#### Question

In an auditorium, seats were arranged in rows and columns. 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 in each row was reduced by 10, the total number of seats increased by 300. Find:

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

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

#### Solution

Let the number of rows in the original arrangement be x.

Then, the number of seats in each row in original arrangement = x

Total number of seats = x × x = x²

From the given information,

2x(x – 10) = x^{2} + 300

2x^{2} – 20x = x^{2} + 300

x^{2} – 20x – 300 = 0

(x – 30) (x + 10) = 0

x = 30, -10

Since, the number of rows or seats cannot be negative. So, x = 30.

1) The number of rows in the original arrangement = x = 30

2) The number of seats after re-arrangement = x^{2} + 300 = 900 + 300 = 1200