Question

A rectangular auditorium seats 1702 people. The number of seats in each row exceeds the number of rows by 9. Find the number of seats in each row.

Answer 1

We have been given that a rectangular auditorium seats 1702 people. Also, the number of seats in each row exceeds the number of rows by 9.

Let us assume that there are x rows in the auditorium. Therefore, there must be (x+9) seats in the each row. This will also be equal to the number of columns in the auditorium.

Therefore, we can set up an equation as:

$x(x+9)=1702\\x^{2}+9x=1702\\x^{2}+9x-1702=0$

We can now factor this quadratic as shown below:

$x^{2}+9x-1702=0\\(x-37)(x+46)=0$

Upon solving this equation, we get:

$x=-46,37$

Since number of rows cannot be negative. Therefore, there must be 37 rows in the auditorium. And hence, the number of seats in each row will be 37+9=46 seats.