Question

In a school auditorium, the number of chairs in a row is the same as the number of rows of chairs. There are 7033 students in the school. If we know that less than 31 chairs were empty, how many chairs were empty after all the students sat down?

Answer 1

Let chairs in row be x
so number of rows also , x
total number of chairs = x * x
= x^2
there are 7033 students in school and each student gets chair . some chairs leaves empty
so , x^2 > 7033 , such that x^2 is not exceeds than 7033 + 31 = 7064
so finding the square number just below 7064
80^2 = 6400
90^2 = 8100
the number is between 80 and 90
now 85^2 = 7225
so expected number is below 85
83^2 = 688984^2 = 7056
so x^2 = 7056 = 84^2
which is total number of chairs in auditorium
so 7056 - 7033 = 23
23 chairs remains empty.