A rectangular classroom seats 252 students.

Question

A rectangular classroom seats 252 students. When the seats are rearranged with three more seats in each row, the classroom has two fewer rows. Find the original amount of seats in each row.

Answer 1

r is the number of rows

 

c is the number of seats in each row (or COLUMNS)

 

r x c = 252 ---> r = 252/c

 

(r-2) x (c + 3) = 252

 

(252/c - 2) x (c + 3) = 252

 

(252 - 2c)(c+3) = 252c <--- multiplies everything by c to clear the fractions

 

 

252c - 2c^2 + 756 - 6c = 252c

 

-2c^2 + 756 - 6c = 0 <--- subtracts 252c from both sides

 

2c^2 + 6c - 756 = 0 <--- multiplies both sides by -1 to change the signs so that the leading coefficient is positive

 

c^2 + 3c - 378 = 0

 

(  c -18    )( c +21      ) = 0

 

c-18 = 0 --> c = 18

 

c+21 = 0 ---> c=-21  <--- rejected; negative # of seats not allowed

 

There were 18 seats in each row, so 252/18 = 14 rows

 

If there are 21 seats in each row, then 252/21 = 12 rows.