Pipes & Cistern Questions and Answers

Answer: D

The tank is half filled in 1 hour, 40 minutes.

Pipe M fills \(\frac{1}{40}\) of the tank per minute and pipe N empties \(\frac{1}{50}\) per minute. Together:

\[ \begin{aligned} \frac{1}{40} - \frac{1}{50} &= \frac{5 - 4}{200} \\ &= \frac{1}{200} \end{aligned} \]

So the full tank would take 200 minutes, and half the tank takes 100 minutes, which is 1 hour 40 minutes.

3 hours 20 minutes (200 minutes) is the time to fill the whole tank, the most likely slip.

Answer: C

Pipe C alone fills the tank in 2 2/3 hours. In 3 hours A and B together fill \[3\left(\frac{1}{8} + \frac{1}{10}\right)\] \[= \frac{27}{40}\] of the tank, leaving 13/40. A and C together fill this in 39 minutes, which is 13/20 hour, so their combined rate is \[\frac{13/40}{13/20} = \frac{1}{2}\] tank per hour. Then C's rate is \[\frac{1}{2} - \frac{1}{8} = \frac{3}{8}\] and C alone takes 8/3 = 2 2/3 hours. The options 3, 3 1/2 and 4 1/2 give rates of 1/3, 2/7 and 2/9 for C, which do not fit.

Answer: B

The two inlet pipes fill 1/40 + 1/48 = 11/240 of the tank every hour, and the outlet empties 20 gallons per hour. All three together fill the tank in 40 hours, so in each hour they add 1/40 of the capacity C.

\[ \frac{C}{40} = \frac{11C}{240} - 20 \]

\[ 20 = \frac{5C}{240} \]

So C = 960 gallons, option (B). Check: 960 / 40 = 24, and 960 x 11 / 240 = 44, and 44 - 20 = 24. The values 840, 900 and 1000 gallons do not satisfy the equation.

Answer: C

Pipes A and C together can empty a full tank in 30 hours. In one hour, A fills 1/15 and B fills 1/18, which together make 11/90 of the tank. With C also open, 1/9 of the tank is filled in 5 hours, which is 1/45 per hour.

\[ \frac{11}{90} - c = \frac{1}{45} \]

\[ c = \frac{9}{90} = \frac{1}{10} \]

So C empties the tank at 1/10 per hour. A and C together then change the tank by 1/15 - 1/10 = -1/30 of it per hour, so they empty a full tank in 30 hours. The options 25, 28 and 35 hours do not follow from these rates.

Answer: D

Pipe C alone empties one-third of the tank in 6 hours. A and B together fill \(\frac{1}{20} + \frac{1}{30} = \frac{1}{12}\) of the tank per hour. With C also open, \(\frac{2}{9}\) of the tank fills in 8 hours, so the net rate is \[ \frac{2/9}{8} = \frac{1}{36} \] C's emptying rate is \[ \begin{aligned} \frac{1}{12} - \frac{1}{36} &= \frac{2}{36} \\ &= \frac{1}{18} \end{aligned} \] so C empties the whole tank in 18 hours and one-third of it in \(\frac{18}{3} = 6\) hours. The answer 18 hours would be for the whole tank; 3, 4 and 5 hours do not follow from the rate.

Answer: C

The tank holds 120 litres.

Let the capacity be \(C\) litres. P fills at \(C/10\) litres per minute and Q at \(C/6\), while R removes 34 litres per minute. With all three running, the full tank empties in 60 minutes, so the net outflow is \(C/60\) litres per minute:

\[ \begin{aligned} 34 - \frac{C}{10} - \frac{C}{6} &= \frac{C}{60} \\ 34 &= \frac{6C + 10C + C}{60} = \frac{17C}{60} \\ C &= 120 \end{aligned} \]

Check: P adds 12, Q adds 20 and R removes 34 litres per minute, a net loss of 2 litres per minute, which empties 120 litres in exactly 60 minutes. With 60 litres, P and Q would add only 16 litres per minute against R’s 34, emptying the tank in under 4 minutes rather than an hour; 26.8 and 127.5 do not satisfy the equation either.

Answer: C

The tank is filled in 24 hours.

In one hour the two inlets fill 1/10 and 1/15 of the tank and the outlet empties 1/8:

\[ \begin{aligned} \tfrac{1}{10} + \tfrac{1}{15} - \tfrac{1}{8} &= \tfrac{12 + 8 - 15}{120} \\ &= \tfrac{5}{120} = \tfrac{1}{24} \end{aligned} \]

So the net filling rate is 1/24 of the tank an hour, and the tank is full in 24 hours.

The two inlets alone would fill it in 6 hours; the outlet slows this sharply. 20, 23 and 25 hours do not match the net rate of 1/24.

Answer: D

Neither 1 nor 2 is correct.

Statement 1: together X and Y fill \(\tfrac{1}{30} + \tfrac{1}{45} = \tfrac{5}{90} = \tfrac{1}{18}\) of the tank an hour, so they need 18 hours, not 8.

Statement 2: with all three open the net rate is

\[ \begin{aligned} \tfrac{1}{15} + \tfrac{1}{20} - \tfrac{1}{25} &= \tfrac{20 + 15 - 12}{300} = \tfrac{23}{300} \\ 10 \text{ h} &\to \tfrac{230}{300} \text{ filled} \end{aligned} \]

The remaining \(\tfrac{70}{300}\) is filled by P and Q at \(\tfrac{35}{300}\) an hour in 2 hours, not 12. Both statements are false.

Answer: B

The tank cannot be filled.

Pipe X fills \(1/3\) of the tank per hour while pipe Y empties \(1/2\) per hour. Together:

\[ \frac{1}{3} - \frac{1}{2} = -\frac{1}{6}\ \text{per hour} \]

The net rate is negative: the emptying pipe is faster than the filling pipe, so starting from empty the tank never fills. 3 Hrs is X alone, and the other times come from adding the rates as if both pipes filled.

Answer: A

The filling capacity is 3000 m³/hr.

Let the filling rate be \(f\) m³/minute, so emptying is \(f + 10\). Emptying takes 8 minutes less:

\[ \begin{aligned} \frac{2400}{f} - \frac{2400}{f+10} &= 8 \\ 2400 \times 10 &= 8f(f + 10) \\ f^{2} + 10f - 3000 &= 0 \end{aligned} \]

\[ (f + 60)(f - 50) = 0 \Rightarrow f = 50 \]

The pump fills 50 m³ a minute, which is \(50 \times 60 = 3000\) m³ an hour. Check: filling takes 48 minutes and emptying (60 m³/min) takes 40, a difference of 8. The options are in m³ per hour, so the trap is to stop at 50.