Question

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

Answer 1

Hint: In order to solve the problem first assume the number of students going to the picnic in terms of some unknown variable. Then with the help of the problem statement find some equations in order to convert the problem into quadratic equations. Then solve the quadratic equation to find the number of students.

Complete step-by-step solution -

Let x be the number of students planned for the picnic.
Total budget of the picnic = Rs. 480
Thus cost of food per person $ = \dfrac{{{\text{Total cost}}}}{{{\text{Number of students}}}} = Rs.\dfrac{{480}}{x}$
After 8 students could not go to the picnic
Number of students in the picnic = x – 8
Total budget of the picnic = Rs. 480
Thus cost of food per person $ = \dfrac{{{\text{Total cost}}}}{{{\text{Number of students}}}} = Rs.\dfrac{{480}}{{x - 8}}$
As given in the question new cost per person is increased by Rs. 10
Which means New cost – old cost = Rs. 10
$
  Rs.\dfrac{{480}}{{x - 8}} - Rs.\dfrac{{480}}{x} = Rs.10 \\
   \Rightarrow \dfrac{{480}}{{x - 8}} - \dfrac{{480}}{x} = 10 \\
 $
Now let us solve the above equation in order to find the value of x
$
   \Rightarrow \dfrac{{480}}{{x - 8}} - \dfrac{{480}}{x} = 10 \\
   \Rightarrow \dfrac{{480}}{{x - 8}} = 10 + \dfrac{{480}}{x} \\
 $
Now let us take LCM on RHS and proceed further by cross multiplying
\[
   \Rightarrow \dfrac{{480}}{{x - 8}} = \dfrac{{10x + 480}}{x} \\
   \Rightarrow 480x = \left( {x - 8} \right)\left( {10x + 480} \right) \\
   \Rightarrow 480x = 10{x^2} + 480x - 80x - 3840 \\
   \Rightarrow 10{x^2} + 80x - 3840 = 0 \\
 \]
Now, let us solve the quadratic equation by dividing the equation by 10 and simplifying the middle term
\[
   \Rightarrow \dfrac{1}{{10}} \times \left( {10{x^2} + 80x - 3840} \right) = 0 \\
   \Rightarrow {x^2} + 8x - 384 = 0 \\
   \Rightarrow {x^2} - 24x + 16x - 384 = 0 \\
   \Rightarrow x\left( {x - 24} \right) + 16\left( {x - 24} \right) = 0 \\
   \Rightarrow \left( {x - 24} \right)\left( {x + 16} \right) = 0 \\
   \Rightarrow \left( {x - 24} \right) = 0{\text{ or }}\left( {x + 16} \right) = 0 \\
   \Rightarrow x = 24{\text{ or }}x = - 16 \\
 \]
Since, x represents the number of students and the number of students cannot be negative so x = 24 is the only solution.
So 24 students were initially ready to go for a picnic out of which 8 could not come. So the total number of students who went for the picnic is = 24 – 8 = 16 students.
Hence, 16 students attended the picnic.

Note: These types of practical problems are easier to solve if they are converted into mathematical problems with the help of some mathematical tool. As in the above case we have converted the problem into quadratic problems. Students must remove the part of the solution at the end which is not practically possible.