Question

All students of the tenth class planned for a picnic. The entire package with food was of Rs 960. But eight students failed to go and thus, the cost of each member is increased by Rs 6.
a.) How many of students attended the picnic.
b.) Find the total number of the students in tenth class.

Answer 1

Hint: The above question is a very simple question of linear equation in one variable. So, we will try to convert the given statements into Mathematical form.

Complete step by step answer:
Let us assume that there are x students in the class so initially according to the question amount each student has to pay is equal to Rs $\dfrac{960}{x}$.
Now, again in question, it is given that 8 students failed to go to the picnic. So, now each student has to pay Rs. $\left( \dfrac{960}{x}+6 \right)$ and total number of students going to the picnic will be (x -8).
Also, the entire package with food remains same. Hence, we can write $\left( \dfrac{960}{x}+6 \right)\times \left( x-8 \right)=960$


The above is the question of linear equation in one variable.
So, we will first assume that there are x students in the class ${{10}^{th}}$.
Initially, when the whole class is going to the picnic the package for the picnic is Rs 960.
So, we can say that each student will pay Rs. $\dfrac{960}{x}$ .
According to the next part of the question, we can say that 8 students failed to go to the picnic. So, the total number of students going to the picnic now is (x-8).
Also, cost of each member is increased by Rs 6. So, we will say each student will now pay Rs.$\left( \dfrac{960}{x}+6 \right)$ instead of Rs $\dfrac{960}{x}$.
Now, according to the question again we will say that package for the picnic is Rs 960 even if 8 students failed to go to the picnic.
Hence, we can say that (cost per student after 8 students failed to go to picnic) multiplied by (Total number of students going to the picnic after 8 failed to go) is equal to the entire package with food of the picnic.
So, $\left( \dfrac{960}{x}+6 \right)\times \left( x-8 \right)=960$
$\Rightarrow \left( \dfrac{960+6x}{x} \right)\left( x-8 \right)=960$
$\Rightarrow \left( 960+6x \right)\left( x-8 \right)=960x$
$\Rightarrow 960x+6{{x}^{2}}-48x-7680=960x$
$\Rightarrow 6{{x}^{2}}-48x-7680=0$
$\Rightarrow {{x}^{2}}-8x-1280=0$
We will use the method of splitting to solve the above quadratic equation.
$\Rightarrow {{x}^{2}}-40x+32x-1280=0$
$\Rightarrow x\left( x-40 \right)+32\left( x-40 \right)=0$
$\Rightarrow \left( x+32 \right)\left( x-40 \right)=0$
$\therefore x=-32,40$
Since we know that number of students can’t be negative so, x must be greater than zero.
Hence x = 40.
Hence, the total number of students in the tenth class is 40. It is the answer of part (b) of the question.
Now, in part a) of the question we have to find the total number of students who have attended the picnic. It is equal to the total students in the class minus the number of students who failed to go.
Hence, the number of students who have attended the picnic = (40 - 8) = 32. It is the answer of part (b) of the question.
a.) The total number of students attended the picnic = 32.
b.) The total number of students in the tenth class = 40.
This is our required solution.

Note:
 We can also solve the above question by assuming two variables and converting the question into a linear equation of two variables. Let ‘x’ be the total number of students in the class and ‘y’ be the amount each student has paid for the picnic.
Then, the equation will be like: xy = 960 and (x - 8)(y + 6) = 960 and after solving these equations we will also get the same result as our above solution.