Question

Choose the correct option from the provided options to the following question:
The table shows the number of fillings a class of $40$ pupils had at the time of a dental inspection.

No. of fillings	$0$	$1$	$2$	$3$	$4$	$5$	$6$
No. of pupils	$1$	$4$	$8$	$x$	$9$	$y$	$2$

If the mean number of fillings per pupil is $3.2$, then find the values of $x$ and$y$.
A. $x = 5,y = 4$
B. $x = 10,y = 6$
C. $x = 9,y = 6$
D. $x = 12,y = 4$

Answer 1

Find the total number of pupils, i.e., the total frequency. Given the mean. Find the mean which we get in terms of the variables. After that, we get two simultaneous equations, which needs to be solved to find the value of the variables.

Complete step-by-step solution:
Number of total pupils $=$ $40$
Add the total number of pupils and equate it to $40$
$\Rightarrow 1 + 4 + 8 + x + 9 + y + 2 = 40$
Add the individual terms which simplifies as follows;
$\Rightarrow 24 + x + y = 40$
Rearrange the terms and further simplify, we get;
$\Rightarrow x + y = 16$
Let us take the above equation as equation $1$
We have mean as given
Mean $= 3.2$
$\Rightarrow \dfrac{{\sum {f(x)} }}{{\sum f }} = 3.2$
\Rightarrow \dfrac{{0 \times 1 + 1 \times 4 + 2 \times 8 + 3 \times x + 4 \times 9 + 5 \times y + 6 \times 2}}{{40}}$$$$ = 3.2
Simplifying the equation, we get;
$\Rightarrow \dfrac{{0 + 4 + 16 + 3x + 36 + 5y + 12}}{{40}} = 3.2$
Now, adding the terms and taking the denominator to the right-hand side, we get;
$\Rightarrow 3x + 5y + 68 = 3.2 \times 40$
Multiplying and rearranging the terms, we get;
$\Rightarrow 3x + 5y = 128 - 68$
Now, subtracting the left-hand side terms, we get;
$\Rightarrow 3x + 5y = 60$
Let us consider the above equation as $2$
Taking equations $1$ and $2$, by solving them as a simultaneous equation, it is given as follows.
First, we multiply the equation $1$ with $3$ to make both the equations uniform in one variable and eliminate that variable.
By multiplying it with $3$, we get;
$\Rightarrow 3x + 3y = 48$
Now subtracting equation $1$ from equation $2$, we get;

{\text{ }}3x + 5y = 60 \\\ \- 3x - 3y = - 48 \\\ \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ \\\ {\text{ }}2y = 12 \\\ \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ \\\

$\Rightarrow y = 6$
Now substituting $\Rightarrow y = 6$ in equation $1$, we get;
$x + 6 = 16$
Rearranging and simplifying the terms, we get;
$x = 10$
Hence, we have $x = 10,y = 6$.

$\therefore$ The correct option is B.

Note: The total number of fillings/observations multiplied by the frequency respectively divided by the total frequency gives you the mean. The simultaneous equations are solved by making one term of it equal to the term in another equation in order to subtract those terms and eliminate a variable. Then the acquired variable is substituted in any one of the equations to get another variable.