Some students planned a picnic. The budget for the food was ₹ $2400$. Since $8$ of them failed to come, the cost of food for each member increased by ₹ $50$. Find the number of students who went for the picnic.
Let the original number of students to go on a picnic be $= x$
Original cost of food for each student $= $ ₹ $\left(\dfrac{2400}{x}\right)$
New number of students who went for the picnic $= x - 8$
New cost of food per student $= $ ₹ $\left(\dfrac{2400}{x - 8}\right)$
As per question,
New cost of food per student $= $ Old cost of food per student $+ $ ₹ $50$
i.e. $\;$ $\dfrac{2400}{x - 8} = \dfrac{2400}{x} + 50$
i.e. $\;$ $2400 \left[\dfrac{1}{x - 8} - \dfrac{1}{x}\right] = 50$
i.e. $\;$ $48 \left[\dfrac{x - x + 8}{x \left(x - 8\right)}\right] = 1$
i.e. $\;$ $48 \times 8 = x^2 - 8x$
i.e. $\;$ $x^2 - 8x - 384 = 0$
i.e. $\;$ $x^2 - 24x + 16x - 384 = 0$
i.e. $\;$ $x \left(x - 24\right) + 16 \left(x - 24\right) = 0$
i.e. $\;$ $\left(x - 24\right) \left(x + 16\right) = 0$
i.e. $\;$ $x = 24$ $\;$ or $\;$ $x = -16$
$\therefore \;$ Original number of students $= 24$
$\therefore \;$ Number of students who went for a picnic $= 24 - 8 = 16$