Pipes & Cisterns

PrepBank · পাতা ১ / ৫ · ১–১০০ / ৪০৮

১.

Two taps, P and Q, together can fill a tank in 12 minutes. If Tap Q takes 10 minutes more than Tap P to fill the tank separately, how much time will be taken by Tap P alone to fill the tank?

ক
15 minutes
খ
20 minutes
গ
24 minutes
ঘ
40 minutes

ব্যাখ্যা

Question: Two taps, P and Q, together can fill a tank in 12 minutes. If Tap Q takes 10 minutes more than Tap P to fill the tank separately, how much time will be taken by Tap P alone to fill the tank?

সমাধান:
ধরি,
নল P একা ট্যাঙ্কটি পূর্ণ করতে সময় নেয় x মিনিট।
তাহলে, নল Q একা ট্যাঙ্কটি পূর্ণ করতে সময় নেবে (x + 10) মিনিট।

প্রশ্নমতে, তারা একসাথে 12 মিনিটে পূর্ণ করে। অর্থাৎ,
(1/x) + 1/(x + 10) = 1/12
⇒ (x + 10 + x)/{x(x + 10)} = 1/12
⇒ (2x + 10)/(x² + 10x) = 1/12
⇒ x² + 10x = 12(2x + 10)
⇒ x² + 10x - 24x - 120 = 0
⇒ x² - 14x - 120 = 0
⇒ x² - 20x + 6x - 120 = 0
⇒ x(x - 20) + 6(x - 20) = 0
⇒ (x - 20)(x + 6) = 0

যেহেতু সময় ঋণাত্মক হতে পারে না, তাই x = 20
∴ নল P একা ট্যাঙ্কটি পূর্ণ করতে 20 মিনিট সময় নেবে।

২.

Water flows through a cylindrical pipe of an internal diameter of 7cm at the rate of 5m/s. The time, in minutes, the pipe would take to fill an empty rectangular tank of 4m × 3m × 2.31m is -

ক
20 min
খ
18 min
গ
28 min
ঘ
24 min

ব্যাখ্যা

Question: Water flows through a cylindrical pipe of an internal diameter of 7cm at the rate of 5m/s. The time, in minutes, the pipe would take to fill an empty rectangular tank of 4m × 3m × 2.31m is -

Solution:
the total volume of the tank is = 400 × 300 × 231 cc
total water flow per second through the pipe is = πr²h
= (22/7) × (7/2)² × 500

∴ total time = (400 × 300 × 231)/{(22/7) × (7/2)² × 500}
= (400 × 300 × 231 × 4 × 7)/(22 × 49 × 500)
= 1440 s
= 24 min

৩.

A water tank can be filled in 15 minutes by two pipes A and B working together. Both pipes are opened simultaneously, but after 9 minutes, pipe A is closed. If pipe B takes an additional 10 minutes to fill the remaining part of the tank, how many minutes would pipe B alone take to fill the entire tank?

ক
15 minutes
খ
18 minutes
গ
25 minutes
ঘ
30 minutes

ব্যাখ্যা

Question: A water tank can be filled in 15 minutes by two pipes A and B working together. Both pipes are opened simultaneously, but after 9 minutes, pipe A is closed. If pipe B takes an additional 10 minutes to fill the remaining part of the tank, how many minutes would pipe B alone take to fill the entire tank?

Solution:
পাইপ A ও B একত্রে 15 মিনিটে পূর্ণ করে 1 অংশ
∴ পাইপ A ও B একত্রে 1 মিনিটে পূর্ণ করে (1/15) অংশ
∴ পাইপ A ও B একত্রে 9 মিনিটে পূর্ণ করে (9/15) অংশ = 3/5 অংশ

ট্যাংকটির অবশিষ্ট অংশ = {1 - (3/5)} অংশ = 2/5 অংশ

প্রশ্নমতে, পাইপ A বন্ধ করার পর পাইপ B অবশিষ্ট 2/5 অংশ পূর্ণ করে 10 মিনিটে।
∴ পাইপ B দ্বারা 2/5 অংশ পূর্ণ হয় 10 মিনিটে
∴ পাইপ B দ্বারা 1 বা সম্পূর্ণ অংশ পূর্ণ হয় (10 × 5) / 2 মিনিটে
= 50 / 2 মিনিট
= 25 মিনিটে

সুতরাং, পাইপ B একাকী 25 মিনিটে সম্পূর্ণ ট্যাংকটি পূর্ণ করতে পারবে।

৪.

An outgoing pipe pours water at half the amount of an ingoing pipe. After 4 hours of running both pipes, a tank was filled. If the outgoing pipe was closed, how much time would it take to fill the tank with the ingoing pipe?

ক
4 hours
খ
8 hours
গ
1 hours
ঘ
2 hours

ব্যাখ্যা

Question: An outgoing pipe pours water at half the amount of an ingoing pipe. After 4 hours of running both pipes, a tank was filled. If the outgoing pipe was closed, how much time would it take to fill the tank with the ingoing pipe?

Solution:
Let,
ingoing pipe needs X hours,
The outgoing pipe needs 2X hours.

together in one hour, these pipes can fill = 1/X - 1/2X = 1/2X

ATQ,
2X = 4
X = 2
∴ Ingoing pipe will take 2 hours to fill the tank.

৫.

12 buckets of water fill a tank when the capacity of each bucket is 13.5 litres. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 9 litres?

ক
14
খ
16
গ
18
ঘ
20
ঙ
None of the above

ব্যাখ্যা

Question: 12 buckets of water fill a tank when the capacity of each bucket is 13.5 litres. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 9 litres?

Solution:
Capacity of the tank
= (12 × 13.5) litres
= 162 litres

Capacity of each bucket = 9 litres

∴ Number of buckets needed
= 162/9
= 18

৬.

A cistern measures 5 meters in length, 2.5 meters in breadth, and 3 meters in height. Find its capacity in liters.

ক
37.5 liters
খ
375000 liters
গ
3750 liters
ঘ
37500 liters

ব্যাখ্যা

Question: A cistern measures 5 meters in length, 2.5 meters in breadth, and 3 meters in height. Find its capacity in liters.

Solution:
দেওয়া আছে,
চৌবাচ্চাটির দৈর্ঘ্য = 5 মিটার
চৌবাচ্চাটির প্রস্থ = 2.5 মিটার
চৌবাচ্চাটির উচ্চতা = 3 মিটার

আমরা জানি,
চৌবাচ্চাটির আয়তন = (5 × 2.5 × 3) ঘন মিটার
= 37.5 ঘন মিটার
= (37.5 × 1000) লিটার [১০০০ লিটার = ১ ঘন মিটার]
= 37500 লিটার

∴ চৌবাচ্চাটির পানি ধারণক্ষমতা = 37500 লিটার।

৭.

Half of the water tank is filled manually. Tap A can fill the tank in 20 minutes and B can empty the tank in 12 minutes. If A and B are opened together, then the time taken to empty or fill the tank is -

ক
ক) 30 minutes
খ
খ) 15/2 minutes
গ
গ) 60 minutes
ঘ
ঘ) 45/2 minutes

ব্যাখ্যা

Given that,
A takes 20 minutes to fill and B takes 12 minutes to empty
Clearly,
tap B is faster than tap A.
And so, the tank will be emptied.
Half of the tank or 1/2 part of the tank is already filled.
Therefore,
we have to find the time taken to empty that 1/2 part.
Part filled by A in 1 minute = 1/20
Part emptied by B in 1 minute = 1/12.
Part emptied by (A + B) in 1 minute
= (1/12) – (1/20)
= (5 - 3)/60
= 2/30.
Therefore,
The time taken by (A + B) to empty the full tank is 15 minutes.
Time taken to empty 1/2 part of the tank is 30/2
= (30/2) × (1/2) minutes.
= 15/2 minutes.

৮.

Two pipes can fill a tank in 6 hours and 8 hours respectively while a third pipe empties the full tank in 12 hours. If all the three pipes operate simultaneously, in how much time will the tank be filled?

ক
4 hrs
খ
18/7 hrs
গ
24/5 hrs
ঘ
3 hrs
ঙ
None

ব্যাখ্যা

Question: Two pipes can fill a tank in 6 hours and 8 hours respectively while a third pipe empties the full tank in 12 hours. If all the three pipes operate simultaneously, in how much time will the tank be filled?

Solution:
If one pipe fills the tank in 'x' hrs, another pipe fills the same tank in 'y' hrs but the third pipe empties the tank in 'z' hrs and all of them are opened together, then

The net part filled in 1hr = (1/x) + (1/y) - (1/z)
From the given data, net part filled in 1 hour = (1/6) + (1/8) - (1/12)
= (4 + 3 - 2)/24
= 5/24

So, the total time to fill the tank with all pipes open = 24/5 hrs

৯.

A pump can fill a tank with water in 2 hours. Because of a leak, it took (7/3) hours to fill the tank. The leak can drain all the water of the tank in:

ক
(13/3)hrs.
খ
7 hrs.
গ
8 hrs.
ঘ
14 hrs.

ব্যাখ্যা

Question: A pump can fill a tank with water in 2 hours. Because of a leak, it took (7/3) hours to fill the tank. The leak can drain all the water of the tank in:

Solution:
pump fills 1/2 tank in 1 hour

because of leak, pump fills 3/7 tank in 1 hour

leak drains (1/2) - (3/7) = 1/14 tank in 1 hour
leak drains full tank in 14 hours

১০.

In 1 minute 3/7 of a bucket is filled. The rest of the bucket can be filled in -

ক
4/7 minutes
খ
4/3 minutes
গ
7/4 minutes
ঘ
None

ব্যাখ্যা

Question: In 1 minute 3/7 of a bucket is filled. The rest of the bucket can be filled in -

Solution:
Filled of the bucket 3/7 part.
Remaining = 1 - (3/7) = 4/7 part.

3/7 part is filled in = 1 min
∴ 1 part is filled in = 7/3 min
So, 4/7 part is filled in = (7/3) × (4/7) min
= 4/3 minutes

১১.

A reservoir has two pipes, A and B. A can fill the reservoir 5 hours faster than B. If both together fill the reservoir in 6 hours, the reservoir will be filled by A alone in-

ক
8 hours
খ
10 hours
গ
11 hours
ঘ
12 hours

ব্যাখ্যা

Question: A reservoir has two pipes, A and B. A can fill the reservoir 5 hours faster than B. If both together fill the reservoir in 6 hours, the reservoir will be filled by A alone in-

Solution:
Let, A alone can fill the reservoir in x hours
B can fill in x + 5 hours

Both complete in 1 hour = (1/x) + (1/ x + 5)
= (2x + 5)/(x² + 5x)

In 6 hour Both together fill the full part or 1
In 1 hour Both together fill 1/6 part

Now
(2x + 5)/(x² + 5x) = 1/6
⇒ x² + 5x = 12x + 30
⇒ x² - 7x - 30 = 0
⇒ x² - 10x + 3x - 30 = 0
⇒ x(x - 10) + 3(x - 10) = 0
⇒ (x - 10) (x + 3) = 0
∴ x = 10 or, x = - 3 , negative value not possible

A alone can fill the reservoir in 10 hours

১২.

A booster pump can be used to fill as well as to empty the tank. The capacity of the tank is 1200 m³. The emptying capacity of the tank is 10 m³ per minute higher than its filling capacity and the pump requires 4 minutes less to vacant the tank than it requires to fill it. Calculate the filling capacity of the pump is-

ক
40 m³/min
খ
45 m³/min
গ
55 m³/min
ঘ
50 m³/min

ব্যাখ্যা

Question: A booster pump can be used to fill as well as to empty the tank. The capacity of the tank is 1200 m³. The emptying capacity of the tank is 10 m³ per minute higher than its filling capacity and the pump requires 4 minutes less to vacant the tank than it requires to fill it. Calculate the filling capacity of the pump is-

Solution:
Let, the filling capacity of the pump = x m³/min

Given that,
Capacity of the tank = 1200 m³
Emptying rate is 10 m³/min more than filling rate
Emptying time is 4 minutes less than filling time

Then,
Filling time = 1200/x minutes
Emptying capacity = x+10 m³/min
Emptying time = 1200/(x + 10)

According to question,
(1200/x) - {1200/(x + 10)} = 4
(1/x) - {1/(x + 10)} = 1/300
(x + 10 - x)/x(x + 10) = 1/300
x2 + 10x - 3000 = 0
(x + 60) (x - 50) = 0

So, possible values is,
x = 50 And x = - 60 [not valid]

So, the filling capacity of the pump is 50 m³/min

১৩.

3/4 part of the tank is full of water. When 18 liters of water is taken out, the tank becomes empty. The capacity of the tank is -

ক
20 liters
খ
24 liters
গ
27 liters
ঘ
30 liters

ব্যাখ্যা

Question: 3/4 part of the tank is full of water. When 18 liters of water is taken out, the tank becomes empty. The capacity of the tank is -

Solution:
3/4 part of the tank is 18 liters

Capacity of the tank is = 18 × 4/3
= 24 liters

১৪.

Two pipes P and Q can fill a cistern in 15 and 20 minutes respectively. Both pipes are opened together, after how many minutes should Q be turned off, so that the cistern be fill in 12 minutes?

ক
9 minutes
খ
6 minutes
গ
4 minutes
ঘ
8 minutes

ব্যাখ্যা

Question: Two pipes P and Q can fill a cistern in 15 and 20 minutes respectively. Both pipes are opened together, after how many minutes should Q be turned off, so that the cistern be fill in 12 minutes?

Solution:
P can fill the cistern in 15 minutes
So in 1 min P can fill the cistern = 1/15 th part
In 12 min, P can fill the cistern = 12/15
= 4/5 part

Remaining part = 1- (4/5) part
= 1/5 part

As Q can fill full cistern in 20 minutes
So it will fill 1/5 part in = (1/5) × 20 = 4 minutes.

∴ Pipe Q should be turned off after 4 minutes.

১৫.

To fill a tank, 30 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to two-third of its present capacity?

ক
42.5
খ
38
গ
37.5
ঘ
45

ব্যাখ্যা

Question: To fill a tank, 30 buckets of water are required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to two-third of its present capacity?

Solution:
Let,
The capacity of 1 bucket be n
∴ The capacity of tank be 30n
Capacity of the new bucket = 2n/3

∴ Required number of buckets = (30n)/(2n/3)
= (30n × 3)/2n
= 90/2
= 45

১৬.

A cistern can be filled by two pipes X and Y in 8 hours and 12 hours respectively. When full, the tank can be emptied by a third pipe Z in 24 hours. If all the taps be turned on at the same time, the cistern will be full in?

ক
8 hrs.
খ
9 hrs.
গ
4 hrs.
ঘ
6 hrs.

ব্যাখ্যা

Question: A cistern can be filled by two pipes X and Y in 8 hours and 12 hours respectively. When full, the tank can be emptied by a third pipe Z in 24 hours. If all the taps be turned on at the same time, the cistern will be full in?

Solution:
Net filling in 1 hour = (1/8) + (1/12) - (1/24)
= (3 + 2 - 1)/24
= 4/24
∴ Time taken to fill the cistern = (24/4) hrs.
= 6 hrs.

১৭.

Pipe A alone can fill a tank in 8 hours. Pipe B can fill it in 6 hours. If both the pipes are opened and after 2 hours pipe A is closed, then the other pipe will fill the tank in-

ক
ক) 5/2 hours
খ
খ) 7/2 hours
গ
গ) 4 hours
ঘ
ঘ) 6 hours

ব্যাখ্যা

A ও B 1 ঘণ্টায় পূর্ণ করে = (1/8) + (1/6) অংশ
= (3 + 4)/24 অংশ
= 7/24 অংশ
A ও B 2 ঘণ্টায় পূর্ণ করে = (7 × 2)/24 = 7/12 অংশ

বাকি থাকে = 1 - (7/12) অংশ
= (12 - 7)/12 অংশ
= 5/12 অংশ

B 1 অংশ পূর্ণ করে = 6 ঘণ্টায়
B 5/12 অংশ পূর্ণ করে = (6 × 5)/12 ঘণ্টায়
= 5/2 ঘণ্টায়

১৮.

Two tanks of the same capacity will be filled by two pipes that have the capacity to fill up a tank in 12 hours and 16 hours respectively. If after filling one tank the pipe is attached to the other one how much time will it take to fill both tanks?

ব্যাখ্যা

Question: Two tanks of the same capacity will be filled by two pipes that have the capacity to fill up a tank in 12 hours and 16 hours respectively. If after filling one tank the pipe is attached to the other one how much time will it take to fill both tanks?

Solution:
the first pipe can fill the first tank in 12 hours.
after that, the pipe will be attached to the second tank.

in 12 hours, the second pipe can fill = 12/16 = 3/4
remaining = 1/4

in one hour, both pipes can fill
= 1/12 + 1/16
= 7/48
so, to fill 1/4 of a tank it will take
= 1/4 × 48/7
= 12/7 hours

total time = 12 + 12/7
=

১৯.

A cistern is to be filled by a pipe of capacity 12 hours per cistern. But after every 2 hours, 1/20th of the cistern got empty. How much time will take to fill the full cistern?

ক
120/7 hours
খ
60/7 hours
গ
80/7 hours
ঘ
120/9 hours

ব্যাখ্যা

Question: A cistern is to be filled by a pipe of capacity 12 hours per cistern. But after every 2 hours, 1/20th of the cistern got empty. How much time will take to fill the full cistern?

Solution:
in two hours total fill up
= 2/12 - 1/20
= 1/6 - 1/20
= 7/60

in one hour = 7/120

total time = 120/7 hours

২০.

A Pipe P can fill a tank in 16 minutes and the other pipe Q can empty the whole tank in 32 minutes. If both P and Q are opened simultaneously then the time taken to fill the tank is -

ক
ক) 16 minutes
খ
খ) 32 minutes
গ
গ) 48 minutes
ঘ
ঘ) 40 minutes

ব্যাখ্যা

Let X hours be the time taken to fill a tank by P.
Let Y hours be the time taken to empty the tank by Q.
Then the time taken to fill the tank when P and Q are switched together : XY/(Y - X) hours.
Here, X = 16 minutes And Y = 32 minutes
Therefore,
Required time = (16 × 32)/(32 - 16)
= (32 × 16)/16
= 32 minutes.

২১.

Two pipes A and B can fill a tank in 20 and 30 minutes respectively. If both the pipes are open in how many hours will the tank?

ক
ক) 12 minutes
খ
খ) 30 minutes
গ
গ) 15 minutes
ঘ
ঘ) 60 minutes

ব্যাখ্যা

Part filled by (A and B) pipes in 1 min = (1/20) + (1/30)
= (3 + 2)/60
= 5/60
= 1/12

Time taken to fill the tank = (1 × 12)/1 minutes
= 12 minutes

২২.

Two pipes A and B would fill the tank in 20 and 30 minutes respectively. Both pipes being open, find when A must be turned off so that the tank may be just filled in 15 minutes.

ক
ক) After 10 mins
খ
খ) After 15 mins
গ
গ) After 20 mins
ঘ
ঘ) After 13 mins

ব্যাখ্যা

Question: Two pipes A and B would fill the tank in 20 and 30 minutes respectively. Both pipes being open, find when A must be turned off so that the tank may be just filled in 15 minutes.

Solution:
Let, after x minutes pipe A must be turned off.
Part fill by (A + B ) in 1 minutes = (1/20+ 1/30) = 1/12
Part fill by (A + B ) in x minutes = x/12

Then, pipe B does the job = (15 - x) minutes
In (15 - x) minutes Pipe B can fill the tank (15 - x) part

ATQ,
x/12 + (15 - x)/30 = 1
⇒ (5x + 30 - 2x)/60 = 1
⇒ 3x + 30 = 60
⇒ 3x = 30
⇒ x = 10

২৩.

An outgoing pipe pours water at half the amount of an ingoing pipe. After 6 hours of running both pipes, a tank was filled. If the outgoing pipe was closed, how much time would it take to fill the tank with the ingoing pipe?

ক
2 hours
খ
1.5 hours
গ
3 hours
ঘ
4 hours

ব্যাখ্যা

Question: An outgoing pipe pours water at half the amount of an ingoing pipe. After 6 hours of running both pipes, a tank was filled. If the outgoing pipe was closed, how much time would it take to fill the tank with the ingoing pipe?

Solution:
Let,
ingoing pipe needs X hours,
The outgoing pipe needs 2X hours.

together in one hour, these pipes can fill = (1/X) - (1/2X) = 1/2X

ATQ,
2X = 6
X = 3
∴ Ingoing pipe will take 3 hours to fill the tank.

২৪.

A pipe can fill 1/4th of a tank in 30 minutes. How much time will it take to fill two tanks?

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

ব্যাখ্যা

Question: A pipe can fill 1/4th of a tank in 30 minutes. How much time will it take to fill two tanks?

Solution:
A pipe can fill 1/4th of a tank in 30 minutes
∴ A pipe can fill 1 part or full of a tank in = (4 × 30)
= 120 minutes = 2 hours

One tank is filled in 2 hours
∴ two tanks is filled in = 4 hours

২৫.

An outgoing pipe is attached to a tank that can empty it in 10 hours. An ingoing pipe is connected to the tank that can fill it in 4 hours. How much time will it take to fill the half-full tank?

ক
10/3 hours
খ
10/6 hours
গ
20/3 hours
ঘ
3 hours

ব্যাখ্যা

Question: An outgoing pipe is attached to a tank that can empty it in 10 hours. An ingoing pipe is connected to the tank that can fill it in 4 hours. How much time will it take to fill the half-full tank?

Solution:
ingoing pipe in one hour can fill = 1/4
outgoing pipe in one hour can empty = 1/10

in one hour total fill up = 1/4 - 1/10 = 3/20

to fill half the tank it will take = 20/6 = 10/3 hours.

২৬.

3/4 part of the tank is full of water. When 30 litres of water is taken out, the tank becomes empty. The capacity of the tank is -

ক
36 litres
খ
32 litres
গ
44 litres
ঘ
40 liters

ব্যাখ্যা

Question: 3/4 part of the tank is full of water. When 30 litres of water is taken out, the tank becomes empty. The capacity of the tank is -

Solution:
Let us consider,
The tank has 4x litres of total capacity and holds 3x litres of water.
And if 30 litres of water is taken out, then the tank becomes empty.

It means 3x litres of water is taken out.
∴ 3x = 30 litres
⇒ x = 10 litres

Capacity of tank
= 4x
= 4 × 10
= 40 liters.

২৭.

Time required by two pipes A and B working separately to fill a tank is 36 seconds and 45 seconds respectively. Another pipe C can empty the tank in 30 seconds. Initially, A and B are opened and after 7 seconds, C is also opened. In how much more time the tank would be completely filled?

ক
47 seconds
খ
43 seconds
গ
39 seconds
ঘ
35 seconds

ব্যাখ্যা

Question: Time required by two pipes A and B working separately to fill a tank is 36 seconds and 45 seconds respectively. Another pipe C can empty the tank in 30 seconds. Initially, A and B are opened and after 7 seconds, C is also opened. In how much more time the tank would be completely filled?

Solution:
Let the capacity of the tank be LCM (36, 45, 30) = 180 units
∴ Efficiency of pipe A = 180/36 = 5 units/second
Efficiency of pipe B = 180/45 = 4 units/second
Efficiency of pipe C = - 180 / 30 = - 6 units/second

Now,
for the first 7 seconds, A and B were open.
Combined efficiency of A and B = 5 + 4 = 9 units/second
∴ Part of the tank filled in 7 seconds = 7 × 9 = 63 units

Part of tank empty = 180 - 63 = 117 units

Now, all pipes are opened.
Combined efficiency of all pipes = 5 + 4 - 6 = 3 units/second
Therefore, more time required = 117/3 = 39 seconds.

২৮.

A tank is 30% full with water. If 18 liters of water is added the tank becomes 3/4 full. What is the capacity of the tank?

ক
ক) 20 liters
খ
খ) 35 liters
গ
গ) 40 liters
ঘ
ঘ) 45 liters

ব্যাখ্যা

Question: A tank is 30% full with water. If 18 liters of water is added the tank becomes 3/4 full. What is the capacity of the tank?

Solution:
Let, Capacity of the tank is x Liters.

ATQ,
30% of x + 18 = (3/4) × x
⇒ (30x/100) + 18 = 3x/4
⇒ (3x/10) + 18 = 3x/4
⇒ (3x/4) - (3x/10) = 18
⇒ (15x - 6x)/20 = 18
⇒ 9x = 18 × 20
⇒ 9x = 360
∴ x = 40

∴ The capacity of tank is 40 Liters.

২৯.

A tank was 20% full of oil. The oil was poured into an empty 100-liter bucket, filling it halfway. What is half of the tank’s total capacity (in liters)?

ক
50 liters
খ
150 liters
গ
80 liters
ঘ
None

ব্যাখ্যা

Question: A tank was 20% full of oil. The oil was poured into an empty 100-liter bucket, filling it halfway. What is half of the tank’s total capacity (in liters)?

Solution:
Let the total capacity of the tank = T liters.
The tank was 20% full
now, oil volume = 20% of T = 0.2 × T

The oil fills 50% of the bucket
50% of 100 liters = 50 liters

So, 0.2 × T = 50 → T = 50 / 0.2 = 250 liters

Half of the tank = 250 ÷ 2 = 125 liters.

৩০.

Pipes A and B can fill a tank in 5 and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then the tank will be filled in:

ক
60/17 hours
খ
67/17 hours
গ
53/17 hours
ঘ
45/17 hours
ঙ
None of the above

ব্যাখ্যা

Question: Pipes A and B can fill a tank in 5 and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then the tank will be filled in:

Solution:
Net part filled in 1 hour = (1/5) + (1/6) - (1/12)
= {(12 + 10) - 5}/60
= 17/60

Hence, the tank will be filled in 60/17 hours.

৩১.

Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A to fill the cistern separately?

ক
6 hours
খ
8 hours
গ
9 hours
ঘ
10 hours

ব্যাখ্যা

Question: Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A to fill the cistern separately?

Solution:
Let,
the cistern be filled by pipe A alone in x hours.
Then, pipe B will fill it in (x + 6) hours

ATQ,
{1/x + 1/(x + 6)} = 1/4
⇒ (x + 6 + x)/{x(x + 6)} = 1/4
⇒ (2x + 6)/(x² + 6x) = 1/4
⇒ (x² + 6x) = 8x + 24
⇒ x² + 6x - 8x - 24 = 0
⇒ x² - 2x - 24 = 0
⇒ x² - 6x + 4x - 24 = 0
⇒ x(x - 6) + 4(x - 6) = 0
⇒ (x - 6)(x + 4) = 0
⇒ x = 6 or - 4
∴ x = 6 [neglecting the negative value of x]

∴ the cistern be filled by pipe A alone in 6 hours

৩২.

A and B two taps fill in a tank in 5 hours and 10 hours respectively. If both tapes are opened together, the tank will be full in:

ক
ক) 3 hours
খ
খ) 5 hours
গ
গ) 10/3 hours
ঘ
ঘ) 20/3 hours

ব্যাখ্যা

Question: A and B two taps fill in a tank in 5 hours and 10 hours respectively. If both tapes are opened together, the tank will be full in:

Solution:
A's 1 hour's work = 1/5
B's 1 hour's work = 1/10

(A+B)'s 1 hour's work = (1/5) + (1/10)
= (2 + 1)/10
= 3/10

Now,
3/10 parts both the taps can fill the tank in 1 hour
∴ 1 part both the taps can fill the tank in 10 × (1/3) hours
= 10/3hours

∴ The tank will be full in 10/3 hours.

৩৩.

Two inline pipes are used to fill a tank in 6 hours. The first inline pipe has double efficiency than the second one. How much time will the second pipe take to fill the tank alone?

ক
ক) 12 hours
খ
খ) 18 hours
গ
গ) 9 hours
ঘ
ঘ) 24 hours

ব্যাখ্যা

Question: Two inline pipes are used to fill a tank in 6 hours. The first inline pipe has double efficiency than the second one. How much time will the second pipe take to fill the tank alone?

Solution:
ধরি,
দ্বিতীয় পাইপ ক সময়ে সম্পূর্ণ চৌবাচ্চা একা পূর্ণ করতে পারে।
যেহেতু
প্রথম পাইপের দক্ষতা দ্বিতীয় পাইপের চেয়ে বেশি সেহেতু প্রথম পাইপের সময় লাগবে ক/২।

১ ঘন্টায় ২য় পাইপ পূর্ণ করে ১/ক অংশ
১ম পাইপ পূর্ণ করে ২/ক অংশ

১ ঘন্টায় মোট পূর্ণ করে = (১/ক + ২/ক) অংশ
= ৩/ক অংশ

∴ সম্পূর্ণ অংশ পূর্ণ করতে মোট সময় লাগবে = ক/৩ সময়

প্রশ্নমতে,
ক/৩ = ৬
ক = ১৮

∴ ২য় পাইপের সম্পূর্ণ চৌবাচ্চা পূর্ণ করতে সময় লাগবে ১৮ ঘন্টা।

৩৪.

Two pipes can fill a tank in 18 and 27 minutes respectively and a waste pipe can empty 4 gallons per minute. All the three pipes working together can fill the tank in 12 minutes. The capacity of the tank is:

ক
352 gallons.
খ
385 gallons.
গ
432 gallons.
ঘ
472 gallons.

ব্যাখ্যা

Question: Two pipes can fill a tank in 18 and 27 minutes respectively and a waste pipe can empty 4 gallons per minute. All the three pipes working together can fill the tank in 12 minutes. The capacity of the tank is:

Solution:
Let
the capacity of the tank be x gallons.

Then,
First pipe fills = x/18 gallons per minute
Second pipe fills = x/27 gallons per minute
Waste pipe empties = 4 gallons per minute

All three working together fill the tank in 12 minutes,

ATQ,
x/18 + x/27 - 4 = x/12
⇒ x/18 + x/27 - x/12 = 4
⇒ (6x + 4x - 9x)/108 = 4
⇒ x/108 = 4
∴ x = 432

∴ Capacity of the tank = 432 gallons.

৩৫.

A pump can fill a tank with water in 4 hours. Because of a leak, it took 9/2 hours to fill the tank. The leak can drain all the water of the tank in:

ক
ক) 18 hours
খ
খ) 24 hours
গ
গ) 36 hours
ঘ
ঘ) 38 hours

ব্যাখ্যা

Part of the tank filled by the pump in 1 hour = 1/4
Part of the tank filled by the pump in 1 hour because of the leak = 2/9
∴ Part of the tank emptied by the leak in 1 hour = 1/4 - 2/9
= (9 - 8)/36
= 1/36
∴ Leak will empty the tank in 36 hours

৩৬.

Two pipes P and Q can fill a tank in 12 hours and 24 hours, respectively. If both pipes are opened together, how long will it take to fill the tank?

ক
6 hours
খ
16 hours
গ
8 hours
ঘ
12 hours

ব্যাখ্যা

Question: Two pipes P and Q can fill a tank in 12 hours and 24 hours, respectively. If both pipes are opened together, how long will it take to fill the tank?

Solution:
Part filled by P in 1 hour = 1/12
Part filled by Q in 1 hour = 1/24

Part filled by (P + Q) in 1 hour
= (1/12) + (1/24)
= (2 + 1)/24
= 3/24
= 1/8

∴ Time to fill the tank = 1/(1/8) = 8 hours

∴ Both pipes can fill the tank in 8 hours

৩৭.

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
25 hours
খ
30 hours
গ
35 hours
ঘ
40 hours

ব্যাখ্যা

Question: A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

Solution:
Suppose,
pipe A alone takes x hours to fill the tank.
Then, pipes B and C will take x/2 and x/4 hours respectively to fill the tank.

Now,
1/x + 2/x + 4/x = 1/5
⇒ 7/x = 1/5
∴ x = 35

∴ Pipe A alone takes 35 hours to fill the tank.

৩৮.

A pipe can fill a tank in 4 hours but an outlet B can empty the tank in 8 hours. If both the pipes are opened simultaneously when the tank is half full, then the tank will be filled in -

ক
ক) 4 hours
খ
খ) 5 hours
গ
গ) 6 hours
ঘ
ঘ) 8 hours

ব্যাখ্যা

Question: A pipe can fill a tank in 4 hours but an outlet B can empty the tank in 8 hours. If both the pipes are opened simultaneously when the tank is half full, then the tank will be filled in -

Solution:
in 1 hour, A fills = 1/4
but B reject = 1/8

so, in 1 hour the net fill-up is = 1/4 - 1/8 = 1/8

hence,
It will take 8 hours to fill the full tank if both the pipes are opened.

so, 8/2 or, 4 hours is required to fill the tank when it's half full.

৩৯.

One pipe can fill a tank four times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then the slower pipe alone will be able to fill the tank in -

ক
ক) 144 min
খ
খ) 126 min
গ
গ) 114 min
ঘ
ঘ) 180 min

ব্যাখ্যা

Suppose the slower pipe alone can fill the tank in x minutes.
Then the faster pipe can fill the tank in x/4 minutes.

Part filled by the slower pipe in 1 minute = 1/x
Part filled by the faster pipe in 1 minute = 4/x
Part filled by both the pipes in 1 minute = 1/x + 4/x

Given that both the pipes together can fill the tank in 36 minutes.
Part filled by both the pipes in 1 minute = 1/36

According to the question,
1/x + 4/x = 1/36
5/x = 1/36
x = 180

৪০.

A cistern is normally filled in 8 hours but takes two hours longer to fill because of a leak in its bottoms. If the cistern is full, the leak will empty it in?

ক
ক) 28 hrs
খ
খ) 20 hrs
গ
গ) 40 hrs
ঘ
ঘ) 36 hrs

ব্যাখ্যা

ধরি,
ট্যাংকটি x ঘণ্টায় খালি হবে
ছিদ্র থাকা অবস্থায় পানি ভর্তি হতে সময় লাগে =(8 + 2) ঘণ্টা
= 10 ঘণ্টা
প্রশ্নমতে,
1/ 8 - 1/x = 1/10
⇒ 1/x = 1/8 - 1/10
⇒ 1/x =(5 - 4) /40
⇒ 1/x = 1/40
∴ x = 40

৪১.

A water tap fills a tub in 'p' hours and a sink at the bottom empties it in 'q' hours. If p < q and both tap and sink are opened the tank is filled in 'r' hours, then the relation between p, q, r :

ক
r = (1/p) - (1/q)
খ
r = p - q
গ
r = p + q
ঘ
1/r = (1/p) + (1/q)
ঙ
None of the above

ব্যাখ্যা

Question: A water tap fills a tub in 'p' hours and a sink at the bottom empties it in 'q' hours. If p < q and both tap and sink are opened the tank is filled in 'r' hours, then the relation between p, q, r :

Solution:
Total unit of work = pq unit

The efficiency of the water tap that fills a tub = pq/p = q
The efficiency of the sink at the bottom that empties the tub = pq/q = p

Net efficiency = q - p [q > p]

Hence, Time required to fill the tub, r = pq/(q - p)
⇒ 1/r = (1/p) - (1/q)

৪২.

A pump can fill a tank with water in 2 hours. Because of a leak, it took 140 minutes to fill the tank. The leak can drain all the water of the tank in-

ক
ক) 4.5 hr
খ
খ) 9 hr
গ
গ) 12 hr
ঘ
ঘ) 14 hr

ব্যাখ্যা

Question: A pump can fill a tank with water in 2 hours. Because of a leak, it took 140 minutes to fill the tank. The leak can drain all the water of the tank in-

Solution:
একটি পাইপ চৌবাচ্চা পূর্ণ করতে পারে ২ ঘণ্টায় বা ১২০ মিনিটে
১ মিনিটে পূর্ণ করে ১/১২০ অংশ

একটি ছিদ্র থাকায় তা পূর্ণ করতে পারে ১৪০ মিনিটে
১ মিনিটে পূর্ণ হয় ১/১৪০ মিনিটে

ছিদ্র দিয়ে ১ মিনিটে খালি হয় = (১/১২০) - (১/১৪০)
= (৭ - ৬)/৮৪০
= ১/৮৪০ অংশ

সম্পূর্ণ অংশ খালি করতে সময় লাগে = ১/১/৮৪০ মিনিট
= ৮৪০ মিনিটে
= ৮৪০/৬০ ঘণ্টায়
= ১৪ ঘণ্টায়

৪৩.

A large tanker can be filled by two pipes A and B in 60 minutes and 40 minutes, respectively. How many minutes will it take to fill the empty tanker if only B is used in the first-half of the time and A and B are both used in the second-half of the time?

ক
ক) 27.5 minutes
খ
খ) 15 minutes
গ
গ) 30 minutes
ঘ
ঘ) 20 minutes

ব্যাখ্যা

Question: A large tanker can be filled by two pipes A and B in 60 minutes and 40 minutes, respectively. How many minutes will it take to fill the empty tanker if only B is used in the first-half of the time and A and B are both used in the second-half of the time?

Solution:
Part filled by (A + B) in 1 minute = 1/60 + 1/40
= (2 + 3)/120
= 1/24

(A + B) will take half of total time= (x/2)× (1/24)
The rest half will be filled by B only in half of total time = (x/2) × (1/40)

Now
x/48 + x/80 = 1
(5x + 3x)/240 = 1
8x/240 = 1
x/30 = 1
x = 30

৪৪.

One pipe can fill a tank four times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then the faster pipe alone will be able to fill the tank in:

ক
40 minutes
খ
45 minutes
গ
50 minutes
ঘ
55 minutes

ব্যাখ্যা

Question: One pipe can fill a tank four times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then the faster pipe alone will be able to fill the tank in:

Solution:
Let,
the slower pipe alone fill the tank in x minutes.
Then, Faster pipe alone will fill it in x/4 minutes.

ATQ,
(1/x) + (4/x) = 1/36
⇒ 5/x = 1/36
∴ x = 180

The slower pipe alone fill the tank in 180 minutes.
the faster pipe alon will be able to fill the tank in = (180 ÷ 4) minutes.
= 45 minutes

৪৫.

A cistern can be filled by two taps X and Y in 15 hours and 20 hours respectively. The full cistern can be emptied by a third tap Z in 10 hours. If all the taps are turned on at the same time, in how much time will the empty cistern be filled completely?

ক
50 hours
খ
60 hours
গ
70 hours
ঘ
None

ব্যাখ্যা

Question: A cistern can be filled by two taps X and Y in 15 hours and 20 hours respectively. The full cistern can be emptied by a third tap Z in 10 hours. If all the taps are turned on at the same time, in how much time will the empty cistern be filled completely?

Solution :

X’s 1 hour work = 1/15
Y’s 1 hour work = 1/20
Z’s 1 hour work = 1/10 (emptying → negative)

Net 1 hour work = 1/15 + 1/20 – 1/10

Find LCM of denominators (15, 20, 10) → 60

Convert:
1/15 = 4/60, 1/20 = 3/60, 1/10 = 6/60

Net work = 4/60 + 3/60 – 6/60 = 1/60

Time taken = 1 ÷ (1/60) = 60 hours.

৪৬.

Pipe A can fill a tank in 22.5 minutes while a diametrically bigger Pipe B can do it in 15 minutes. Initially we open both the pipes together for some time but after how much time, should we close Pipe B so that the tank is full in 18 minutes?

ক
2.5 minutes
খ
3 minutes
গ
4 minutes
ঘ
4.5 minutes

ব্যাখ্যা

Given,
A fills the tank in 22.5 minutes and A remains open for 18 minutes.
So total tank filled by A = Tank filled in 1 min × 18 minutes = 1/(22.4) × 18 = 4/5
This is the entire work done by A.

So whatever is the remaining work, it is done by only B
Let B be open for T minutes.
Total tank filled by B = 1 - (4/5) = 1/5 = Tank filled in 1 min × T minutes = (1/15) × T
1/5 = (1/15) × T

∴ T = 3 minutes = B should be closed after this much time.

৪৭.

A supply pipe can fill a cistern in 6 hours, while a drainage pipe empties it in 9 hours. If both pipes are opened together, how long will it take to fill the cistern?

ক
12 hours
খ
18 hours
গ
20 hours
ঘ
22 hours

ব্যাখ্যা

Question: A supply pipe can fill a cistern in 6 hours, while a drainage pipe empties it in 9 hours. If both pipes are opened together, how long will it take to fill the cistern?

Solution:
The cistern fill in 1 hour = (1/6) - (1/9) part
= 1/18 part

The cistern fill 1/18 part
= 1 hour

The cistern fill full = (1 × 18) /1 hour
= 18 hours

৪৮.

A tank is filled in 8 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
56 hours
খ
28 hours
গ
36 hours
ঘ
45 hours
ঙ
42 hours

ব্যাখ্যা

Quesation: A tank is filled in 8 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

Solution:
Suppose,
pipe A alone takes x hours to fill the tank. Then, pipes B and C will take (x/2) and (x/4) hours respectively to fill the tank.

Now,
⇒ (1/x) + (2/x) + (4/x) = 1/8
⇒ 7/x = 1/8
⇒ x = 56
∴ Pipe A alone takes 56 hours to fill the tank.

৪৯.

A tank is filled in 8 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
ক) 36 hours
খ
খ) 56 hours
গ
গ) 46 hours
ঘ
ঘ) 52 hours

ব্যাখ্যা

Let
Pipe A can fill the tank in x hours. Then,
Pipe B can fill the tank in x/2 hours,
Pipe C can fill the tank in x/4 hours.

Part filled by pipe A in 1 hour = 1/x
Part filled by pipe B in 1 hour = 2/x
Part filled by pipe C in 1 hour = 4/x

Now
1/x + 2/x + 4/x = 1/8
(1 + 2 + 4)/x = 1/8
7/x = 1/8
x = 56 hours

৫০.

Two pipes A and B can fill a tank in 25 and 50 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?

ক
50/3 minutes
খ
47/3 minutes
গ
53/3 minutes
ঘ
44/3 minutes

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 25 and 50 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?

Solution:
pipe A fill a tank in 25 minutes
so, it fills in one minute (1/25) part

pipe B fill a tank in 50 minutes
so, it fills in one minute (1/50) part

both pipes fill in one minute = (1/25) + (1/50) part
= 3/50 part

So, it will take to fullfill the tank = 1/(3/50) minutes
= 50/3 minutes

৫১.

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
35 hours
খ
32 hours
গ
25 hours
ঘ
40 hours
ঙ
None of the above

ব্যাখ্যা

Question: A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

Solution:
Suppose,
Pipe A alone takes x hours to fill the tank.
So, pipes B and C will take x/2 and x/4 hours respectively to fill the tank.

Now,
1/x + 2/x + 4/x = 1/5
⇒ (1 + 2 + 4)/x = 1/5
⇒ 7/x = 1/5
∴ x = 35

∴ Pipe A alone takes 35 hours to fill the tank.

৫২.

A pump can fill a tank with water in 2 hours. Because of a leak, it took 7/3 hours to fill the tank. The leak can drain all the water from the tank in:

ক
16 hours
খ
14 hours
গ
10 hours
ঘ
12 hours

ব্যাখ্যা

Question: A pump can fill a tank with water in 2 hours. Because of a leak, it took 7/3 hours to fill the tank. The leak can drain all the water from the tank in:

Solution:
ধরি, ছিদ্র দিয়ে ট্যাঙ্ক খালি হয় x ঘণ্টায়
১ ঘণ্টায় খালি হয় ১/x ঘন্টায়

একটি পাম্প ২ ঘন্টায় পূর্ণ হয় ১ অংশ
১ ঘণ্টায় পূর্ণ হয় ১/২ অংশ

প্রশ্নমতে,
(১/২) - (১/x) = ৩/৭
⇒ ১/x = (১/২) - (৩/৭)
= ১/১৪
x = ১৪ ঘণ্টা

৫৩.

A tank is 1/4 parts full with water. If 15 liters of water is added, the tank becomes 2/3 parts full. What is the capacity of the tank?

ক
42 liters
খ
36 liters
গ
30 liters
ঘ
28 liters

ব্যাখ্যা

Question: A tank is 1/4 parts full with water. If 15 liters of water is added, the tank becomes 2/3 parts full. What is the capacity of the tank?

Solution:
Let the total capacity of the tank be x liters

ATQ,
(x/4) + 15 = 2x/3
⇒ (2x/3) - (x/4) = 15
⇒ (8x - 3x)/12 = 15
⇒ 5x = 12 × 15
⇒ x = 180/5
∴ x = 36

So the total capacity of the tank is 36 liters.

৫৪.

Two pipes can fill a tank in 8 and 12 minutes respectively and a waste pipe can empty 4 gallons per minute. All the three pipes working together can fill the tank in 24 minutes. The capacity of the tank is:

ক
24 gallons
খ
20 gallons
গ
18 gallons
ঘ
15 gallons

ব্যাখ্যা

Question: Two pipes can fill a tank in 8 and 12 minutes respectively and a waste pipe can empty 4 gallons per minute. All the three pipes working together can fill the tank in 24 minutes. The capacity of the tank is:

Solution:
Work done by the waste pipe in 1 minute
= (1/24) - [(1/8) + (1/12)]
= (1 - 3 - 2)/24
= - 1/6 [ - ve sign means emptying ]

∴ Volume of (1/6) part = 4 gallons
Volume of whole
= (6 × 4) gallons
= 24 gallons

৫৫.

Two pipes A and B can fill a tank in 18hrs and 6hrs respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?

ক
ক) 7/2 hours
খ
খ) 11/2 hours
গ
গ) 5/2 hours
ঘ
ঘ) 9/2 hours

ব্যাখ্যা

Part filled by A in 1 hour =1/18

Part filled by B in 1 hour =1/6

Part filled by (A + B) in 1 hour =(1/18) + (1/6)
= (1 + 3)/18
= 4/18
= 2/9

Hence, both the pipes together will fill the tank in 9/2 hours

৫৬.

One pipe can fill a tank four times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then the slower pipe alone will be able to fill the tank in:

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

ব্যাখ্যা

Question: One pipe can fill a tank four times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then the slower pipe alone will be able to fill the tank in:

Solution:
Let the faster pipe can fill it in X minutes
in one minute it can fill up = 1/X of the tank

so the slower pipe can do it in 4X minutes
in one minute it can fill up = 1/4X of the tank

so in one minute both can fill = (1/X) + (1/4X)
= 5/4X
the full tank will be filled in = 4X/5 minutes
ATQ,
4X/5 = 36
X = 45
so the slower pipe can do it in = 4 × 45 = 180 minutes
= 3 hours

৫৭.

Pipe A can fill a tank in 10 minutes, and pipe B can fill it in 20 minutes. If both are opened together into an empty tank, when should pipe A be turned off so that the tank gets filled in exactly 12 minutes?

ক
3 minutes
খ
4 minutes
গ
8 minutes
ঘ
10 minutes

ব্যাখ্যা

Question: Pipe A can fill a tank in 10 minutes, and pipe B can fill it in 20 minutes. If both are opened together into an empty tank, when should pipe A be turned off so that the tank gets filled in exactly 12 minutes?

Solution:
২য় নল দ্বারা,
20 মিনিটে পূর্ণ হয় = 1 অংশ
∴ 1 মিনিটে পূর্ণ হয় = 1/20 অংশ
∴ 12 মিনিটে পূর্ণ হয় = 12/20 অংশ
= 3/5 অংশ

∴ অবশিষ্ট থাকে = 1 - (3/5) অংশ
= 2/5 অংশ

১ম নল দ্বারা
1 বা সম্পূর্ণ অংশ পূর্ণ হয় = 10 মিনিটে
∴ 2/5 অংশ পূর্ণ হয় = (10 × 2)/5 মিনিটে
= 4 মিনিটে

∴ 4 মিনিট পর প্রথম নলটি বন্ধ করতে হবে।

৫৮.

A pipe can fill a cistern in 8 hours. Due to an accident, the water flow became half after pouring half of the cistern. How much time will it take to fill the whole cistern?

ক
10 hours.
খ
16 hours.
গ
12 hours
ঘ
14 hours.

ব্যাখ্যা

Question: A pipe can fill a cistern in 8 hours. Due to an accident, the water flow became half after pouring half of the cistern. How much time will it take to fill the whole cistern?

Solution:
pouring alf of the cistern will take 4 hours.
after that, the flow became half, which means it will take double the time to fill the rest half.
so the next half will be filled in 8 hours.

total time = 4 + 8 = 12 hours.

৫৯.

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
35 hours
খ
38 hours
গ
42 hours
ঘ
45 hours

ব্যাখ্যা

Suppose pipe A alone takes x hours to fill the tank.
Then, pipes B and C will take x/2 and x/4 hours respectively to fill the tank.
∴ 1/x + 2/x + 4/x = 1/5
⇒ 7/x = 1/5
⇒ x = 35 hours

৬০.

An outgoing pipe is attached to a tank that can empty it in 18 hours. An ingoing pipe is connected to the tank that can fill it in 9 hours. How much time will it take to fill the half-full tank?

ক
13 hours
খ
9 hours
গ
18 hours
ঘ
20 hours

ব্যাখ্যা

Question: An outgoing pipe is attached to a tank that can empty it in 18 hours. An ingoing pipe is connected to the tank that can fill it in 9 hours. How much time will it take to fill the half-full tank?

Solution:
ingoing pipe in one hour can fill = 1/9
outgoing pipe in one hour can empty = 1/18

in one hour, the total fill up = 1/9 - 1/18 = 1/18

∴ To fill the tank it will take = 18 hours.
∴ To fill half the tank it will take = 18/2
= 9 hours

৬১.

Two pipes, Pipe A and Pipe B, can fill a tank in 15 hours and 20 hours, respectively. If both pipes are opened together, after how many hours should Pipe B be closed so that the tank is completely filled in 12 hours?

ক
2 hours
খ
4 hours
গ
6 hours
ঘ
8 hours
ঙ
5 hours

ব্যাখ্যা

Question: Two pipes, Pipe A and Pipe B, can fill a tank in 15 hours and 20 hours, respectively. If both pipes are opened together, after how many hours should Pipe B be closed so that the tank is completely filled in 12 hours?

Solution:
ধরি, ট্যাঙ্কটির ধারণক্ষমতা হলো LCM (15, 20) = 60 ইউনিট
পাইপ A-এর কর্মদক্ষতা = 60/15 = 4 ইউনিট/ঘন্টা
পাইপ B-এর কর্মদক্ষতা = 60/20 = 3 ইউনিট/ঘন্টা
পাইপ A এবং B-এর মিলিত কর্মদক্ষতা = 4 + 3 = 7 ইউনিট/ঘন্টা

ধরি, পাইপ A এবং পাইপ B একত্রে চলে n ঘন্টা।
∴ পাইপ B বন্ধ করার পর পাইপ A একা (12 - n) ঘন্টা চলে।

প্রশ্নানুসারে,
7n + 4(12 - n) = 60
⇒ 7n + 48 - 4n = 60
⇒ 3n + 48 = 60
⇒ 3n = 60 - 48
⇒ 3n = 12
⇒ n = 12/3
⇒ n = 4

সুতরাং, পাইপ B-কে 4 ঘন্টা পর বন্ধ করা উচিত।

৬২.

Three pipes A, B and C can fill a tank in 6 hours. After working at it together for 2 hours, C is closed and A and B can fill the remaining part in 6 hours. The number of hours taken by C alone to fill the tank is:

ক
ক) 8 hours
খ
খ) 10 hours
গ
গ) 14 hours
ঘ
ঘ) 18 hours

ব্যাখ্যা

Question: Three pipes A, B and C can fill a tank in 6 hours. After working at it together for 2 hours, C is closed and A and B can fill the remaining part in 6 hours. The number of hours taken by C alone to fill the tank is

Solution:
A, B and C together can fill in one hour = 1/6
in two hours = 2/6 = 1/3

the remaining part is = 1 - 1/3 = 2/3

in 6 hours, A and B can do = 2/3
so, in one hour A and B can do = 2/(3 × 6) = 2/18

so, C can do in one hour = (A, B and C in one hour) - (A and B in one hour)
= 1/6 - 2/18
= 1/18

hence, C take 18 hours to fill the tank

৬৩.

A container is 2/3 full of oil. After removing 10 liters, it becomes 1/3 full. What is the capacity of the container?

ক
20 liters
খ
50 liters
গ
45 liters
ঘ
30 liters

ব্যাখ্যা

Question: A container is 2/3 full of oil. After removing 10 liters, it becomes 1/3 full. What is the capacity of the container?

Solution:
Given that,
The container is initially 2/3 full
After removing 10 liters, it becomes 1/3 full

Let the total capacity be x liters
Then,
According to the question,
⇒ (2x/3) - 10 = x/3
⇒ (2x/3) - (x/3) = 10
⇒ (2x - x)/3 = 10
∴ x = 30

So the capacity of the container is 30 liters.

৬৪.

A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
3 hrs 15 min
খ
3 hrs 45 min
গ
4 hrs 15 min
ঘ
4 hrs 1 min

ব্যাখ্যা

Question: A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

Solution:
Time taken by one tap to fill half of the tank = 3 hrs.
Part filled by the four taps in 1 hour = 4 × (1/6) = 2/3
Remaining part =(1 - 1/2) = 1/2

Four taps filled 2/3 part in 60 min
∴ Four taps filled full part in (60 × 3)/2 min
∴ Four taps filled 1/2 part in (60 × 3)/(2 × 2) min
= 45 min

So, total time taken = 3 hrs. 45 mins.

৬৫.

15 buckets of water fill a tank when the capacity of each bucket is 16 liters. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 12 liters?

ক
20
খ
18
গ
25
ঘ
12

ব্যাখ্যা

Question: 15 buckets of water fill a tank when the capacity of each bucket is 16 liters. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 12 liters?

Solution:
total capacity of the tank is = (15 × 16) = 240 liters.

total buckets of 12 liters = 240/12 = 20 buckets

৬৬.

A tap can fill a tank in 4 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
ক) 2 h 30 m
খ
খ) 3 h 20 m
গ
গ) 3 h 45 m
ঘ
ঘ) 4 h 30 m

ব্যাখ্যা

A tap can fill a tank in 4 hours.
Therefore the tap can fill half the tank in 2 hours.

Remaining = 1/2

After half the tank is filled, three more similar taps are opened.
Hence, the total number of taps becomes 4.

Part filled by one tap in 1 hour = 1/4
Part filled by four taps in 1 hour = 4 × (1/4) = 1
i.e., 4 taps can fill the remaining half in 30 minutes.

Total time taken
= 2 hour + 30 minute = 2 hour 30 minutes.

৬৭.

A tank is 30% full with water. If 18 liters of water is added the tank become 3/4 full. What is the capacity of the tank?

ক
ক) 45 Liters
খ
খ) 40 Liters
গ
গ) 42 Liters
ঘ
ঘ) 35 Liters

ব্যাখ্যা

Question: A tank is 30% full with water. If 18 liters of water is added the tank become 3/4 full. What is the capacity of the tank?

Solution:
Let, Capacity of tank is x Liters.

ATQ,
30% of x + 18 = (3/4) × x
⇒ (30x/100) + 18 = 3x/4
⇒ (3x/10) + 18 = 3x/4
⇒ (3x/4) - (3x/10) = 18
⇒ (15x - 6x)/20 = 18
⇒ 9x = 18 × 20
⇒ 9x = 360
∴ x = 40

∴ Capacity of tank is 40 Liters.

৬৮.

Some tanks of capacity 720 liters are meant to be filled by three pipes of capacity of loading water at 50 liters, 25 liters, and 15 liters in 10 minutes respectively. In one day they together can fill -

ক
18 tanks
খ
20 tanks
গ
9 tanks
ঘ
25 tanks

ব্যাখ্যা

Question: Some tanks of capacity 720 liters are meant to be filled by three pipes of capacity of loading water at 50 liters, 25 liters, and 15 liters in 10 minutes respectively. In one day they together can fill -

Solution:
in 10 minutes total fill-up = (50 + 25 + 15) = 90 liters

in 1 hour = 540 liters.
in 24 hours = (540 × 24) = 12960 liters.

total tank = 12960/720 = 18

৬৯.

A cistern has two pipes. Both working together can fill the cistern in 12 minutes. First pipe is 10 minutes faster than the second pipe. How much time would it take to fill the cistern if only second pipe is used?

ক
30 minutes
খ
20 minutes
গ
40 minutes
ঘ
60 minutes

ব্যাখ্যা

Question: A cistern has two pipes. Both working together can fill the cistern in 12 minutes. First pipe is 10 minutes faster than the second pipe. How much time would it take to fill the cistern if only second pipe is used?

Solution:
Let the time taken by first pipe working alone be ‘t’ minutes.
Time taken by second pipe working alone = t + 10 minutes.

Part of tank filled by pipe A in one hour working alone = 1/t
Part of tank filled by pipe B in one hour working alone = 1/(t + 10)

∴ Part of tank filled by pipe A and B in one hour working together = (1/t) + (1/t+10) = (2t + 10)/[t × (t + 10)]
But we are given that it takes 12 minutes to completely fill the cistern if both pipes are working together.
∴ (2t + 10)/[t × (t + 10)] = 1/12
⇒ t × (t + 10)/(2t + 10) = 12
⇒ t² + 10t = 24t + 120
⇒ t² - 14t - 120 = 0
⇒ (t - 20) (t + 6) = 0
∴ t = 20 minutes (Time cannot be negative)

Therefore, time taken by second pipe working alone = 20 + 10 = 30 minutes

৭০.

A cistern can be filled by a tap in 6 hours while it can be emptied by another tap in 9 hours. If both the taps are opened simultaneously, then after how much time the cistern will get filled?

ক
18 hours
খ
20 hours
গ
16 hours
ঘ
12 hours

ব্যাখ্যা

Question: A cistern can be filled by a tap in 6 hours while it can be emptied by another tap in 9 hours. If both the taps are opened simultaneously, then after how much time the cistern will get filled?

Solution:
The cistern fill in 1 hour = ( 1/6 ) - ( 1/9 ) part = 1/18 part
The cistern fill 1/18 part = 1 hour
The cistern fill full = ( 1 × 18 ) /1 hour = 18 hours

৭১.

A tank can be filled in 9 minutes by two pipes together. After keeping both pipes open for 6 minutes, the first pipe is closed. If it then takes another 7 minutes to completely fill the tank, how many minutes will the second pipe alone take to fill the tank?

ক
14 min
খ
18 min
গ
21 min
ঘ
28 min

ব্যাখ্যা

Question: A tank can be filled in 9 minutes by two pipes together. After keeping both pipes open for 6 minutes, the first pipe is closed. If it then takes another 7 minutes to completely fill the tank, how many minutes will the second pipe alone take to fill the tank?

Solution:
দুইটি পাইপ একত্রে 9 মিনিটে পূর্ণ করে 1 অংশ
∴ দুইটি পাইপ একত্রে 1 মিনিটে পূর্ণ করে (1/9) অংশ
∴ দুইটি পাইপ একত্রে 6 মিনিটে পূর্ণ করে (6/9) অংশ
= 2/3 অংশ

∴ অবশিষ্ট অংশ = {1 - (2/3)} অংশ
= (3 - 2)/3 অংশ
= 1/3 অংশ

২য় পাইপ দ্বারা 1/3 অংশ পূর্ণ হয় 7 মিনিটে
∴ ২য় পাইপ দ্বারা 1 অংশ পূর্ণ হয় (3 × 7) মিনিটে
= 21 মিনিটে

৭২.

Two pipes A and B can fill a tank in 10 and 15 hours respectively. If both the pipes are used together, then how long will it take to fill the tank

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

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 10 and 15 hours respectively. If both the pipes are used together, then how long will it take to fill the tank?

Solution:
Part filled by A in 1 hour = 1/10

Part filled by B in 1 hour = 1/15

Part filled by (A + B) in 1 hour
= (1/10) + (1/15)
= (3 + 2)/30
= 1/6

∴ Both pipes can fill the tank in 6 hours

৭৩.

A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?

ক
ক) 20 hours
খ
খ) 25 hours
গ
গ) 35 hours
ঘ
ঘ) None

ব্যাখ্যা

Suppose pipe A alone takes x hours to fill the tank.
Then, pipes B and C will take x/2 and x/4 hours respectively to fill the tank
∴ 1/x + 2/x + 4/x = 1/5
7/x = 1/5
x = 35 hours.

৭৪.

Two pipes A and B can fill a tank in 15 and 30 hours respectively. If both the pipes are used together, then how long will it take to fill the tank?

ক
8 hours
খ
9 hours
গ
10 hours
ঘ
12 hours

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 15 and 30 hours respectively. If both the pipes are used together, then how long will it take to fill the tank?

Solution:
Part filled by A in 1 hour = 1/15

Part filled by B in 1 hour = 1/30

Part filled by (A + B) in 1 hour
= (1/15) + (1/30)
= (2 + 1)/30
= 3/30
= 1/10

∴ Both pipes can fill the tank in 10 hours

৭৫.

Two pipes A and B can fill a tank in 20 and 30 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?

ক
15 minutes
খ
25 minutes
গ
50 minutes
ঘ
12 minutes
ঙ
9 minutes

ব্যাখ্যা

Part filled by A in 1 minute = 1/20
Part filled by B in 1 minute = 1/30

Part filled by (A + B) in 1 minute
= 1/20 + 1/30
= 1/12

∴ Both pipes can fill the tank in 12 minutes

৭৬.

12 buckets of water fill a tank when the capacity of each tank is 13.5 litres How many buckets will be needed to fill the same tank, if the capacity of each bucket is 9 liters.

ক
8
খ
18
গ
16
ঘ
4

ব্যাখ্যা

ট্যাংকের ধারণক্ষমতা = 12 × 13.5 = 162
দেয়া আছে, প্রতিটি বাকেটের ধারণক্ষমতা = ৯ লিটার
মোট বাকেট লাগবে = 162/9 = 18 টি
উত্তরঃ 18 টি

৭৭.

Two pipes A and B can fill a tank in 6 hours and 4 hours respectively. If they are opened on alternate hours and if pipe A is opened first, then the tank shall be filled in:

ক
ক) 4.8 hr
খ
খ) 5 hr
গ
গ) 5.5 hr
ঘ
ঘ) 7 hr

ব্যাখ্যা

Question: Two pipes A and B can fill a tank in 6 hours and 4 hours respectively. If they are opened on alternate hours and if pipe A is opened first, then the tank shall be filled in:

Solution:
নল A ৬ ঘণ্টায় করে ১ অংশ
১ ঘণ্টায় করে ১/৬ অংশ

নল B ৪ ঘণ্টায় করে ১ অংশ
১ ঘণ্টায় করে ১/৪ অংশ

প্রথম ঘণ্টায় নল A, ২য় ঘণ্টায় নল B এভাবে ক্রমান্বয়ে নলগুলো কাজ করে।

দুটি নল একসাথে ২ ঘণ্টায় করে (১/৬) + (১/৪) অংশ
= ৫/১২ অংশ

৪ ঘণ্টায় করে ১০/১২ অংশ

বাকি থাকে ১ - ১০/১২
= ১/৬ অংশ ; যা নল A ১ ঘন্টায় সম্পূর্ণ করে।

সময় লাগবে = ৪ + ১ ঘণ্টা
= ৫ ঘণ্টা

৭৮.

Two pipes A and B can fill the tank in 24 and 36 minutes, respectively. Both the pipes are opened together. After how many minutes should the pipe B be turned off, so that the tank be fill in 18 minutes?

ক
9 minutes
খ
12 minutes
গ
10 minutes
ঘ
16 minutes

ব্যাখ্যা

Question: Two pipes A and B can fill the tank in 24 and 36 minutes, respectively. Both the pipes are opened together. After how many minutes should the pipe B be turned off, so that the tank be fill in 18 minutes?

Solution:
Given that,
Pipe A fills the tank in 24 minutes.
Pipe B fills the tank in 36 minutes.
Total time to fill the tank = 18 minutes.
Now,
LCM of 24 and 36 = 72 (Total capacity of the tank).
Efficiency of pipe A = 72/24 = 3 units/minute.
Efficiency of pipe B = 72/36 = 2 units/minute.

Let,
pipe B be turned off after x minutes.
Pipe A works for 18 minutes.
Pipe B works for x minutes.
Work done by A in 18 minutes = 3 × 18 = 54 units.
Work done by B in x minutes = 2x = 2x units.

Total work done = 54 + 2x = 72
⇒ 2x = 72 - 54
⇒ 2x = 18
⇒ x = 18/2
∴ x = 9

∴ Pipe B should be turned off after 9 minutes.

৭৯.

A tap can fill a tank in 4 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

ক
ক) 1 hr 30
খ
খ) 2 hr 30 min
গ
গ) 2 hr
ঘ
ঘ) 3 hr

ব্যাখ্যা

A tap can fill a tank in 4 hours.
Therefore the tap can fill half the tank in 2 hours.

Remaining = 1/2

After half the tank is filled, three more similar taps are opened.
Hence, the total number of taps becomes 4.

Part filled by one tap in 1 hour = 1/4
Part filled by four taps in 1 hour = 4 × (1/4) = 1
i.e., 4 taps can fill the remaining half in 30 minutes.

Total time taken
= 2 hour + 30 minute = 2 hour 30 minutes.

৮০.

A tap can fill a tank in 10 minutes and another can empty it in 6 minutes. If the tank is already two-fifths full and both the tapes are opened together, how long will it take before the tank is either filled completely or emptied completely?

ক
12 min
খ
10 min
গ
8 min
ঘ
6 min

ব্যাখ্যা

Question: A tap can fill a tank in 10 minutes and another can empty it in 6 minutes. If the tank is already two-fifths full and both the tapes are opened together, how long will it take before the tank is either filled completely or emptied completely?

Solution:
Part to be emptied = 2/5 part

Net part emptied in 1 minute = 1/6 - 1/10 = (5 - 3)/30 = 2/30 = 1/15

1/15 part emptied in 1 minute
2/5 part emptied in 15 × (2/5) minute = 6 min

৮১.

Two pipes P and Q, when opened alone can fill the tank in 20 and 30 hours respectively. If both pipes are opened together, then in how many hours will the tank be filled?

ক
ক) 6 hours
খ
খ) 10 hours
গ
গ) 12 hours
ঘ
ঘ) 16 hours

ব্যাখ্যা

Question: Two pipes P and Q, when opened alone can fill the tank in 20 and 30 hours respectively. If both pipes are opened together, then in how many hours will the tank be filled?

Solution:
Part of tank filled by pipe P in 1 hour = 1/20
Part of tank filled by pipe Q in 1 hour = 1/30

Tank filled by both pipes in 1 hour = 1/20 + 1/30
= 5/60
= 1/12

∴ Complete tank will be filled by both in 1/1/12 hours
= 12 hours

৮২.

An outlet pipe can empty a cistern in 2 hours and 30 minutes. In what time will it empty 3/5 of the cistern?

ক
1.5 hours
খ
2 hours
গ
2.5 hours
ঘ
3 hours
ঙ
5 hours

ব্যাখ্যা

Question: An outlet pipe can empty a cistern in 2 hours and 30 minutes. In what time will it empty 3/5 of the cistern?

Solution:
The outlet pipe empties the one complete cistern in 2 hours and 30 minutes or 2.5 hours

∴ Time taken to empty 3/5 Part of the cistern = (3/5) × 2.5
= 1.5 hours

৮৩.

A pipe was used to fill a cistern in 20 hours but after working for 14 hours it stopped. Another pipe that has the capacity to fill the tank in 30 hours was replaced to fill the rest of the tank. How much time will it take to fill the rest of the tank by the second pipe?

ক
10 hours
খ
9 hours
গ
8 hours
ঘ
7 hours

ব্যাখ্যা

Question: A pipe was used to fill a cistern in 20 hours but after working for 14 hours it stopped. Another pipe that has the capacity to fill the tank in 30 hours was replaced to fill the rest of the tank. How much time will it take to fill the rest of the tank by the second pipe?

Solution:
In 14 hours, first pipe fill-up = 14/20 = 7/10 part
∴ remaining = (1 - 7/10) part
= 3/10 part

Given,
The second pipe fills 1/30 of the tank per hour.
∴ Time taken to fills 3/10 of the tank by the second pipe = 3/10 × 30
= 9 hours

৮৪.

An air conditioner can cool the hall in 40 miutes while another takes 60 minutes to cool under similar conditions. If both air conditioners are switched on at same instance then how long will it take to cool the room?

ক
ক) 20 min
খ
খ) 22 min
গ
গ) 24 min
ঘ
ঘ) 26 min

ব্যাখ্যা

Question: An air conditioner can cool the hall in 40 miutes while another takes 60 minutes to cool under similar conditions. If both air conditioners are switched on at same instance then how long will it take to cool the room?
Solution:
৪০ মিনিটে ঠান্ডা হয় সম্পূর্ণ অংশ
১ মিনিটে পূর্ণ হয় ১/৪০ অংশ

৬০ মিনিটে পূর্ণ হয় সম্পূর্ণ অংশ
১ মিনিটে পূর্ণ হয় ১/৬০ অংশ

দুটি মিলে পূর্ণ হয় ১/৪০ + ১/৬০
= ৩ + ২ / ১২০
= ৫/১২০ মিনিট
= ১/২৪ মিনিট

সম্পূর্ণ অংশ ঠান্ডা হতে সময় লাগে ২৪ মিনিট।

৮৫.

A tank has a leak which would empty the completely filled tank in 10 hours. If the tank is full of water and a tap is opened which admits 4 litres of water per minute in the tank , the leak takes 15 hours to empty the tank. How many litres of water does the tank hold?

ক
2400 litres
খ
4500 litres
গ
1200 litres
ঘ
7200 litres

ব্যাখ্যা

Let the total capacity of the tank is 30 units.

The efficiency of Leakage(Pipe A) will be = 30/10 = 3

And the efficiency of the leakage (Pipe A) and another Pipe (B) which is filling the tank will be = 30/15 = 2
Pipe A is emptying at 3 units/hr and when filling pipe B started then the emptying rate will come down to 2 units/hr.

∴ Filling Pipe B efficiency is 3 - 2 = 1unit/hr
Pipe B will be fill the tank in = 30/1 = 30 hrs.
Filling rate of Pipe B per minute is 4 litter

∴ Total Capacity of tank will be = (4 × 60) × 30 = 7200 litters.

৮৬.

2/5 of a tank is filled with water. After 36 litres are added, the tank becomes full. What is the total capacity of the tank?

ক
60 litres
খ
52 litres
গ
46 litres
ঘ
36 litres

ব্যাখ্যা

Question: 2/5 of a tank is filled with water. After 36 litres are added, the tank becomes full. What is the total capacity of the tank?

Solution:
Let,
the total capacity of the tank be x litres.

Already filled = (2/5) × x = (2x/5)

ATQ,
After adding 36 litres, the tank is full
∴ x - 2x/5 = 36
⇒ (5x - 2x)/5 = 36
⇒ 3x/5 = 36
⇒ x = (36 × 5)/3
∴ x = 60

∴ the total capacity of the tank be 60 litres.

৮৭.

A tank is 7 metre long and 4 meter wide wide. At what speed should water run through a pipe 5 cm broad and 4 cm deep so that in 6 hours and 18 minutes water level in the tank rises by 4.5 meter?

ক
8 km/hours
খ
10 km/hours
গ
12 km/hours
ঘ
14 km/hours
ঙ
None of these

ব্যাখ্যা

Question: A tank is 7 metre long and 4 meter wide wide. At what speed should water run through a pipe 5 cm broad and 4 cm deep so that in 6 hours and 18 minutes water level in the tank rises by 4.5 meter?

Solution:
Rate of flow of water = x cm/minute
∴ Volume of water that flowed in the in 1 minutes
= (5 × 4 × x)
= 20 x cu.cm.

∴ Volume of water that flowed in the tank in 6 hours 18 minutes.
= (6 × 60 + 18) = 378 minutes
= 20x × 378 cu. cm.

According to question,
20x × 378 = 700 × 400 × 4 50
⇒ x = (700 × 400 × 450)/(20 × 378)cm/minutes
⇒ x = (700 × 400 × 450 × 60)/(100000 × 20 × 378)km/hours
⇒ x = 10 km/hours

৮৮.

Three pipes A, B and C are connected to a tank. Out of the three, A and B are the inlet pipes and C is the outlet pipe. If opened separately, A fills the tank in 10 hours and B fills the tank in 30 hours. If all three are opened simultaneously, it takes 30 minutes extra than if only A and B are opened. How much time does it take to empty the tank if only C is opened?

ক
100 hours
খ
90 hours
গ
130 hours
ঘ
120 hours

ব্যাখ্যা

Question: Three pipes A, B and C are connected to a tank. Out of the three, A and B are the inlet pipes and C is the outlet pipe. If opened separately, A fills the tank in 10 hours and B fills the tank in 30 hours. If all three are opened simultaneously, it takes 30 minutes extra than if only A and B are opened. How much time does it take to empty the tank if only C is opened?

Solution:
Let the capacity of tank be LCM (10, 30) = 30 units
⇒ Efficiency of pipe A = 30/10 = 3 units/hour
⇒ Efficiency of pipe B = 30/30 = 1 units/hour
∴ Combined efficiency of pipes A and B = 4 units/hour

Therefore, time taken to completely fill the tank if only A and B are opened = 30/4 = 7 hours 30 minutes
∴ Time taken to completely fill the tank if all pipes are opened = 7 hours 30 minutes + 30 minutes = 8 hours
∴ Combined efficiency of all pipes = 30/8 = 3.75 units/hour
Now,
efficiency of pipe C = Combined efficiency of all three pipes - Combined efficiency of pipes A and B
Therefore, efficiency of pipe C = 4 - 3.75 = 0.25 units/hour

Thus, time taken to empty the tank if only C is opened = 30/0.25 = 120 hours.

৮৯.

Two pipes working together can fill a fish tank in 12 minutes. If one pipe fills the fish tank 10 minutes faster than the second pipe, at what time the second pipe alone can fill the fish tank?

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

ব্যাখ্যা

Let the first pipe fill the reservoir in X minutes

So, the second pipe will fill the reservoir in (X+10) minutes

As per question;
(1/X) + 1/(X + 10) = 1/12
⇒ (X + 10 + X)/X(X + 10) = 1/12
⇒ 12X + 120 + 12X = X² + 10X
⇒ X² + 10X - 24X -120 = 0
⇒ X² - 14X -120 =0
⇒ X² - 20X + 6X - 120=0
⇒ X(X - 20) + 6(X - 20) =0
⇒ (X + 6) (X - 20) = 0
⇒ X = 20

∴Second pipe will fill the reservoir in 20 + 10= 30 minutes

৯০.

3/4 part of the tank is full of water. When 30 litres of water is taken out, the tank becomes empty. The capacity of the tank is -

ক
44 litres
খ
32 litres
গ
36 litres
ঘ
38 litres
ঙ
None of the above

ব্যাখ্যা

Question: 3/4 part of the tank is full of water. When 30 litres of water is taken out, the tank becomes empty. The capacity of the tank is -

Solution:
Let us consider,
The tank has 4x litres of total capacity and holds 3x litres of water.
And if 30 litres of water is taken out, then the tank becomes empty.

It means 3x litres of water is taken out.
∴ 3x = 30 litres
⇒ x = 10 litres

Capacity of tank
= 4x
= 4 × 10
= 40 liters.

৯১.

An outgoing pipe is attached to a tank that can empty it in 20 hours. An ingoing pipe is connected to the tank that can fill it in 8 hours. How much time will it take to fill the half-full tank?

ক
10/3 hours
খ
20/7 hours
গ
20/3 hours
ঘ
40/7 hours

ব্যাখ্যা

Question: An outgoing pipe is attached to a tank that can empty it in 20 hours. An ingoing pipe is connected to the tank that can fill it in 8 hours. How much time will it take to fill the half-full tank?

Solution:
ingoing pipe in one hour can fill = 1/8
outgoing pipe in one hour can empty = 1/20

in one hour total fill up = 1/8 - 1/20 = 3/40

to fill half the tank it will take = 40/6 = 20/3 hours.

৯২.

A tap fills a cistern in 6 hours, while another empties it in 9 hours. How much time will it take to fill the cistern if both taps are opened at once?

ক
18 hours
খ
28 hours
গ
16 hours
ঘ
10 hours

ব্যাখ্যা

Question: A tap fills a cistern in 6 hours, while another empties it in 9 hours. How much time will it take to fill the cistern if both taps are opened at once?

Solution:
The cistern fill in 1 hour = (1/6) - (1/9) part
= 1/18 part

The cistern fill 1/18 part
= 1 hour

The cistern fill full = (1 × 18) /1 hour
= 18 hours

৯৩.

A leak in the bottom of a tank can empty the whole tank in 8 hours. An inlet pipe fills water are the rate of 6 liters a minute. When the tank is full, the inlet is opened, and due to the leak, the tank is empty in 12 hours. How many liters does the tank hold?

ক
6840 liters
খ
8290 liters
গ
8640 liters
ঘ
6890 liters

ব্যাখ্যা

Question: A leak in the bottom of a tank can empty the whole tank in 8 hours. An inlet pipe fills water are the rate of 6 liters a minute. When the tank is full, the inlet is opened, and due to the leak, the tank is empty in 12 hours. How many liters does the tank hold?

Solution:
Work done by the inlet pipe in 1 hour = (1/8 - 1/12) = (3 - 2)/24 = 1/24
Work done by the inlet pipe in 1 minute = (1/24 × 1/60) = 1/1440

Volume of 1/1440 part = 6 liters
Volume of the whole tank = (1440 × 6) = 8640 liters

৯৪.

3/4 part of the tank is full of water. When 30 litres of water is taken out, the tank becomes empty. The capacity of the tank is -

ক
ক) 36 litres
খ
খ) 42 litres
গ
গ) 40 litres
ঘ
ঘ) 32 litres

ব্যাখ্যা

If the tank has 4x liters of total capacity and holds 3x liters of water and if 30 liters of water is taken out, then the tank becomes empty.
It means 3x liters of water is taken out
3x = 30 liters
x = 10 liters
Capacity of tank
= 4x
= 4 × 10
= 40 liters.

৯৫.

Three pipes A, B, and C are connected to a tank. Out of the three, A and B are the inlet pipes and C is the outlet pipe. If opened separately, A fills the tank in 10 hours, B fills the tank in 12 hours and C empties the tank in 30 hours. If all three are opened simultaneously, how much time does it take to fill / empty the tank?

ক
6 hours 40 minutes to empty
খ
6 hours 40 minutes to fill
গ
6 hours 30 minutes to empty
ঘ
6 hours 30 minutes to fill
ঙ
6 hours 20 minutes to fill

ব্যাখ্যা

Question: Three pipes A, B, and C are connected to a tank. Out of the three, A and B are the inlet pipes and C is the outlet pipe. If opened separately, A fills the tank in 10 hours, B fills the tank in 12 hours and C empties the tank in 30 hours. If all three are opened simultaneously, how much time does it take to fill / empty the tank?

Solution:
Part of tank filled by pipe A in one hour working alone = 1/10
Part of tank filled by pipe B in one hour working alone = 1/12
Part of tank emptied by pipe C in one hour working alone = 1/30

Part of tank filled by pipes A, B and C in one hour working together = (1/10) + (1/12) - (1/30) = (6 + 5 - 2)/60 = 9/60 = 3/20
Therefore, time taken to completely fill the tank if A, B and C work together = 20/3 hours = 6 hours 40 minutes

৯৬.

ক
8650 liters
খ
6840 liters
গ
8460 liters
ঘ
8640 liters

ব্যাখ্যা

৯৭.

A swimming pool is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is?

ক
12 hours
খ
13 hours
গ
14 hours
ঘ
15 hours

ব্যাখ্যা

Question: A swimming pool is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is?

Solution:
Let,
the first pipe alone takes x hours to fill the swimming pool.
Then, the second and third pipes will take (x - 5) and (x - 9) hours respectively to fill the swimming pool.

ATQ,
1/x + 1/(x - 5) = 1/(x - 9)
⇒ (x - 5 + x)/x(x - 5) = 1/(x - 9)
⇒ (2x - 5)/(x² - 5x) = 1/(x - 9)
⇒ 2x² - 18x - 5x + 45 = x² - 5x
⇒ 2x² - 23x + 45 - x² + 5x = 0
⇒ x² - 18x + 45 = 0
⇒ x² - 15x - 3x + 45 = 0
⇒ x(x - 15) - 3(x - 15) = 0
⇒ (x - 15)(x - 3) = 0
∴ x = 15 [neglecting x = 3 ]

∴ the first pipe alone takes 15 hours to fill the swimming pool.

৯৮.

Two pipes A and B can fill a pool in 3 hours and 6 hours respectively. If both pipes work together, how long it will take to fill the pool?

ক
ক) 2 hours
খ
খ) 4 hours
গ
গ) 5 hours
ঘ
ঘ) 7 hours

ব্যাখ্যা

Question: Two pipes A and B can fill a pool in 3 hours and 6 hours respectively. If both pipes work together, how long it will take to fill the pool?

Solution:
A can fill in 1 hour = 1/3 part
B can fill in 1 hour = 1/6 part
Both pipe in 1 hour can fill = ( 1/3 + 1/6 ) = 3/6 = 1/2 part
Again, 1/2 part can be filled in 1 hour
∴ 1 part ( full ) can be filled in 2 hour

৯৯.

A pipe can fill up an empty tank in 12 minutes, Another pipe flows out 8 liters of water per minute. If the two pipes are opened together and the empty tank is filled up in 60 minutes. How much water does the tank contain?

ক
120 liters
খ
115 liters
গ
112 liters
ঘ
108 liters

ব্যাখ্যা

Question: A pipe can fill up an empty tank in 12 minutes, Another pipe flows out 8 liters of water per minute. If the two pipes are opened together and the empty tank is filled up in 60 minutes. How much water does the tank contain?

Solution:
Let the tank empty in x minute

ATQ,
(1/12) - (1/x) = 1/60
(1/12) - (1/60) = 1/x
4/60 = 1/x
x = 60/4
x = 15

So the tank emptied by the other pipe in 15 minute

∴ The tank contain = 15 × 8 liter
= 120 liter

১০০.

3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?

ক
8 hours
খ
10 hours
গ
12 hours
ঘ
14 hours

ব্যাখ্যা

Question: 3 pumps working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?

Solution:
3 pumps need 2 × 8 hours = 16 hours
1 pump needs 16 × 3 hours
4 pumps need (16 × 3)/4 hours
= 12 hours