Pipes and Cisterns Concepts

1. Inlet & Outlet Staggered Timings

• Formula:

$$\text{Net rate}=\sum\text{Inlet rates}-\sum\text{Outlet rates}$$ $$\text{Work}=\text{Rate}\times\text{Time}$$

For different operating intervals, calculate the work done in each interval separately.

• Example: Pipe A fills a tank in 8 hours and Pipe B in 12 hours. Both are opened for 2 hours, after which B is closed. How long does the tank take to fill? Solution: Combined rate:

$$\frac18+\frac1{12}=\frac5{24}$$

Work done in 2 hours:

$$2\times\frac5{24}=\frac5{12}$$

Remaining:

$$1-\frac5{12}=\frac7{12}$$

A alone fills at (\frac18) tank/hour.

Time required:

$$\frac{7/12}{1/8}=\frac{14}{3}=4\frac23\text{ hours}$$

Total time:

$$2+4\frac23=\boxed{6\frac23\text{ hours}}$$

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2. Efficiency-Based Filling

• Formula: Efficiency is directly proportional to rate and inversely proportional to time:

$$\text{Rate}\propto\text{Efficiency},\qquad \text{Time}\propto\frac1{\text{Efficiency}}$$

If A is (k) times as efficient as B:

$$T_A=\frac{T_B}{k}$$

• Example: Pipe A is twice as efficient as Pipe B. Together they fill a tank in 4 hours. Find the time taken by A alone. Solution: Let B’s rate be (x).

A’s rate:

$$2x$$

Combined rate:

$$3x=\frac14$$

Therefore:

$$x=\frac1{12}$$

A’s rate:

$$2x=\frac16$$

Hence A alone takes:

$$\boxed{6\text{ hours}}$$

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3. Partial Tank Capacity Scenarios

• Formula:

$$\text{Remaining work}=1-\text{Initial fraction}$$ $$\text{Time}=\frac{\text{Remaining work}}{\text{Net rate}}$$

• Example: A tank is (\frac25) full. Pipe A fills it in 6 hours and Pipe B empties it in 10 hours. If both are opened, how long will the tank take to fill? Solution: Net rate:

$$\frac16-\frac1{10} =\frac{5-3}{30} =\frac1{15}$$

Remaining tank:

$$1-\frac25=\frac35$$

Time:

$$\frac{3/5}{1/15} =\frac35\times15 =\boxed9\text{ hours}$$

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4. Net Discharge & Emptying Failures

• Formula: If outlet rate exceeds inlet rate:

$$\text{Net outflow}=\text{Outlet rate}-\text{Inlet rate}$$ $$\text{Time to empty}=\frac{\text{Tank capacity}}{\text{Net outflow}}$$

• Example: A pipe fills a tank in 8 hours, while a leak empties the full tank in 6 hours. If both operate simultaneously, how long will the tank take to empty? Solution: Inlet rate:

$$\frac18$$

Outlet rate:

$$\frac16$$

Net outflow:

$$\frac16-\frac18 =\frac{4-3}{24} =\frac1{24}$$

Time to empty:

$$\frac1{1/24}=\boxed{24\text{ hours}}$$

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5. Two or More Inlets with One Outlet

• Formula:

$$R_{\text{net}}=\frac1{T_1}+\frac1{T_2}+\cdots-\frac1{T_o}$$ $$T=\frac1{R_{\text{net}}}$$

• Example: Pipes A and B fill a tank in 12 and 18 hours respectively. Pipe C empties it in 36 hours. If all are opened together, how long will the tank take to fill? Solution:

$$R=\frac1{12}+\frac1{18}-\frac1{36}$$

Taking LCM (36):

$$R=\frac3{36}+\frac2{36}-\frac1{36} =\frac4{36}=\frac19$$

Therefore:

$$T=\boxed9\text{ hours}$$

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6. Pipes Opened or Closed After a Fixed Time

• Formula: Calculate work done during each interval:

$$W=R\times T$$

Then:

$$W_{\text{remaining}}=1-W_{\text{completed}}$$

• Example: A fills a tank in 10 hours and B fills it in 15 hours. Both are opened for 3 hours, then A is closed. How much longer does B take to fill the tank? Solution: Combined rate:

$$\frac1{10}+\frac1{15}=\frac16$$

Work in 3 hours:

$$3\times\frac16=\frac12$$

Remaining:

$$1-\frac12=\frac12$$

B’s rate:

$$\frac1{15}$$

Time:

$$\frac{1/2}{1/15}=\boxed{7.5\text{ hours}}$$

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7. Leak Starts After the Tank Is Partially Filled

• Formula: Treat each time interval separately:

$$\text{Work done}=\text{Rate}\times\text{Time}$$

Then subtract the leak’s work after it starts.

• Example: A pipe fills a tank in 8 hours. After 2 hours, a leak that can empty the tank in 16 hours is opened. How much total time is required to fill the tank? Solution: Work done in first 2 hours:

$$2\times\frac18=\frac14$$

Remaining:

$$1-\frac14=\frac34$$

Net rate after leak:

$$\frac18-\frac1{16}=\frac1{16}$$

Time for remaining (\frac34):

$$\frac{3/4}{1/16}=12\text{ hours}$$

Total:

$$2+12=\boxed{14\text{ hours}}$$

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8. Alternate Opening of Pipes

• Formula: Calculate work done during one complete cycle, then determine the number of complete cycles and remaining work.
• Example: A fills a tank in 6 hours and B fills it in 12 hours. They are opened alternately for 1 hour each, starting with A. How long will the tank take to fill? Solution: In 2 hours:

$$\frac16+\frac1{12}=\frac14$$

After 3 complete cycles:

$$3\times\frac14=\frac34$$

Time used:

$$3\times2=6\text{ hours}$$

Remaining:

$$\frac14$$

Next is A:

$$\frac{1/4}{1/6}=\frac32\text{ hours}$$

Total:

$$6+1.5=\boxed{7.5\text{ hours}}$$

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9. Tank Already Partially Emptying While Filling

• Formula:

$$\text{Net rate}=\text{Inlet rate}-\text{Outlet rate}$$

For an initially (f)-full tank:

$$T=\frac{1-f}{R_{\text{net}}}$$

• Example: A tank is (\frac34) full. An inlet fills it in 10 hours and a leak empties it in 20 hours. Find the time to fill the remaining portion. Solution: Net rate:

$$\frac1{10}-\frac1{20}=\frac1{20}$$

Remaining:

$$1-\frac34=\frac14$$

Time:

$$\frac{1/4}{1/20}=\boxed5\text{ hours}$$

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10. Filling and Emptying Time Comparison

• Formula: If a pipe fills a tank in (x) hours and a leak empties it in (y) hours, then:

$$R_{\text{net}}=\frac1x-\frac1y$$

If (y>x), filling is possible; if (y<x), emptying occurs.

• Example: A pipe fills a tank in 5 hours and a leak empties it in 10 hours. How long does it take to fill the tank when both operate? Solution:

$$R=\frac15-\frac1{10} =\frac2{10}-\frac1{10} =\frac1{10}$$

Therefore:

$$T=\boxed{10\text{ hours}}$$

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Advanced Variants

11. Staggered Inlets and Outlets with Multiple Intervals

• Formula: For every interval:

$$W_i=R_iT_i$$

and total work is:

$$W_{\text{total}}=\sum W_i$$

Stop when (W_{\text{total}}=1).

• Example: A fills a tank in 10 hours, B in 15 hours, and C empties it in 30 hours. A and B operate for 2 hours; then C is also opened for 3 hours. How much of the tank is filled after these 5 hours? Solution: First 2 hours:

$$R=\frac1{10}+\frac1{15}=\frac16$$

Work:

$$2\times\frac16=\frac13$$

Next 3 hours:

$$R=\frac1{10}+\frac1{15}-\frac1{30} =\frac2{15}$$

Work:

$$3\times\frac2{15}=\frac25$$

Total filled:

$$\frac13+\frac25 =\frac5{15}+\frac6{15} =\boxed{\frac{11}{15}}$$

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12. Outlet Capacity Greater Than Combined Inlet Capacity

• Formula: If

$$\sum\text{outlet rates}>\sum\text{inlet rates},$$

the tank empties:

$$T=\frac1{\sum\text{outlet rates}-\sum\text{inlet rates}}$$

• Example: A and B fill a tank in 12 and 18 hours, while C empties it in 6 hours. If all operate together, how long will the full tank take to empty? Solution: Inlet rate:

$$\frac1{12}+\frac1{18}=\frac5{36}$$

Outlet rate:

$$\frac16=\frac6{36}$$

Net outflow:

$$\frac6{36}-\frac5{36}=\frac1{36}$$

Time:

$$\boxed{36\text{ hours}}$$

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13. Pipe Efficiency Ratio with a Third Pipe

• Formula: If efficiency ratio is (a:b:c), rates are in the same ratio:

$$R_A:R_B:R_C=a:b:c$$

Divide the total rate according to the ratio.

• Example: A, B and C have efficiencies in the ratio (2:3:5). Together they fill a tank in 5 hours. Find the time taken by C alone. Solution: Total efficiency:

$$2+3+5=10$$

C contributes:

$$\frac5{10}=\frac12$$

of the combined rate.

Combined rate:

$$\frac15$$

C’s rate:

$$\frac12\times\frac15=\frac1{10}$$

Therefore C alone takes:

$$\boxed{10\text{ hours}}$$

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14. Find Unknown Pipe Rate from Combined Time

• Formula: If A’s rate and combined rate are known:

$$R_B=R_{\text{combined}}-R_A$$

Then:

$$T_B=\frac1{R_B}$$

• Example: Pipe A fills a tank in 12 hours. A and B together fill it in 4 hours. Find B’s time alone. Solution: A’s rate:

$$\frac1{12}$$

Combined rate:

$$\frac14$$

B’s rate:

$$\frac14-\frac1{12} =\frac3{12}-\frac1{12} =\frac16$$

Therefore:

$$\boxed{6\text{ hours}}$$

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15. Time Saved by Using Two Pipes Together

• Formula: If A takes (x) hours and B takes (y) hours:

$$T_{\text{together}}=\frac{xy}{x+y}$$

Time saved compared with A alone:

$$x-T_{\text{together}}$$

• Example: A fills a tank in 12 hours and B in 18 hours. How much time is saved by using both instead of A alone? Solution:

$$T=\frac{12\times18}{12+18} =\frac{216}{30}=7.2\text{ hours}$$

Time saved:

$$12-7.2=\boxed{4.8\text{ hours}}$$

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16. Find the Time for a Tank to Become Empty from a Fraction

• Formula:

$$T=\frac{\text{Initial fraction of tank}}{\text{Net outflow rate}}$$

• Example: A tank is (\frac35) full. An outlet empties a full tank in 12 hours, while an inlet fills it in 20 hours. Find the time to empty the tank. Solution: Net outflow:

$$\frac1{12}-\frac1{20} =\frac{5-3}{60} =\frac1{30}$$

Initial quantity:

$$\frac35$$

Time:

$$\frac{3/5}{1/30} =\frac35\times30 =\boxed{18\text{ hours}}$$