The hotel bill for the number of people for overnight stay is $Rs.4,800,$ if there were $4$ more; the bill each person had to pay would have been reduced by $Rs.200.$ Find the number of people staying overnight.

The correct option is D: $8$ people

Let the number of people staying overnight be ${}^{\prime }x{.}^{\prime }$

The total bill for ${}^{\prime }{x}^{\prime }$ people is given as $Rs.4800.$

Bill for each people $=Rs.\frac{4800}{x}$

When $4$ more people were to the number of people already have, then new number of people staying overnight $=x+4$

Then, Bill for each people when $4$ were added $=\frac{4800}{x+4}$

It is given that the bill for each people will reduce by $Rs.200$ when 4 people were added.

$\therefore \frac{4800}{x}-\frac{4800}{x+4}=200$

$\Rightarrow 4800[\frac{1}{x}-\frac{1}{x+4}]=200$

$\Rightarrow 4800\left[\frac{x+4-x}{x(x+4)}\right]=200$

$\Rightarrow \frac{4800\times 4}{200}=x^2+4x$

$\Rightarrow 96=x^2+4x$

$\Rightarrow x^2+4x-96=0$

$\Rightarrow x^2+12x-8x-96=0$

$\Rightarrow x(x+12)-8(x+12)=0$

$\Rightarrow (x-8)(x+12)=0$

$\Rightarrow (x-8)=0or,(x+12)=0$

$\therefore x=8or,x=-12$

Number of people cannot be negative, so $x=8$

Hence, the number of people staying overnight is $8.$