NIFT PG 2020 Question Paper with solutions

Let the time to finish the work together $=\mathit{x}$ h
$\mathit{A}$ alone will finish the work in $\left(\mathit{x}+8\right)\mathit{h}$.
$\mathit{B}$ alone will finish the work in $\left(\mathit{x}+45\right)$ h.
$\therefore \frac{1}{\mathit{x}+8}+\frac{1}{\mathit{x}+45}=\frac{1}{\mathit{x}}$
$\Rightarrow \frac{\mathit{x}+8+\mathit{x}+45}{\left(\mathit{x}+8\right)\left(\mathit{x}+4.5\right)}=\frac{1}{\mathit{x}}$
$\Rightarrow \frac{2\mathit{x}+12.5}{{\mathit{x}}^2+4.5\mathit{x}+8\mathit{x}+36}=\frac{1}{\mathit{x}}$
$\Rightarrow 2{\mathit{x}}^2+12.5\mathit{x}={\mathit{x}}^2+12.5\mathit{x}+36$
$\Rightarrow {\mathit{x}}^2=36\Rightarrow \mathit{x}=6$
$\therefore$ Time required to finish the work together is $6\mathit{h}$.
Alternate Method
Here, $\mathit{a}=8\mathit{h},\mathit{b}=4\frac{1}{2}\mathit{h}$
$\therefore$ Time required to complete the work by both of the workers together
$=\sqrt{\mathit{a}\mathit{b}}=\sqrt{8\times \frac{9}{2}}=\sqrt{36}=6\mathit{h}$