Question

Students of a class are made to stand in rows. If $4$ students are extra in each row then there would be $2$ rows less. If four students are less in each row then there would be $4$ more rows. What is the number of students in the class?
A.$36$
B.$106$
C.$86$
D.$96$

Answer 1

Hint: Assume the number of students in each row to be x and the total rows to be y. Then form two linear equations using the statements (which tell the relation between the rows and the number of students) and equate then to the total number of students. Solve the equations to find the total number of students.

Complete step by step answer:

Given, students of a class are made to stand in rows. Then let the number of the students in each row be x and the number of rows is y.
So the total number of students=no. of students in each row× no. of rows=xy
Now given if four students are extra in each row then the rows will become less by two
So no. of students in each row will become$\left( {x + 4} \right)$ and the number of rows will become$\left( {y - 2} \right)$
Then according to question,
$ \Rightarrow \left( {x + 4} \right)\left( {y - 2} \right) = xy$
On solving we get,
$ \Rightarrow x\left( {y - 2} \right) + 4\left( {y - 2} \right) = xy$
On multiplication we get,
$ \Rightarrow xy - 2x + 4y - 8 = xy$
On cancelling xy from both sides we get,
$ \Rightarrow - 2x + 4y - 8 = 0$ --- (i)
Now given If four students are less in each row then there would be $4$ more rows
Then the number of students in each row becomes$\left( {x - 4} \right)$ and the number of rows becomes $\left( {y + 4} \right)$
Then we can write-
$ \Rightarrow \left( {x - 4} \right)\left( {y + 4} \right) = xy$
On solving we get,
$ \Rightarrow 4x - 4y - 16 + xy = xy$
On cancelling xy both side we get,
$ \Rightarrow 4x - 4y - 16 = 0$ --- (ii)
Now we have to find the total number of students.
On adding eq. (i) and (ii) we get,
$ \Rightarrow - 2x + 4y + 4x - 4y - 8 - 16 = 0$
On cancelling the common terms and subtracting the terms of common coefficient we get,
$ \Rightarrow 2x - 24 = 0$
$ \Rightarrow 2x = 24$
Then we get,
$ \Rightarrow x = \dfrac{{24}}{2} = 12$
On putting this value in eq. (i) we get,
$ \Rightarrow - 2\left( {12} \right) + 4y - 8 = 0$
On simplifying we get,
$ \Rightarrow $ $4y = 32$
Then we get,
$ \Rightarrow y$ = $\dfrac{{32}}{4}$ = 8
Now we know the value of x and y then we can find the total number of students in the class.
$ \Rightarrow xy = 12 \times 8 = 96$
Hence the number of students=$96$
Answer- the correct answer is D.

Note: Here, the students may get confused as to why are we equating the relation formed by the two statements to the total number of students. So remember the number of students in each row and the number of rows is changing but the number of students in the class is the same. So when we multiply the number of students in each row to the number of rows we will always get the total number of students, hence we equate their multiple to the total number of students in the class.