400 m in 100 seconds and 60 seconds respectively. The length of the train is-
Solution:
Let the length of the train is x m and speed is s.
ATQ,
s = (x + 800)/100 and,
s = (x + 400)/60
∴ (x + 800)/100 = (x + 400)/60
or, 60x + 48000 = 100x + 40000
or, 40x = 8000
or, x = 200 m
Question: A boat moves at a speed of 30 km/h in still water. The speed of the current is 5 km/h. How far will the boat travel downstream in 12 minutes?
Solution:
Speed of the boat in still water = 30 km/h
Speed of the current = 5 km/h
∴ Speed downstream = 30 + 5 = 35 km/h
Time = 12 minutes = 12/60 hours = 1/5 hours
∴ Distance travelled downstream = 35 × (1/5) km
= 7 km
∴ The distance travelled downstream is 7 km.
Cyclist:Jogger
Ratio of distance→ 2:1
Ratio of time→ 1:2
Ratio of their speed (Jogger:Cyclist)
= (1/2):(2/1)
= 1:4
Question: The ratio between the speeds of two buses is 5 : 6. If the second bus runs 450 km in 5 hours, then the speed of the first bus is:
Solution:
Given that,
Ratio of speeds of first bus : second bus = 5 : 6
Second bus covers 450 km in 5 hours
Now, Speed of second bus = distance/time
= 450/5
= 90 km/h
Let speed of first bus = 5x km/h
Speed of second bus = 6x km/h
We know speed of second bus = 90 km/h
So, 6x = 90
⇒ x = 90/6
∴ x = 15
∴ Speed of first bus = 5x = 5 × 15 = 75 km/h
So the speed of the first bus is 75 km/h.
Let,
The distance between A and destination be X
The total time taken by the car to cover X = 8 hours
Since X/2 by 80km/hr and remaining X/2 by 100km/hr
Then by Time = distance/speed, we have
(X/2)/80 + (X/2)/100 = 8
⇒ 1/2 {(X/80 + X/100)} = 8
⇒ X/80 + X/100 = 16
⇒ (5X + 4X)/400 = 16
⇒ 5X + 4X = 6400
⇒ 9X = 6400
⇒ X = 6400/9
⇒ X = 711.11
Hence 711.11 km is the required answer.
Question: A boat travels 24 km downstream in 40 minutes. If the speed of the stream is 6 km/h, what is the speed of the boat in still water?
Solution:
স্রোতের অনুকূলে 40 মিনিটে যায় 24 কিমি
স্রোতের অনুকূলে 1 মিনিটে যায় 24/40 কিমি
স্রোতের অনুকূলে 1 ঘণ্টা বা 60 মিনিটে যায় (24 × 60)/40 কিমি
= 36 কিমি
∴ স্রোতের অনুকূলে বেগ = 36 কিমি/ঘণ্টা
দেওয়া আছে,
স্রোতের বেগ = 6 কিমি/ঘণ্টা।
∴ স্থির পানিতে নৌকার বেগ = স্রোতের অনুকূলে বেগ - স্রোতের বেগ
= 36 - 6 = 30 কিমি/ঘণ্টা।
Let length and speed of the train be x metre and v kmph respectively.
x/9 = (v − 2) × 5/18 ⋯ ( 1 )
x/10 = (v − 4) × 5/18 ⋯ ( 2 )
Dividing (1) by (2) gives,
10/9 = (v − 2)/(v − 4)
⇒ 10v − 40 = 9v − 18
⇒ v = 22
Substituting the value of v in (1)
x/9 = 100/18
⇒ x = 50
To travel 10 miles at a speed of 50 mph, the train needs = 10/50 × 60 = 12 minutes
As the given time is 20 minutes
Then for the round trip the train has to travel 10 miles within 8 minutes
∴ Required speed = 10 × 8/60 = 75 mph
Question: A person swimming in a stream that flows 3 km/hr finds that in a given time, he can swim four times as far with the stream as he can against it. At what rate does he swim?
Solution:
ধরি,
স্রোতের প্রতিকূলে গতিবেগ = x কিমি/ঘন্টা
এবং স্রোতের অনুকূলে গতিবেগ = 4x কিমি/ঘন্টা।
স্রোতের গতিবেগ = (স্রোতের অনুকূলে গতিবেগ - স্রোতের প্রতিকূলে গতিবেগ)/2
= (4x - x)/2
= 3x/2 কিমি/ঘন্টা
প্রশ্নমতে, স্রোতের গতিবেগ 3 কিমি/ঘন্টা।
⇒ 3x/2 = 3
⇒ 3x = 6
⇒ x = 2
স্থির পানিতে সাঁতার কাটার গতিবেগ = (স্রোতের অনুকূলে গতিবেগ + স্রোতের প্রতিকূলে গতিবেগ)/2
= (4x + x)/2
= 5x/2
= (5 × 2)/2 [x এর মান বসিয়ে]
= 10/2
= 5 কিমি/ঘন্টা
সুতরাং, লোকটি 5 কিমি/ঘন্টা গতিতে সাঁতার কাটে।
Question: In covering a distance of 48 km, Robin takes 2 hours more than Karim. If Robin triples his speed, he would take 2 hours less than Karim. What is Robin's original speed in km/h?
সমাধান:
ধরি, Robin-এর মূল গতিবেগ = x কিমি/ঘন্টা
∴ Robin-এর 48 কিমি অতিক্রম করতে সময় লাগে = 48/x ঘন্টা
ধরি, Karim-এর 48 কিমি অতিক্রম করতে সময় লাগে = t ঘন্টা
প্রথম শর্ত অনুযায়ী:
Robin, Karim-এর চেয়ে 2 ঘন্টা বেশি সময় নেয়,
⇒ 48/x = t + 2 ........ (i)
দ্বিতীয় শর্ত অনুযায়ী:
Robin যদি তার গতিবেগ তিনগুণ করে (3x কিমি/ঘন্টা), তাহলে সে Karim-এর চেয়ে 2 ঘন্টা কম সময় নেয়,
⇒ 48/(3x) = t - 2 ........ (ii)
সমীকরণ (i) থেকে: t = 48/x - 2
সমীকরণ (ii)-তে বসিয়ে পাই,
48/(3x) = (48/x - 2) - 2
⇒ 16/x = 48/x - 4
⇒ 48/x - 16/x = 4
⇒ 32/x = 4
⇒ x = 32/4
⇒ x = 8 কিমি/ঘন্টা
সুতরাং, Robin-এর মূল গতিবেগ হলো 8 কিমি/ঘন্টা।
Distance = X
Distance covered at 5km and 4 km = x/3 + 2x/5
= (5x + 6x)/15
= 11x/15
Rest of the distance = 1 - 11x/15 = 4x/15
ATQ,
4x/15 = 12
or, x = (12 × 15)/4
or, x = 45.
Given that,
Two thirds of the 6 km was covered at 4 km/hr
i.e. 4 km distance was covered at 4 km/hr.
Time taken to cover 4 km = 4 km/(4 km/hr)
= 1 hr
= 60 minutes
Time left = (84 – 60) minutes.
= 24 minutes
Now,
The man has to cover remaining 2 km in 24 minutes
or 24/60
= 2/5 hours
Speed required for remaining 2 km
= 2 km/(2/5)hr
= 5 km/hr.
Question: A train 180 meters long passes a pole in 12 seconds. How long will it take to pass a platform that is 420 meters long?
Solution:
Train's speed = Distance/Time
= 180/12 = 15 m/s
Total distance to pass the platform,
= Length of train + Length of platform
= 180 m + 420 m
= 600 m
∴ Required time = Distance/Speed
= 600/15
= 40 seconds
∴ The train will take 40 seconds to pass platform.
We have to find the speed of a current.
t (downstream) = 3 h 45 min = 3.75 h
t (upstream) = 2 h 30 min = 2.5 h
Speed downstream = 22.5 km / 3.75 h = 6 km/h
Speed upstream = 10 km / 2.5 h = 4 km/h
So, Speed of the current = (Speed downstream - speed upstream) / 2
= (6 - 4) / 2
= (2 / 2) km/h
= 1 km/h
Question: Rahman is a boatman. He can row a boat at the speed of 5 km/hr upstream and 15 km/hr downstream. Find the speed of the stream.
Solution:
Let’s denote:
B as Speed of the boat in still water (km/h)
S as Speed of the stream (km/h)
Speed of the boat upstream is the speed of the boat in still water minus the speed of the stream:
B - S = 5 km/hr -------- (1)
Speed of the boat downstream is the speed of the boat in still water plus the speed of the stream:
B + S = 15 km/hr -------- (2)
(1) + (2)
B - S = 5
B + S = 15
2B = 20
∴ B = 10 km/hr
Putting the value of B in (2)
∴ S = (15 - 10) km/hr = 5 km/hr
∴ The speed of the stream is 5 km/hr
Question: A boatman goes 4 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 7 km in stationary water?
Solution:
Speed along current = 60/10 km/h
= 6 km/h
Speed against current = 4 km/h
∴ Speed in still water = (Speed against current + Speed along current)/2
= (4 + 6)/2
= 10/2
= 5
∴ Required time = 7/5 h
= [(7/5) × 60] min
= 84 min
= 60 min + 24 min
= 1 h 24 min