Question

Some students planned a picnic. The total budget for food was ₹2000. But, 5 students failed to attend the picnic and thus the cost for food for each member increased by ₹20. How many students attended the picnic and how much did each student pay for the food? [CBSE 2010]

Answer 1

Let x be the number of students who planned a picnic.

∴ Original cost of food for each member = ₹$\frac{2000}{x}$

Five students failed to attend the picnic. So, (x − 5) students attended the picnic.

∴ New cost of food for each member = ₹$\frac{2000}{\left(x-5\right)}$

According to the given condition,
₹$\frac{2000}{x-5}$ − ₹$\frac{2000}{x}$ = ₹20
$$\begin{gathered} \Rightarrow \frac{2000x-2000x+10000}{x\left(x-5\right)}=20 \\ \Rightarrow \frac{10000}{x^2-5x}=20 \\ \Rightarrow x^2-5x=500 \\ \Rightarrow x^2-5x-500=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x^2-25x+20x-500=0 \\ \Rightarrow x\left(x-25\right)+20\left(x-25\right)=0 \\ \Rightarrow \left(x-25\right)\left(x+20\right)=0 \\ \Rightarrow x-25=0\mathrm{or}x+20=0 \end{gathered}$$
$\Rightarrow x=25\mathrm{or}x=-20$
∴ x = 25 (Number of students cannot be negative)

Number of students who attended the picnic = x − 5 = 25 − 5 = 20

Amount paid by each student for the food = ₹$\frac{2000}{\left(25-5\right)}$ = ₹$\frac{2000}{20}$ = ₹100