If 3 students are made to sit in a row, then 9 students do not have a seat. Alternatively, if 9 students are made to sit in a row, then 5 rows are empty. How many students have to be seated in a row so that an equal number of students sit in all the rows and all the students are seated?
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Irrespective of the seating arrangement two things remain unchanged.
Equating the number of students in class when seated in arrangement 1 to that in arrangement 2 should get you the answer.
Let the number of students in the class be 'n'
Let the number of rows in the classroom be 'r'
Arrangement 1: If 3 students are made to sit in a row, then 9 students do not have a seat.
3 students are made to sit in a row.
So, 3r students will be seated.
9 students do not have seats.
The total number of students in the class will be sum of those who are seated and those who do not have seats.
So, n = 3r + 9
Arrangement 2: If 9 students are made to sit in a row, then 5 rows are empty.
Each row has 9 students. We know there are ‘r’ rows.
However, not all rows are occupied when each row accommodates 9 students. 5 rows are empty.
So, only (r – 5) rows are occupied.
So, number of students in the class = all students who have been seated.
i.e., n = 9(r – 5)
Arrangement 1: n = 3r + 9
Arrangement 2: n = 9(r - 5)
Equating the two we get 3r + 9 = 9(r – 5)
3r + 5 = 9r – 45
6r = 54
or r = 9
The number of rows in the classroom is 9.
Substitute r = 9 in the equation {n = 3r + 9} obtained in Step 1.
Number of students in the class = 3(9) + 9 = 27 + 9 = 36.
We need to find out the number of students to be seated per row if each row seated equal number of students.
36 students to be seated in 9 rows.
So, number of students per row = \36 \over 9 \\)
Number of students per row = 4.