Question

The students of a class are made to stand in (complete) rows. If one student is extra in a row, there would be 2 rows less, and if one student is less in a row, there would be 3 rows more. Find the number of students in the class.

Answer 1

Hint: We will take the total number of rows as x and the number of students in each row as y. So, the total number of students will be xy. Then by applying the conditions given in the question, find the values of xy which is the total number of students.

Complete step-by-step answer:
In the question we are asked to find the total number of students in the class with some conditions given. The students are made to stand in rows. We are given the question that, if a student is extra in a row, then there will be two rows less and if a student is less in a row, then there will be three more rows. Now, let us take the number of rows in which the students are lined up as x and the number of students in each row as y. So, we get the total number of students as $x\times y$ that is xy.
Now, from the first condition, that is, for each extra student in a row there will be two rows less, we get the number of rows as (x - 2) and the number of students in each row as (y + 1). Hence, the total number of students is, xy = (x - 2) (y + 1).
On simplifying we get, xy = xy + x -2y + (-2).
On simplifying further, we get the total number of students as, \[x=2y+2\ldots \ldots \ldots (i)\]
Now, from the second condition, that is, for each student less in a row, there would be three rows more, we get, the number of rows as (x + 3) and the number of students in each row as (y - 1). Hence the total number of students, xy = (x + 3) (y - 1).
On simplifying we get, xy = xy – x + 3y – 3.
On simplifying further, we get the total number of students as, $x=3y-3\ldots \ldots \ldots (ii)$
Now, on equating equation (i) and (ii), we get,
2y + 2 = 3y – 3
Taking y terms to one side and constants to the other side we get,
2 + 3 = 3y – 2y
y = 5
Now, on substituting the value of y in equation (ii), we get,
x = 3(5) – 3
x = 15 – 3
x = 12
Now, we know that the total number of students = xy = 12 $\times $ 5 = 60.
Hence, the total number of students = 60.

Note: Students must read the question twice before answering it specially the condition as they are a little tricky. Students must also do the calculations properly to avoid any mistakes.