Pipes and cisterns-Hard test 1

Answer: Option A

Explanation

Let us suppose the fester pipe fills the tank in x hours

and then the slower pipe fills the tank in x+10 hours

When both the pipes are opened together, the reserviour will be filled in 12 hours

 $$\begin{gathered} \frac{\mathrm{x}\left(\mathrm{x}+10\right)}{x+(x+10)}=12 \\ \frac{x^2+10x}{x+x+10}=12 \end{gathered}$$

Upon cross multiplication,

$$\begin{gathered} {\mathrm{x}}^2+10\mathrm{x}=12\left[\mathrm{x}+\mathrm{x}+10\right] \\ {\mathrm{x}}^2+10\mathrm{x}=12\mathrm{x}+12\mathrm{x}+120 \\ {\mathrm{x}}^2-14\mathrm{x}-120=0 \\ \therefore x=20 or -6 \end{gathered}$$

Because x cannot take negative value, the faster pipe will fill the reservoir in 20 hours.

Hence option A is correct.

Answer: Option D

Explanation

Let the faster pipe fill the tank in x hours.

then the slower pipe fills in (x+5) hours

According tot he question, both the pipes together can fill the tank in 6 hours.

$$\begin{gathered} \Rightarrow \frac{\mathrm{x}\left(\mathrm{x}+5\right)}{\mathrm{x}+\mathrm{x}+5}=6 \mathrm{hours} \\ \Rightarrow {\mathrm{x}}^2+5\mathrm{x}=12\mathrm{x}+30 \\ \Rightarrow {\mathrm{x}}^2-7\mathrm{x}-30=0 \end{gathered}$$

X can take value either 10 or -3

As the question is about the pipe that fills faster, we will neglect negative value

Therefore, the faster tap fills the tank in 10 hours.

Hence option D is correct.

Answer: Option C

Explanation

Leak in the tank can empty the full tank in 6 hours

Inlet can fill the tank at the rate of 4 liters per minute

Work done by inlet in 1 hour$=\frac{1}{6}-\frac{1}{8 }=\frac{4-3}{24}=\frac{1}{24}$ 

Work done by inlet in 1 min$=\frac{1}{24}\times \frac{1}{60 }=\frac{1}{1440}$ 

Volume of $\frac{1}{1440}$ parts of the tank is 4 liters

$\frac{1}{1440}=4$

Upon cross multiplication,

Total capacity of the tank is 1440 x 4=5760 liters

Hence option C is correct. 

Answer: Option D

Explanation

Let the capacity of the tank be x L

As per the question,

Quantity of water emptied by the leak in 1 h $=\frac{\mathrm{x}}{10}\mathrm{L}$

And to empty 4L in 1h$=\frac{\mathrm{x}}{15}$

Quantity of water filled by the tap in 1 min = 4L

∴1h=4L

As per the question,

$$\begin{gathered} \frac{\mathrm{x}}{10}-\frac{\mathrm{x}}{15}=240 \\ \Rightarrow \frac{3\mathrm{x}-2\mathrm{x}}{30}=240 \\ \Rightarrow \frac{\mathrm{x}}{30}=240 \\ \therefore \mathrm{x}=240\times 30=7200\mathrm{L} \end{gathered}$$

Hence option D is correct.

Answer: Option A

Explanation

Let us consider the third pipe to be a filling pipe which fills the tank in x hours

then, 

According to the If three pipes A, B and C can fill the tank separately in x hours, y hours and z hours respectively then, the time taken by all the three pipes together to fill the tank is  $\frac{\mathrm{xyz}}{\mathrm{xy}+\mathrm{yz}+\mathrm{zx}}$ hours.

So,

 $$\begin{gathered} \frac{10\times 15\times x}{(10\times 15)+(10\times x)+(15\times x)}=12 \\ \frac{150x}{150+10x+15x}=12 \end{gathered}$$

Upon cross multiplication,

$$\begin{gathered} 150x=12 \left(150+10x+15x\right) \\ 150x=\left(150\times 12\right)+\left(25x\times 12\right) \\ 150x=1800+300x \\ 1800=300x-150x \\ 1800=150x \\ x=\frac{1800}{150}=-12 \end{gathered}$$

Here negative sign implies that the pipe is a waste pipe and it emties the water in the tank in 12 hours.   

Hence option A is correct.