Question

Some students planned a picnic. The total budget for food was Rs 2000. But 5 students failed to attend the picnic and thus the cost of food for each member increased by Rs 20. How many students attended the picnic and how much did each student pay for the food.

Answer 1

Hint: Assume a variable x that will represent the number of students that had planned the picnic and assume a variable y that will represent the cost that each student had to pay for the food. In the question, there are two situations that are given and using these two situations, we can form two equations. These two equations can be solved together and we can get the value of x and y.

Complete step-by-step answer:
Let us assume that x number of students had planned the picnic and each of these students had to pay Rs y for the food.
In the question, it is given that the budget for the food was Rs 2000. So, we can say that,
xy = 2000 . . . . . . . . (1)
It is also given in the question that if there were 5 students who failed to attend the picnic and due to this, the budget of each member has increased by Rs 20. But in this case, the budget will still remain the same i.e. Rs 2000. So, we can say,
(x - 5) (y + 20) = 2000 . . . . . . . . . (2)
From equation (1), we can say,
y = $\dfrac{2000}{x}$
Substituting y in equation (2), we get,
$\begin{align}
  & \left( x-5 \right)\left( \dfrac{2000}{x}+20 \right)=2000 \\
 & \Rightarrow 2000+20x-\dfrac{10000}{x}-100=2000 \\
 & \Rightarrow \dfrac{20{{x}^{2}}-10000-100x}{x}=0 \\
 & \Rightarrow 20{{x}^{2}}-10000-100x=0 \\
 & \Rightarrow 20\left( {{x}^{2}}-5x-500 \right)=0 \\
 & \Rightarrow {{x}^{2}}-5x-500=0 \\
 & \Rightarrow {{x}^{2}}-25x+20x-500=0 \\
 & \Rightarrow x\left( x-25 \right)+20\left( x-25 \right)=0 \\
 & \Rightarrow \left( x+20 \right)\left( x-25 \right)=0 \\
 & \Rightarrow x=-20,x=25 \\
\end{align}$
Since x represents the number of students, it should be positive.
$\Rightarrow x=25$
Substituting x = 25 in equation (1), we get,
$\begin{align}
  & y=\dfrac{2000}{25} \\
 & \Rightarrow y=80 \\
\end{align}$
The number of students that have attended the picnic is x – 5 = 25 - 5 and each of these students had to pay Rs y + 20 = 80 + 20.
Hence, the number of students that attended the picnic is 20 and each student had to pay Rs 100 for the food.

Note: There is a possibility that one may have assumed the variable x for the number of students that had attended the picnic and y as the amount each of these students had to pay. But in this case, we would have got different equations. The two equations that will be formed in this case will be xy = 2000 and (x + 5) (y – 20) = 2000. The answer in this case will be the obtained value of x and y from the two equations.