Question

In an auditorium. seats were arranged in rows and columus. The number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats increased by 300. Find :

(i) the number of rows in the original arrangement.

(ii) the number of seats in the auditorium after re-arrangement.

Answer 1

Let the number of rows in original arrangement = x

also, the number of seats in original arrangement = x

therefore, total number of seats = $x\times x=x^2$

now , acc. to the question,

$2x(x–10)=x^2+300$

$2x^2–20\times x=x^2+300$

$x^2–20\times x–300=0$

$x^2–(30–10)x–300=0$

$x^2–30x+10x–300=0$

$x(x–30)+10(x–30)=0$

$(x–30)(x+10)=0$

Either x – 30 = 0 or x +10 = 0

x = 30 or x =-10

neglect the -ve value, we get, rows = 30

Thus, no. of rows in the original arrangement , x = 30

and

no. of seats after rearrangement = $x^2+300=(30{)}^2+300=900+300=1200$