GATE CH 2022 | GA | Question: 3

Question

Pipes $\text{P}$ and $\text{Q}$ can fill a storage tank in full with water in $10$ and $6$ minutes, respectively. Pipe $\text{R}$ draws the water out from the storage tank at a rate of $34$ litres per minute. $\text{P, Q and R}$ operate at a constant rate.

If it takes one hour to completely empty a full storage tank with all the pipes operating simultaneously, what is the capacity of the storage tank (in litres)?

• A. $26.8$
• B. $60.0$
• C. $120.0$
• D. $127.5$

Answer 1

Let the capacity of the storage tank be $x \;\text{litres}.$

$\begin{array}{lccc} & \textbf{P} & \textbf{Q} &  \textbf{R} \\  \text{Time:} & 10\;\text{minutes} & 6\;\text{minutes} & \\ \text{Capacity of tank:} & x \;\text{litres}  &  & \\ \text{Efficiency (filling/draining) :} & \frac{x}{10} \;\text{litres/minute}  & \frac{x}{6} \;\text{litres/minute} & 34 \;\text{litres/minute} \end{array}$

If it takes one hour to completely empty a full storage tank with all the pipes operating simultaneously.

Now, $x +\frac{x}{10} \times 60 +  \frac{x}{6} \times 60 = 34 \times 60$

$\Rightarrow \frac{x}{60} + \frac{x}{10} +  \frac{x}{6}  = 34$

$\Rightarrow \frac{x+6x + 10x}{60} = 34$

$\Rightarrow 17x = 34 \times 60$

$\Rightarrow {\color{Blue}{\boxed{x = 120\;\text{litres}}}}$

$\therefore$ The capacity of the storage tank (in litres) is $120.$

Correct Answer $:\text{C}$

${\color{Magenta}{\textbf{PS:}}}\;{\color{Green}{\boxed{\text{Total work = Time} \; \times\; \text{Efficiency}}}}$

Comments

so the twist was tank was already full, dont know why someone will open the tap if it was already full

Answer 2

Let tank storage M litres .

P can add water to the tank = M/10 litres per min,

So in 60 min P can add water to the tank = 60*M/10 =6M litres.

Similarly Q can add water to the tank in 60 min = 60*M/6 = 10M litres .

Now tank was already full .

So, M+(6M+10M) = 34*60

> 17M = 2040

> M = 120

i.e: Tank storage was 120 Litres.

Answer 3

Let, $x = $ full capacity of the tank.

We know, $P$ takes $10$ minutes, $Q$ takes $6$ minutes and $R$ takes $\dfrac{x}{34}$ minutes to fill/empty the full tank.

Let, Total work $= LCM(10, 6, \dfrac{x}{34}) = LCM(\dfrac{10}{1}, \dfrac{6}{1}, \dfrac{x}{34}) = \dfrac{30x}{1} = 30x$ units.

$P$ does $\dfrac{30x}{10} = 3x$ units of work per minute.

$Q$ does $\dfrac{30x}{6} = 5x$ units of work per minute.

$R$ does $\dfrac{30x}{\dfrac{x}{34}} = 30 \times 34 = 1020$ units of work per minute.

So, according to given condition,

a) Initially tank was filled means $30x$ units of work already exists without $P, Q$ and $R$ doing work.
b) $P, Q$ and $R$ was doing work for $60$ minutes.
c) After $60$ minutes of work tank was empty means $0$ units of work left.
d) $P$ and $Q$ worked together to reduce the work while $R$ worked against them to increase the work.

$30x + 3x \times 60 + 5x \times 60 -1020 \times 60 = 0$

$30 (x + 3x \times 2 + 5x \times 2 -1020 \times 2) = 0$

$x + 6x + 10x -2040 = 0$

$17x = 2040$

$x = \dfrac{2040}{17}$

$\boxed{x = 120 \text{ litres}}$

Reference: https://www.youtube.com/watch?v=oApzHGJNx38&list=PLG4bwc5fquzj0Rkn0DVWZP9FxF9iG7OgB&index=39

Comments

How can you directly write lcm as 30 units?

 

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@Aadhyaa Gupttaa Formulas for LCM and HCF of fractions:

https://www.cuemath.com/numbers/properties-of-hcf-and-lcm/#:~:text=H.C.F.%20and%20L.C.M.%20of%20Fractions

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Thanks!

Answer 4

Let the tank capacity be x litres.

• P fills at x/10 litres/min
• Q fills at 6/10 litres/min
• R empties at 34 litres/min

Net outflow:  34 - ( x/10 + x/6) you will get  34 - 4x/15

Time to empty = 60 minutes 

x = ( 34 - 4x/15)  x 60 mins

x = 2040 - 240x / 15 

x = 2040 - 16x 

17x = 2040 

x = 120 litres , the correct option is c