Question

If the mean of the following data is $86$ , then what is the value of $p$ ?

Wages (in Rs.)	$50 - 60$	$60 - 70$	$70 - 80$	$80 - 90$	$90 - 100$	$100 - 110$
No. of workers	$5$	$3$	$4$	$p$	$2$	$13$

Answer 1

Here the value of mean is given in the question and we have to find the value of $p$ . Initially we will plot a table with columns of Wages (in Rs.), second column will be the average of wages, third column will be frequency and, in this question, frequency is the number of workers getting the wages. Lastly, it will be the product of average and frequency. Using the mean formula, we will find the value of $p$ .

Complete answer:
The mean formula is:
$Mean = \dfrac{{\sum {xf} }}{{\sum f }}$
Where x = the average of wages (in Rs.) and f = the frequency i.e., the number of workers getting the wages.
First, we will make a table with columns of Wages (in Rs.), average of wages, frequency is the number of workers getting the wages, product of average and frequency.

Wages (in Rs.)	Average of wages( $x$ )	No. of workers($f$ )	$xf$
$50 - 60$	$55$	$5$	$275$
$60 - 70$	$65$	$3$	$195$
$70 - 80$	$75$	$4$	$300$
$80 - 90$	$85$	$p$	$85 \times p$
$90 - 100$	$95$	$2$	$190$
$100 - 110$	$105$	$13$	$1365$
	$\sum {f = 27 + p}$	$\sum {xf = 2325 + 85p}$

Now we will use the mean formula to calculate $p$ by substituting the values of $\sum f$ and $\sum {xf}$ which we calculated in the table,
$\Rightarrow$ $Mean = \dfrac{{\sum {xf} }}{{\sum f }} = \dfrac{{2325 + 85p}}{{27 + p}}$
The mean value given in the question is $86$.
So, substituting this value of mean in the mean formula, we get,
$\Rightarrow$ $86 = \dfrac{{2325 + 85p}}{{27 + p}}$
Solving this equation further we get,
$\Rightarrow$ $86(27 + p) = 2325 + 85p$
$\Rightarrow$ $2322 + 86p = 2325 + 85p$
And hence on doing the simplification,we have
$\Rightarrow$ $86p - 85p = 2325 - 2322$
$\Rightarrow$ $p = 3$
The value of $p$ is $3$.

Note:
A mean is the simple mathematical average of a set of two or more numbers. The mean formula is derived from the average formula i.e., the summation of all the quantities divided by the total number of quantities.