Question

Some students planned a picnic. The budget for food was Rs $480$. But $4$ of these failed to go and thus the cost of food for each member increased by Rs $10$. How many students attended the picnic ?

Answer 1

Step 1:

Form a quadratic equation:

Let the numbers of student in picnic be $x$.

It is given that the total budget is $\mathrm{Rs}.480$.

Thus, the cost of food per person is $\mathrm{Rs}.\frac{480}{x}$.

Since four of student failed to join, so the number of students will become $x-4$.

Thus, the new cost of food per person is $\mathrm{Rs}.\frac{480}{x-4}$.

According to the question, when the number of students drops by four, the cost per person rises by Rs. 10, then,

$\begin{array}{rcl}\frac{480}{x-4}& =& \frac{480}{x}+10\\ & \Rightarrow & \frac{480}{x-4}=\frac{480+10x}{x}\\ & \Rightarrow & 480x=\left(x-4\right)\left(480+10x\right)\left(\mathrm{On}\mathrm{cross}\mathrm{multiplication}\right)\\ & \Rightarrow & 480x=480x+10x^2-1920-40x\\ & \Rightarrow & 10x^2-40x-1920=0\\ & \Rightarrow & x^2-4x-192=0\end{array}$
Step 2:

Determine the number of students:

Use factorization method to solve the obtained quadratic equation.

$$\begin{gathered} x^2-4x-192=0 \\ \begin{array}{rcl}& \Rightarrow & x^2-16x+12x-192=0\\ & \Rightarrow & x\left(x-16\right)+12\left(x-16\right)=0\\ & \Rightarrow & \left(x-16\right)\left(x+12\right)=0\\ & \Rightarrow & x-16=0\mathrm{or}x+12=0\\ & \Rightarrow & x=16\mathrm{or}x=-12\end{array} \end{gathered}$$

But the numbers of student cannot be negative, thus, rejecting $x=-12$.

$\therefore x=16$.

Calculate the number of students going for picnic.

$\begin{array}{rcl}x-4& =& 16-4\\ & =& 12\end{array}$

Hence, the number of students going on picnic is 12.