Question

Students of a class are made to stand in rows. If one student is extra in a row, there would be 2 rows less. If one student is less in row, there would be 3 rows more. Find the total number of students in the class.
A) 55
B) 30
C) 115
D) 60

Answer 1

Hint: We will assume the number of rows to be x and the number of students in each row be y then we will make linear equation in two variables using the given information and then solve them to get the values of x and y and finally find the total number of students by multiplying the values of x and y.

Complete step by step answer:
Let the number of rows be x.
Let the number of students in each row be y.
Then the total number of students is equal to xy.
Now it is given that, when one student is extra in a row, there would be 2 rows less
Hence, the total number of students in this case is given by:-
\[\Rightarrow \left( {x - 2} \right)\left( {y + 1} \right) = xy\]
Simplifying it further we get:-
\[\Rightarrow xy + x - 2y - 2 = xy\]
Cancelling out the terms we get:-
\[\Rightarrow x - 2y = 2\]…………………………… (1)
Now it is given that, when one student is less in row, there would be 3 rows more.
Hence,
The total number of students in this case will be given by:-
\[\Rightarrow \left( {x + 3} \right)\left( {y - 1} \right) = xy\]
Simplifying it further we get:-
\[\Rightarrow xy - x + 3y - 3 = xy\]
Cancelling out the terms we get:-
\[\Rightarrow 3y - x = 3\]…………………………. (2)
Adding equations 1 and 2 we get:-
\[\Rightarrow 3y - 2y = 2 + 3\]
\[\Rightarrow y = 5\]
Putting this value in equation 1 we get:-
\[\Rightarrow x - 2\left( 5 \right) = 2\]
Solving for x we get:-
\[\Rightarrow x = 12\]
Now we know that the total number of students is given by:-
\[\Rightarrow {\text{total number of students}} = xy\]
Putting in the respective values we get:-
\[\Rightarrow {\text{total number of students}} = 12\left( 5 \right)\]
\[\Rightarrow {\text{total number of students}} = 60\]
Hence the total students are 60.

Hence option D is the correct option.

Note:
Students should note that the linear equation in two variables is the equation in which each of the variables has highest power as 1 and has two variables.
Also, students should take care that the total number of students is equal to the number of students in each row multiplied by the total number of rows.