₹ 250 is divided equally among a certain number of children. If there were 25 children more, each would have received 50 paise less. Find the number of children.

Let no. of children be x,

Each student receives = ₹ $\dfrac{250}{x}$

If 25 children are increased then,

No. of children = x + 25

Now each student receives = $\dfrac{250}{x + 25}$

According to question,

$\Rightarrow \dfrac{250}{x} - \dfrac{250}{x + 25} = \dfrac{50}{100} \\[1em] \Rightarrow \dfrac{250(x + 25) - 250x}{x(x + 25)} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{250x + 6250 - 250x}{x^2 + 25x} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{6250}{x^2 + 25x} = \dfrac{1}{2} \\[1em] \Rightarrow 12500 = x^2 + 25x \\[1em] \Rightarrow x^2 + 25x - 12500 = 0 \\[1em] \Rightarrow x^2 + 125x - 100x - 12500 = 0 \\[1em] \Rightarrow x(x + 125) - 100(x + 125) = 0 \\[1em] \Rightarrow (x - 100)(x + 125) \\[1em] \Rightarrow x - 100 = 0 \text{ or } x + 125 = 0 \\[1em] \Rightarrow x = 100 \text{ or } x = -125.$

Since no. of children cannot be negative,

∴ x = 100

Hence, no. of children = 100.