Question

A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left over.  When he increased the size of the square by one student he found he was short of 25 students.  Find the number of students

Answer 1

Ok now we know that area of square = side²
So no. of students in the line if side was x (let)
No. of students = x²+24 (as there were 24 exrltra students)
No. Of students after increasing 1 student in square = (x+1)²-25 (as there were 25 less students)
So x²+24=(x+1)²-25
x²+ 24 = x²+1 +2x -25
x²-x² + 24+25-1=2x
48=2x
So x=24
So no. Of students = x²+24=24²+24=576+24=600
So no. Of students = 600

Answer 2

Teacher arranged the student in a form of a square that means

No. Of students=Area of square formed by students.
(Can't show figure as my pc is not working now)..

So let the side of that square(formed by students)=X

area of square=X^2..
I.e.
NO. OF STUDENTS in the square..= X^2

the teacher found that 24 students are left so that means
X^2 is not total no. Of students..

So let total no. Of students=Y

now, Acc. to question,

X^2+24=y______eq.1

Also...
When he increased the size of square by 1 student. He was short of 25 student
that means....
Now that means


Side of square= (X+1)
So area= (X+1)^2

Now acc. To question,

(X+1)^2 - 25= Y______eq.2


Comparing eq.1 and eq.2 and we get

X^2+24= (X+1)^2 -25

X^2 +24= X^2+1+2X - 25

24= 2x- 24

2X=24+24

2X=48

X=24.

Putting X=24 in Eq.2, we get

(24+1)^2 - 25=Y

(25)^2 -25=Y
625 - 25=y

600=Y
Answer.