Let's walk through the steps to find the missing value in the wages given the arithmetic mean.
1. Set Up the Problem with Given Data:
[tex]\[
\text{Wages (₹)} = [50, 58, 60, 65, 70, ?, 80, 100]
\][/tex]
[tex]\[
\text{Number of Workers} = [2, 20, 5, 35, 8, 10, 16, 4]
\][/tex]
The arithmetic mean of wages is given as ₹68.25.
2. Calculate the Total Number of Workers:
Adding up the number of workers:
[tex]\[
2 + 20 + 5 + 35 + 8 + 10 + 16 + 4 = 100
\][/tex]
So, the total number of workers is 100.
3. Express the Problem Using the Arithmetic Mean Formula:
The formula for the arithmetic mean is:
[tex]\[
\text{Arithmetic Mean} = \frac{\text{Sum of all wages}}{\text{Total number of workers}}
\][/tex]
Given the arithmetic mean (68.25):
[tex]\[
68.25 = \frac{\text{Sum of all wages}}{100}
\][/tex]
Therefore,
[tex]\[
\text{Sum of all wages} = 68.25 \times 100 = 6825
\][/tex]
4. Calculate the Sum of the Known Wages:
Next, we calculate the sum of wages for the known wages:
[tex]\[
\text{Sum of known wages} = (50 \times 2) + (58 \times 20) + (60 \times 5) + (65 \times 35) + (70 \times 8) + (80 \times 16) + (100 \times 4)
\][/tex]
Performing the multiplications and addition:
[tex]\[
= 100 + 1160 + 300 + 2275 + 560 + 1280 + 400
\][/tex]
[tex]\[
= 6075
\][/tex]
5. Determine the Missing Wage Value:
With the sum of all wages known (6825) and the sum of the known wages (6075), we can find the missing wage by subtracting the two:
[tex]\[
\text{Missing wage value} = \text{Sum of all wages} - \text{Sum of known wages}
\][/tex]
[tex]\[
= 6825 - 6075
\][/tex]
[tex]\[
= 750
\][/tex]
6. Calculate the Average Wage for the Missing Group:
The number of workers with the missing wage is given as 10 workers. Thus, the missing wage per worker is:
[tex]\[
\text{Missing wage per worker} = \frac{750}{10}
\][/tex]
[tex]\[
= 75
\][/tex]
Therefore, the missing wage value is ₹75.
