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

Hint: The given question is that some students planned a picnic. The budget for food was \[Rs.{\text{ }}480\]. But eight failed to go thus the cost of food increases by \[Rs.{\text{ }}10.\] per student. We have to find out how many students attended the picnic. We will solve this question by assuming \[x\] students attended the party and later on, we will find \[x\] by given conditions therefore we will get a number of students attending the party.

Complete step-by-step answer:
In the question, the total budget decided on the picnic food was \[Rs.{\text{ }}480\]. But eight students failed to go, therefore \[Rs.{\text{ }}10\] cost increases per student for food, we have to find out how many students attended the picnic. Let us assume the \[x\] number of students is there at the picnic and the total cost for food is \[480\].
Therefore the cost of food per student is given by \[\dfrac{{480}}{x}\].
Now, eight students failed to go.
Therefore the number of students will become \[\left( {x{\text{ }}-{\text{ }}8} \right)\].
Also how cost has been increased by \[Rs.{\text{ }}10\].
So in the cost of food per student, \[Rs.{\text{ }}10\]is added.
Therefore, we get the equation as
\[\dfrac{{480}}{{x - 8}} = \dfrac{{480}}{x} + 10\]
Where the left-hand side represents the cost of food per student after \[8\] students failed to go and the right-hand side represents the cost of food per student where \[10\]is added because \[Rs.{\text{ }}10\] increased per student.
Therefore on solving the equation, we get
\[\dfrac{{480}}{{x - 8}} = \dfrac{{480}}{x} + 10\]
Taking LCM on the right-hand side
$\Rightarrow$ \[\dfrac{{480}}{{x - 8}} = \dfrac{{480 + 10x}}{x}\]
Now doing cross-multiplication, we get
$\Rightarrow$ \[480 \times x = \left( {x - 8} \right)\left( {480 + 10x} \right)\]
\[\]On multiplying in the right-hand side, we get
$\Rightarrow$ \[480x = 480x + 10{x^2} + 3840 + 80x\]
Taking all the terms to the left-hand side
$\Rightarrow$ \[480x - 480x - 10{x^2} + 3840 + 80x = 0\]
On solving, we get
$\Rightarrow$ \[ - 10{x^2} + 3840 + 80x = 0\]
Taking minus common and take it to right
We get \[10{x^2} - 80x - 3840 = 0\]
Taking 10 common & take it to the right side
$\Rightarrow$ \[10\left( {{x^2} - 8x - 384} \right) = 0\]
$\Rightarrow$ \[{x^2} - 8x - 384 = 0\]
Solving it by middle term splitting, we get
$\Rightarrow$ \[{x^2} - \left( {24 - 16} \right)x - 384 = 0\]
$\Rightarrow$ \[{x^2} - 24x + 16x - 384 = 0\]
Taking terms from two and last two
$\Rightarrow$ \[x\left( {x - 24} \right) + 16\left( {x - 24} \right) = 0\]
$\Rightarrow$ \[\left( {x - 24} \right)\left( {x + 16} \right) = 0\]
If \[x{\text{ }} + {\text{ }}16{\text{ }} = {\text{ }}0\]
Which means \[x = - 16\]
And if \[x-24{\text{ }} = 0\]
Which means \[x{\text{ }} = {\text{ }}24\]
But \[x\] means the number of students and number of students can’t be negative. So \[x{\text{ }} = {\text{ }}24\]Therefore \[24\] students planned a picnic and \[8\] do not go means \[\left( {24{\text{ }}-{\text{ }}8} \right) = 16\] attend the picnic.

Note: The quadratic equation formed in the question, we had solved it with the method of middle term splitting in which we had split the middle term \[8\] into \[2\] factors \[24\] and \[16\] which on subtraction gives \[8\] and multiplication give \[384\]. We can also the quadratic equation by the method of completing the square and method of the discriminant.