In an auditorium, seats were arranged in rows and columns. The number of rows was equal to number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by 10, the total 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.

(i) Let no. of rows be x and no. of seats in each row = x.

Total no. of seats = x²

If row is doubled then rows = 2x
No. of seats in each row are reduced by 10 = (x - 10)

Total no. of seats = 2x(x - 10)

According to question,

⇒ 2x(x - 10) - x² = 300

⇒ 2x² - 20x - x² = 300

⇒ x² - 20x - 300 = 0

⇒ x² - 30x + 10x - 300 = 0

⇒ x(x - 30) + 10(x - 30) = 0

⇒ (x - 30)(x + 10) = 0

⇒ x - 30 = 0 or x + 10 = 0

⇒ x = 30 or x = -10.

Since, no. of rows cannot be negative,

∴ x = 30

Hence, no. of rows in original arrangement = 30.

(ii) No. of seats after rearrangement = 2x(x - 10) = 2(30)(30 - 10) = 2 × 30 × 20 = 1200.

Hence, no. of seats after rearrangement = 1200.