Let the no of children is x and amount given to each child is y

As the total amount is 250 ₹

We have,

xy = 250

…[1]

Also, given if no of children is increased by 25, the amount to each get less by 50 paise i.e. 0.5 ₹

So, we have

(x + 25)(y – 0.5) = 250

[By 1]

⇒ 500x – x² + 12500 – 25x = 500x

⇒ x² + 25x – 12500 = 0

⇒ x² + 125x – 100x – 12500 = 0

⇒ x(x + 125) – 100(x + 125) = 0

⇒ (x – 100)(x + 125) = 0

so,

⇒ x – 100 = 0 or x + 125 = 0

⇒ x = 100 or – 125

However, no. of students can't be negative

Hence, x = 100

So, there were 100 students.