Pipes & Cisterns

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।

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।

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।

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.

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

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.

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.

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

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.

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.

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

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

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.

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.

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

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।

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

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।

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

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.

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

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

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

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.

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.

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

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.

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

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.

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.