Question

If the arithmetic average of the data given below be 165 rupees, find the missing area.

Monthly Wages in Rs.	100	150	200	____	300	500
Number of Labourers	30	20	15	10	4	1

Answer 1

Hint: To solve this question, we will first calculate the $$\sum f_i{x}_i$$ by calculating the table of $$\sum f_i$$ and $$\sum x_i$$ and then by using the formula of mean given by $$\text{Arithmetic mean}=\bar{x}={\displaystyle \frac{\sum f_i{x}_i}{\sum f_i}}.$$ Finally, we will assume a variable for the missing term and get its value by the mean formula.

Complete step by step answer:
We are given that the arithmetic average of the data given below is 165 rupees. The data is given as

Monthly Wages in Rs.	100	150	200	____	300	500
Number of Labourers	30	20	15	10	4	1

Let us make the frequency table of the above data where the monthly wages is $$x_i,i=1,2,3,4,5$$ and the number of laborers is $$f_i$$ which is the frequency. Also, calculate $$x_i{f}_i$$ from the same. Let the value to be determined be ‘x’.

$$x_i$$	$$f_i$$	$$x_i{f}_i$$
100	30	3000
150	20	3000
200	15	3000
x	10	10x
300	4	1200
500	1	500
	$$\sum f_i=80$$	$$\sum f_i{x}_i=10700+10x$$

Then the total number of laborers can be calculated by using $$\sum f_i$$ here.
$$\sum f_i=30+15+20+10+4+1=80$$
And the summation of $$f_i{x}_i$$ is calculated by
$$\sum f_i{x}_i=3000+3000+3000+10x+1200+500$$
$$\Rightarrow \sum f_i{x}_i=10700+10x$$
We are given the question that the arithmetic mean is given by 165. Now, the arithmetic mean is calculated by using the below stated formula.
$$\text{Arithmetic mean}=\bar{x}={\displaystyle \frac{\sum f_i{x}_i}{\sum f_i}}$$
The case when the frequency table is given then the arithmetic mean is given by the above formula,
$$\Rightarrow \bar{x}=165$$
$$\Rightarrow 165={\displaystyle \frac{\sum f_i{x}_i}{\sum f_i}}$$
Substituting the value of $$\sum f_i{x}_i$$ as 10700 + 10x and the value of $$\sum f_i$$ as 80 as stated above
$$\Rightarrow 165={\displaystyle \frac{10700+10x}{80}}$$
$$\Rightarrow 165\times 80=10700+10x$$
$$\Rightarrow 165\times 80-10700=10x$$
$$\Rightarrow 1320-1070=x$$
$$\Rightarrow x=250$$
Therefore, the value of x is 250.

Hence, the missing term is given by x = 250.

Note: Remember that if all the values of the given $$x_i$$ are positive then the mean would be definitely positive and if all the values of $$x_i$$ are negative. Then the mean would be definitely negative. Also, if some values of $$x_i$$ are positive and some are negative, then the mean can be any value from positive or negative in this case.