A rectangular auditorium seats 1833 people. The number of seats in each row exceeds the number of rows by 8. Find the number of seats in each row.

Let x be the no. of seats in each row and y be the no. of seats in each row.

we have,

Total no. of people in the auditorium = 1833

=> xy = 1833

=> y = 1833/x - - - (1)

From the given condition, we get,

y = x+8

=> 1833/x = x+8 [from equation (1) ]

=> (1833/x) - x = 8

=> (1833-x^2/x) = 8

=> 1833-x^2 = 8x

=> - x^2 - 8x + 1833 = 0

=> x^2 + 8x - 1833 = 0

=> x^2 + 47x - 39x - 1833 = 0

=> x (x+47) - 39 (x+47) = 0

=> (x-39) (x+47) = 0

i. e, either x-39 = 0

=> x = 39

or x+47=0

=>x=-47 [but no. of seats cannot be negative]

therefore x = 39

Hence there are 39 seats in each row.