The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less.

Solution: 

Let the no. of students in each row be x

let the no. of rows be y

therefore the total no. of students are  students in a row*no. of rows

therefore the no. of students are xy.

According to 1 condition

no. of students in a row if increased by 3 then the rows decrease by 1.

therefore the equation satisfying the condition is (X+3)(Y-1)=XY ---------------as the no.of students is xy let it be 1 eqn

similarly according to 2 condition

(x-3)(y+2)=xy  --------------------as the no. of students is xy let it be 2 eqn

therefore solving the 1 eqn we get

xy-1x+3y-3=xy

here xy onLHS and RHS gets cancelled

therefore the equation is -x+3y-3=0 ------------------let it be 3 eqn

similarly by solving 2 eqn we get

xy+2x-3y-6=xy

here also xy on LHS and RHS gets cancelled.

therefore the eqn is 2x-3y-6=0  --------------------let it be 4 eqn

multiplying 3 eqn by 2 we get

-2x+6y-6=0  --------------------let it be 5 eqn

adding eqn 4 and 5 we get

2x-3y-6=0

-2x+6y-6=0

3y-12=0  -----------------2x and -2x gets cancelled

therefore 3y=12  -------------taking 12 on RHS

y=12/3

y=3

substituting the value of y in 3 eqn

-x+3y-3=0

-x+3*4-3=0 ------------as y=4

-x+12-3=0

-x+9=0

therefore x=9

total no. of students is xy=9*4

=36

therefore the number of students are 36

Let the number of students in a row be ‘a’ and number of rows be ‘b’.

Given,

3 students are extra in a row, there would be 1 row less. If 3 students are less in a row there would be 2 rows more.

Number of students remain constant .

⇒ ab = (a + 3)(b – 1) ab = (a – 3)(b + 2)

So,

ab = ab - a + 3b - 3a - 3b = - 3 ...... (1)

and

ab = ab + 2a - 3b - 62a - 3b = 6 ...... (2)

Subtract 1 from 2 to get,

2a - 3b - (a - 3b) = 6 - (- 3)2a - 3b - a + 3b = 6 + 3a = 9

Put the value of a in 1 to get,

9 - 3b = - 3 - 3b = - 3 - 9 - 3b = - 12b = 4

Solving the above equations we get,

a = 9 and b = 4

Thus, number of students = ab = (9)(4) = 36