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A water tank has inlets of two types $A$ and $B$. All inlets of type $A$ when open, bring in water at the same rate. All inlets of type $B$, when open, bring in water at the same rate. The empty tank is completely filled in $30$ minutes if $10$ inlets of type $A$ and $45$ inlets of type $B$ are open, and in $1$ hour if $8$ inlets of type $A$ and $18$ inlets of type $B$ are open. In how many minutes will the empty tank get completely filled if $7$ inlets of type $A$ and $27$ inlets of type $B$ are open?

If $M_{1}$ person can do $W_{1}$ work in $D_{1}$ days working $T_{1}$ hours in a day and $M_{2}$ person can do $W_{2}$ work in $D_{2}$ days working $T_{2}$ hours in a day then the relationship between them is :
$$\boxed {\frac{M_{1} \ast D_{1} \ast T_{1}}{W_{1}} = \frac{M_{2} \ast D_{2} \ast T_{2}}{W_{2}}}$$

Let $` W\text{’} \; \text{unit}$ be the capacity of the tank.

Now, $\frac{(10A + 45B) \ast 30}{W} = \frac{(8A + 18B) \ast 60}{W}$

$\Rightarrow 10A + 45B = 16A + 36B$

$\Rightarrow 6A = 9B$

$\Rightarrow 2A = 3B$

$\Rightarrow \frac{A}{B} = \frac{3}{2} = k \;(\text{let})$

$\Rightarrow \boxed{ A = 3k, \; B = 2k}$

And, $\frac{(7A +27B) \ast \text{Time}}{W} = \frac{(8A+18B) \ast 60}{W}$

$\Rightarrow [ 7(3k) + 27(2k)] \ast \text{Time} = [ 8(3k) + 18(2k) \ast 60]$

$\Rightarrow (21k + 54k) \ast \text{Time} = (24k + 36k) \ast 60$

$\Rightarrow 75k \ast \text{Time} = 60k \ast 60$

$\Rightarrow \boxed{ \text{Time} = 48 \; \text{minutes}}$

Correct Answer $:48$

$\textbf{PS:}$

• Work Done $=$ Time Taken $\ast$ Rate of work
• Total Work Done $=$ Total Time $\ast$ Efficiency
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