In an auditorium , the number of rows was equal to number of seats in each row. If the number of rows is doubled and the number of seats in each row is reduced by 5, then the total number of seats is increased by 375. How many rows were there ?

Let number of rows be = $x$ = number of seats in each row

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

Given, if the number of rows is doubled and the number of seats in each row is reduced by 5, then the total number of seats is increased by 375

$\therefore 2x \times (x - 5) - x^2 = 375$

$\Rightarrow 2x(x - 5) - x^2 = 375 \\[1em] \Rightarrow 2x^2 - 10x - x^2 = 375 \\[1em] \Rightarrow x^2 - 10x - 375 = 0 \\[1em] \Rightarrow x^2 - 25x + 15x - 375 = 0 \\[1em] \Rightarrow x(x - 25) + 15(x - 25) = 0 \\[1em] \Rightarrow (x - 25)(x + 15) = 0 \\[1em] \Rightarrow x - 25 = 0 \text{ or } x + 15 = 0 \\[1em] \Rightarrow x = 25 \text{ or } x = -15$

Since number of rows cannot be negative hence, x ≠ -15.

Hence the number rows were 25.