41. An inlet pipe takes 8 hours to fill a tank. An outlet pipe takes 12 hours to empty it. If both pipes are opened simultaneously, in how many hours will the tank be filled?
एक टैंक को भरने में एक इनलेट पाइप को 8 घंटे का समय लगता है। एक आउटलेट पाइप को इसे खाली करने में 12 घंटे लगते हैं। यदि दोनों पाइप एकसाथ खोले जाते हैं, तो टैंक कितने घंटों में भरा जाएगा?
Option “C” is correct.
Given:
An inlet pipe takes 8 hours to fill a tank. An outlet pipe takes 12 hours to empty.
Calculation:
⇒ Time is taken to fill in 1 hour = 1/8
⇒ Time taken to empty in 1 hour = -1/12
⇒ If both pipes are opened simultaneously tank be filled in 1 hour = 1/8 – 1/12 = 1/24 hours
∴ The total time taken is 24 hours.
42. Tap A can fill a tank in 6 hours, tap B can fill the same tank in 8 hours and tap C can empty the same tank in 4 hours. If all three taps A, B and C are opened together, then how much time (In hours) will be taken to fill the tank?
नल A एक टंकी 6 घंटे में भर सकता है, नल B उसी टंकी को 8 घंटे में भर सकता है और नल C उसी टंकी को 4 घंटे में खाली कर सकता है। यदि सभी तीन नल A, B और C को एक साथ खोला जाता है, तो टंकी को भरने में कितना समय (घंटों में) लगेगा?
Option “C” is correct.
Given:
Tap A can fill a tank in = 6 hours
Tap B can fill the same tank in = 8 hours
Tap C can empty the same tank in = 4 hours
Formula:
Total work = Efficiency × Time
Calculation:
Total work = 24 units (LCM of 6, 8 and 4)
Efficiency of tap A = 4 units/day
Efficiency of tap B = 3 units/day
Efficiency of tap C = -6 units/day
∴ Time taken by A, B and C to fill the tank = 24/(4 + 3 – 6) = 24 hours
43. A pipe can fill a tank in 10 minutes while another pipe can empty it in 12 minutes. If the pipes are opened alternately each for 1 minute, beginning with the first pipe. the tank will be full after (in minutes):
एक पाइप 10 मिनट में एक टैंक भर सकता है जबकि दूसरा पाइप 12 मिनट में इसे खाली कर सकता है। यदि शुरुआत पहले पाइप से करके, पाइपों को 1 मिनट के लिए बारी-बारी से खोला जाए। टैंक कितनी देर में पूरा भर जाएगा (मिनट में)?
Option “B” is correct.
Given:
A pipe can fill a tank in 10 minutes
Another pipe can empty it in 12 minutes
They are opened alternately
Beginning with the first pipe
Calculation:
A → +10, B → -12 (lcm of 10,12 is 60) than
efficiency of A is +6 and B is -5
They opened alternative so A fill 6 litres in 1 minute and B empty 5 litres in 1 minute
They fill together 1 litre in 2 minutes
They fill 1 × 54 is 54 litre in 2 × 54 is 108 minutes
(∵ total = 60 litre)
Remaining 6 litre fill by the first pipe in 1 minute because his efficiency is 6litre/min
∴ the tank will full after 108 + 1 is 109 minutes
44. Two pipes A and B can fill a tank in 12 minutes and 15 minutes, respectively. When an outlet pipe C is also opened, then the three pipes together can fill the tank in 10 minutes. In how many minutes can C alone empty the full tank?
दो पाइप A और B क्रमशः 12 मिनट और 15 मिनट में एक टैंक भर सकते हैं। जब एक निकासी पाइप C भी खोल दिया जाता है, तो तीनों पाइप एक साथ टैंक को 10 मिनट में भर सकते हैं। C अकेले कितने मिनट में पूरा टैंक खाली कर सकता है?
Option “B” is correct.
GIVEN:
Time taken by pipe A to fill the tank = 12 minutes
Time taken by pipe B to fill the tank = 15 minutes
Total time taken = 10 minutes
FORMULAE USED:
Total Time taken = 1/(1/A + 1/B – 1/C)
CALCULATION:
10 = 1/(1/12 + 1/15 – 1/C)
(1/12 + 1/15 – 1/C) = 1/10
1/C = 3/20 – 1/10
1/C = 1/20
C = 20 minutes
ALTERNATE METHOD
L.C.M of 12, 15 and 10 = 60 = Total work
Efficiency of pipe A = 60/12 =5 units/minute
Efficiency of pipe B = 60/15 = 4 units/minute
Efficiency of pipe (A + B + C) = 60/10 = 6 units/minute
Efficiency of pipe C = 6 – (5 + 4) units/minute
Efficiency of pipe C = -3 units/minute
Here negative sign means tank is emptying
So, Time taken by pipe C to empty the tank = 60/3 minutes
⇒ 20 minutes
∴ Time taken by pipe C to empty the tank is 20 minutes
45. Two pipes X and Y can fill an empty tank in ‘t’ minutes. If pipe X alone takes 6 minutes more than ‘t’ to fill the tank and Y alone takes 54 minutes more than ‘t’to fill the tank, then X and Y together will fill the tank in:
दो पाइप X और Y एक खाली टैंक को ‘t’ मिनट में भर सकते हैं। यदि अकेले पाइप X टैंक को भरने के लिए ‘t’ से 6 मिनट अधिक लेता है और Y अकेले टैंक को भरने में ‘t’ से 54 मिनट अधिक लेता है, तो X और Y मिलकर टैंक को कितने समय में भरेंगे:
Option “A” is correct.
Given:
Two pipes X and Y can fill the tank in “t” min
X alone takes = (t + 6)min
Y alone takes = (t + 54)min
Calculation:
Let the total volume of the tank be xunits
X’s efficiency = [x/(t+6)] units/min
Time taken by both pipes X and Y to fill the tank = [{x/(t+6)} + {x/(t+54)}] × t = x
⇒ (2t + 60) t = (t + 6) (t + 54)
⇒ 2t2 + 60t = t2 + 60t + 324
⇒ t2 = 324
⇒ t = √324 = 18
∴ time taken by X and Y together to fill the tank = 18mins
46. Two pipes can fill a tank in 15 hours and 4 hours, respectively, while a third pipe can empty it in 12 hours. How long (in hours) will it take to fill the empty tank if all the three pipes are opened simultaneously?
दो पाइप क्रमशः 15 घंटे और 4 घंटे में एक टंकी भर सकते हैं, जबकि एक तीसरा पाइप 12 घंटे में इसे खाली कर सकता है। यदि तीनों पाइप एक साथ खोल दिए जाते हैं तो खाली टंकी को भरने में कितना समय (घंटों में) लगेगा?
Option “A” is correct.
Given:
Time to fill the tank by two pipes = 15 hours and 4 hours
Time to empty the tank by third pipe = 12 hours
Concept used:
If Pipe A take ‘x’ hours and Pipe B takes ‘y’ hours to fill a tank then assume the total capacity of the tank is equal to LCM of ‘x’ and ‘y’ or a multiple of them.
Formula used:
Efficiency = Total work/total time
Calculations:
Let the total capacity of the tank be LCM of 15, 4, and 12 i.e. 60 units.
The efficiency of the first pipe = 60/15
⇒ 4 units/hour
The efficiency of the second pipe = 60/4
⇒ 15 units/hour
The efficiency of the third pipe = 60/12
⇒ 5 units/hour
Combined efficiency of three pipes = 4 + 15 – 5
⇒ 14 units/hour
Time to empty the tank by all the pipes together = 60/14
⇒ 30/7 hours
∴ It takes 30/7 hours to fill the empty tank if all three pipes are opened simultaneously.
Shortcut Trick Total work = 60 unit
A
B
C
Efficiency
4
15
-5
Total efficiency = 4 + 15 – 5 = 14
∴ Time is taken to fill the tank = 60/14 = 30/7 hours
47.Pipes X and Y can fill a tank in 30 minutes and 45 minutes, respectively, whereas pipe Z can empty the full tank in 1 hour. Pipes X and Y are opened together for 10 minutes. Then, pipe X is closed, Z is opened instantly and Y continued to fill. The total time (from the beginning) taken to fill the tank is:
पाइप X और Y क्रमशः 30 मिनट और 45 मिनट में एक टैंक भर सकते हैं, जबकि पाइप Z, 1 घंटे में पूरा टैंक खाली कर सकता है। पाइप X और Y एकसाथ 10 मिनट के लिए खोले जाते हैं। फिर, पाइप X बंद हो जाता है, Z को तुरन्त खोला जाता है और Y टैंक को भरना जारी रखता है। टैंक को भरने के लिए लिया गया कुल समय (शुरुआत से) है:
Option “D” is correct.
Given:
Time taken by pipe X to fill the tank = 30 minutes
Time taken by pipe Y to fill the tank = 45 minutes
Time taken by pipe Z to empty the tank = 60 minutes
Working time of (X + Y) = 10 minutes
Concept:
Consider the total work to be 1 unit. Take the sign of Z as negative because it empties the tank.
Formula used:
1 unit work = 1/(Total time taken to complete the work)
Calculation:
(X + Y)’s 1 min work = (1/30) + (1/45)
= (3 + 2)/90
= 5/90
∴ Unit of work done by (X + Y) in 10 min = (10 × 5)/90
= 50/90
Unit of work left = 1 – (50/90)
= (90 – 50)/90
= 40/90
This work will be done by Y and Z together.
(Y – Z)’s 1 min work = (1/45) – (1/60)
= (4 – 3)/180
= 1/180
∴ Time taken by (Y and Z) to complete the whole work = 180 min
∴ Time taken by (Y and Z) to complete 40/90 unit of work = 180 × (40/90)
= 80 minutes.
∴ Total time taken to finish the whole work = (Time taken by X + Y) + (Time taken by Y and Z)
= 10 min + 80 min
= 90 min
48. Three pipes can fill a tank in 15 hours, 12 hours and 10 hours, respectively. If all the three pipes are opened simultaneously for 3 hours, then what percentage of the tank will remain unfilled?
तीन नल क्रमशः 15 घंटे, 12 घंटे और 10 घंटे में एक टंकी को भर सकते हैं। यदि सभी तीन नलों को एक साथ 3 घंटे के लिए खोला जाता है, तो टंकी का कितना प्रतिशत खाली रहेगा?
Option “D” is correct.
Given:
Three pipes can fill a tank in 15 hours, 12 hours and 10 hours, respectively. If all the three pipes are opened simultaneously for 3 hours
Concept Used:
Pipe and cistern
Calculation:
Three pipes can fill a tank in 15 hours, 12 hours and 10 hours, respectively
To find the total capacity take lcm of three
Lcm of 15,12 and 10 is 60
Total capacity = 60 unit which is 100%
Efficiency of Pipe A = 60/15 = 4
Efficiency of Pipe B = 60/12 = 5
Efficiency of Pipe C = 60/10 = 6
As per the question,
All three pipes were opened for 3 hours
⇒ (4 + 5 + 6) × 3
⇒ 45 unit work done
Remaining work
⇒ 60 – 45 = (15/60) × 100 = 25%
49. Two pipes A and B can fill a tank in 12 hours and 18 hours, respectively. Both pipes are opened simultaneously. In how much time will the empty tank be filled completely ?
दो नल A और B एक टैंक को क्रमशः 12 घंटे और 18 घंटे में भर सकते हैं। दोनों नल एक साथ खोले जाते हैं। खाली टैंक को पूरा भरने में कितना समय लगेगा?
Option “D” is correct.
Given:
Time is taken by A to fill the tank = 12 hrs
Time is taken by B to fill the tank = 18 hrs
Concept used:
Total time = Volume/Volume filled in 1 hour
Calculation:
Volume filled by A in 1 hr = 1/12
Volume filled by B in 1 hr = 1/18
Volume filled by both A and B in 1 hr = 1/12 + 1/18
⇒ Volume filled by both A + B in 1 hr = (3 + 2)/36
⇒ Volume filled by both (A + B) in 1 hr = 5/36
Total Time to fill the tank = (Total volume of the tank)/(Volume filled by both (A + B) in 1 hr)
⇒ Total Time to fill the tank = 1/(5/36)
⇒ Total Time to fill the tank = 36/5 hr
⇒ Total Time to fill the tank = 7 hours 12 minutes
∴ The total time to the filled the empty tank is 7 hours 12 minutes.
Alternate Method
Efficiency of A is 3
Efficiency of B is 2
Total efficiency of A & B is 3 + 2 = 5
So total time taken to fill the tank = 36/5 = 71/5 hour = 7 hour 12 minutes
50. A pipe can fill a cistern in 20 minutes whereas the cistern when full can be emptied by a leak in 28 minutes. When both are opened, The time taken to fill the cistern is:
एक पाइप 20 मिनट में एक टंकी को भर सकता है जबकि एक रिसाव द्वारा पूर्ण भरी टंकी को 28 मिनट में खाली किया जा सकता है। यदि दोनों को एक साथ खोला जाता है, तो टंकी को भरने में लगने वाला समय है:
Option “B” is correct.
Given:
A pipe can fill a cistern in 20 minutes.
Time taken by the leak to empty the tank = 28 minutes.
Concept used:
If a pipe can fill a tank in x hours then it can fill 1/x part of the tank in 1 hour.
Calculation:
Let the time taken by a pipe to fill the tank when the leakage arises be 1 minutes is 1/20 part.
And the time taken by a pipe to empty the tank be 1 minutes is 1/28 part.
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Alternate Method
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