Question

6500/- were divided equally among a certain number of persons. had there been 15 more persons, each would have got 30/- less. find the original number of persons?

Answer 1

let x is no of person and y is amount taje by each person.

6500/x=y ----------------(1)
when 15 more person appear then
6500/(x+15)=y-30 ------------(2)

solve both (1) and (2) equation
subtracting (1) to (2)
6500{15/(x^2+15x)}=30
3250=x^2+15x

x^2+15x-3250 =0
solve this by quadratic formula ,
x=(-15+_root (225+13000))/2
=50,-65
but negative not possible
hence 50 number of person.

Answer 2

Answer:

$\boxed{\sf \: Number\:of\:persons = 50 \: }$

Step-by-step explanation:

Let assume that the number of persons be x.

Case :- 1

Amount to be distributed = 6500/-

Number of persons = x

So, Each person share is

$\boxed{ \sf{ \:\sf \: S_1 = \dfrac{6500}{x} \: }} - - - (1)$

Case :- 2

Amount to be distributed = 6500/-

Number of persons = x + 15

So, Each person share is

$\boxed{ \sf{ \:\sf \: S_2 = \dfrac{6500}{x + 15} \: }} - - - (2)$

According to statement, it is given that had there been 15 persons more, each would get 30/- less.

$\bf\implies \:S_1 - S_2 = 30$

$\sf \: \dfrac{6500}{x} - \dfrac{6500}{x + 15} = 30$

$\sf \:6500\bigg( \dfrac{1}{x} - \dfrac{1}{x + 15}\bigg) = 30$

$\sf \:650\bigg(\dfrac{x + 15 - x}{x(x + 15)}\bigg) = 3$

$\sf \:\dfrac{650 \times 15}{x(x + 15)} = 3$

$\sf \: {x}^{2} + 15x = 3250$

$\sf \: {x}^{2} + 15x - 3250 = 0$

$\sf \: {x}^{2} - 50x + 65x - 3250 = 0$

$\sf \: x(x - 50) + 65(x - 50) = 0$

$\sf \: (x - 50) \: (x + 65) = 0$

$\bf\implies \:x = 50\: \: \: \: \: or \: \: \: \: \: x = - 65 \ \: \: \: \{rejected \}$

Hence,

$\implies\sf \: Number\:of\:persons = 50$

$\rule{190pt}{2pt}$

${{ \mathfrak{Additional\:Information}}}$

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

Three cases arises :

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac