₹7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received ₹100 more. Find the original number of children.

Let the number of children be x

₹7500 is divided equally among x children so , each received = ₹$\dfrac{7500}{x}$

Given, if there were 20 less children, each would have received ₹ 100 more.

If there are 20 less children then each would receive = ₹$\dfrac{7500}{x - 20}$

$\therefore \dfrac{7500}{x - 20} - \dfrac{7500}{x} = 100 \\[1em] \Rightarrow \dfrac{75}{x - 20} - \dfrac{75}{x} = 1 \text{ (Dividing the equation by 100) } \\[1em] \Rightarrow \dfrac{75x - 75(x - 20)}{x(x - 20)} = 1 \\[1em] \Rightarrow \dfrac{75x - 75x + 1500}{x(x - 20)} = 1 \\[1em] \Rightarrow \dfrac{1500}{x(x - 20)} = 1 \\[1em] \Rightarrow 1500 = x^2 - 20x \text{ (On cross multiplication) }\\[1em] \Rightarrow x^2 - 20x - 1500 = 0 \\[1em] \Rightarrow x^2 - 50x + 30x - 1500 = 0 \\[1em] \Rightarrow x(x - 50) + 30(x - 50) = 0 \\[1em] \Rightarrow (x + 30)(x - 50) = 0 \\[1em] \Rightarrow x + 30 = 0 \text{ or } x - 50 = 0 \\[1em] x = -30 \text{ or } x = 50$

Since, number of children cannot be negative hence, x ≠ -30

∴ x = 50

Hence, number of children are 50.