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

Let's denote the number of seats in each row as "x" and the number of rows as "y".
According to the given information, we can set up the following equations:

x * y = 1702 (equation 1)
x = y + 9 (equation 2)

To solve this system of equations, we can substitute equation 2 into equation 1:

(y + 9) * y = 1702

Expanding the equation:

y^2 + 9y = 1702

Rearranging the equation:

y^2 + 9y - 1702 = 0

Now, we can solve this quadratic equation by factoring or using the quadratic formula. However, since the coefficients are not easily factorable, let's use the quadratic formula:

y = (-b ± sqrt(b^2 - 4ac))/(2a)

Using the quadratic formula with a = 1, b = 9, and c = -1702:

y = (-9 ± sqrt(9^2 - 4*1*(-1702)))/(2*1)
y = (-9 ± sqrt(81 + 6808))/2
y = (-9 ± sqrt(6889))/2
y = (-9 ± 83)/2

For y to be a positive integer, we can discard the negative solution:

y = (-9 + 83)/2
y = 74/2
y = 37

Now that we have the value for y, we can substitute it back into equation 2 to find x:

x = y + 9
x = 37 + 9
x = 46

Therefore, the number of seats in each row is 46.