Question

A hotel bill for a number of people for overnight stay is Rs. 4800. If there were 4 people more,
the bill each person had to pay, would have reduced by Rs. 200. Find the number of people
staying overnight.
(A) 5
(B) 8
(C) 6
(D) 7

Answer 1

Hint: Above question is based on quadratic equations. So first we will learn some basics about them.
Quadratic equations- These are the Polynomial equations of degree 2 in one variable.
The quadratic equation will always have two roots. The nature of roots may be real or imaginary.
Lets understand them with and example-:
x² + 5x + 6 = 0
There are two roots of this equation. i.e. -2 and -3
When we place \[x = - 2\] or \[x = - 3\]
In the above equation they will satisfy it.
So, roots of equation- Real number will be called a solution or a root if it satisfies the equation.
I.e. a will be the root of equation,
If \[f\left( a \right) = 0\]
The solution or roots of a quadratic equation are given by the quadratic formula:

Complete step by step solution: Let number of dipole staying overnight $ = $ x
For x people the bill will be $ = $ Rs. 48000/.
So, for 1 person bill will be $ = Rs\left( {\dfrac{{4800}}{x}} \right)$
According to the question :
When 4 persons will more, bill will be $ = Rs\left( {\dfrac{{4800}}{{x + 4}}} \right)$
After adding 4 persons, bill paid by each person will reached by 200/-
Therefore
$\left( {\dfrac{{4800}}{x}} \right) - \left( {\dfrac{{4800}}{{x + 4}}} \right) = 200$
$4800(x + 4) - 4800x = x(x + 4) \times 200$
$4800x + 19200 - 4800x = ({x^2} + 4x)200$
$19200 = ({x^2} + 4x) \times 200$
$96 = {x^2} + 4x$ or ${x^2} + 4x - 96 = 0$
Which is a quadratic equation
So, we will calculate the roots of this equation by quadratic equation.
We know that $x = \dfrac{{ - B \pm \sqrt {{B^2} - 4AC} }}{{2A}}$
Where A $ = $ coefficient of ${x^2} = 1$
B $ = $ coefficient of $x = 4$
C $ = $ coefficient of $ = - 96$
Therefore,
$x = \dfrac{{ - 4 \pm \sqrt {{{(4)}^2} - 4(1)( - 96)} }}{{2 \times 1}}$
$x = \dfrac{{ - 4 \pm \sqrt {16 + 384} }}{2}$
$x = \dfrac{{ - 4 \pm \sqrt {400} }}{2}$
$x = \dfrac{{ - 4 \pm 20}}{2}$
$x = \dfrac{{ - 4 + 20}}{2}$ or $x = \dfrac{{ - 4 - 20}}{2}$
$x = 8$ or $x = - 12$
Since x denotes number of people,
So it can't be negative.
Therefore we will neglect the negative digit.
Hence,
The number of people staying over night \[ = x = 8.\]
Therefore option B i.e. 8 is the correct option.

Note: 1. The roots of the quadratic equation:
\[x = \left( { - b \pm \surd D} \right)/2a\]
where,
 ; and called discriminant of equation.
We can determine the nature of roots by using this Discriminant.
(¡) \[D > 0,\] roots are real and distinct (unequal)
(¡¡)\[D = 0,\] roots are real and equal
(¡¡¡) \[D < 0,\] roots are imaginary and unequal