Question

The numbers $3$, $5$, $6$ and $4$ have frequencies of $x$, $x + 2$, $x - 8$ and $x + 6$ respectively. If their mean is $4$ then the value of $x$ is
A) $5$
B) $6$
C) $7$
D) $4$

Answer 1

Here we are given that the numbers have required frequencies respectively. In this problem we are going to find the value of $x$ with the use of given frequency values. And also here students learn to calculate mean value from the frequency values.

Formula used: Mean =$\dfrac{{{\text{Sum of the product of frequencies and the given numbers }}}}{{{\text{ Sum of the frequencies}}}}$

Complete step-by-step solution:
That is, Mean $\left( {\overline X } \right)$= $\dfrac{{\sum {fx} }}{n}$
If the given number $x$ has frequency $f$ then their product is $fx$
Given that the numbers $3$,$5$,$6$ and $4$ have frequencies of $x$, $x + 2$, $x - 8$ and $x + 6$ respectively.
The number $3$ has frequency $x$ then $fx$=$3x$
The number $5$ has frequency $x$+$2$ then $fx$=$5x$+$10$
The number $6$ has frequency $x$−$8$ then $fx$=$6x$−$48$
The number $4$ has frequency $x$+$6$ then $fx$=$4x$+$24$
Now, Mean $\left( {\overline X } \right)$=$\dfrac{{3x + 5x + 10 + 6x - 48 + 4x + 24}}{{x + x + 2 + x - 8 + x + 6}}$
$\left( {\overline X } \right)$=$\dfrac{{18x - 14}}{{4x}}$
Mean =$4$ is given
Therefore $4$=$\dfrac{{18x - 14}}{{4x}}$
Cross multiply the equation, we get
$\Rightarrow$ $4$×$4x$=$18x$−$14$
$\Rightarrow$ $16x$=$18x$−$14$
Take the $x$ terms into one side.
$\Rightarrow$ $16x$−$18x$=−$14$
Subtracting the terms,
$\Rightarrow$−$2x$=−$14$
Simplifying we get,
$\Rightarrow$ $x$=$\dfrac{{ - 14}}{{ - 2}}$
Solving we get,
$\Rightarrow$ $x$=$7$
Therefore the value of $x$ is $7$

Option C is the correct answer.

Note: To calculate the mean of grouped data, the first step is to determine the midpoint of each interval. These midpoints must then multiply by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean.