Find the average of the workers from the following distribution table. Additionally, find the values of x and y, if the total frequency is 100.

Steps to Solve

1. Identify the Distribution Table The distribution table is provided in two parts. We will first extract the values related to the frequencies of workers.

2. Fill in Missing Values We need to express the unknowns (x) and (y) in terms of the total frequency being 100. Looking at the first half of the table:

[ \begin{array}{|c|c|c|c|c|c|} \hline \text{More than 150} & \text{More than 140} & \text{More than 130} & \text{More than 120} \ \hline 0 & 10 & 29 & 60 \ \hline \end{array} ] From here, the frequencies are:

• More than 150: 0
• More than 140: 10
• More than 130: 29
• More than 120: 60

3. Calculate for the Second Half Next, let's look at the second part of the table:

[ \begin{array}{|c|c|c|c|c|c|} \hline \text{More than 110} & \text{More than 100} & \text{More than 90} & \text{More than 80} \ \hline 104 & 134 & 151 & 160 \ \hline \end{array} ]

4. Establish Total Frequency Condition The total frequency is 100, so we sum up all values, including (x) and (y):

• Total frequencies from the first half: (0 + 10 + 29 + 60 = 99)
• For the second part, we have (104 + 134 + 151 + 160 + y = 100)

From this, we can see that:

$$ x + y = 100 - 99 $$
$$ x + y = 1 $$

5. Solving for x and y We have established one equation. We can substitute one value to find the other.

Let’s take (x) as variable (y):

• If (x = 1 - y)
• Since (x \text{ and } y) are counts, they both must be non-negative integers. Setting (y = 1), we get (x = 0).

6. Average Calculation To find the average number of workers, use the formula for the average from the frequency distribution:

$$ \text{Average} = \frac{\text{Sum of frequencies}}{\text{Total frequency}} $$ Since frequencies from the table correspond to a weighted sum, we compute using averages already in the table.

Insert the values where the weighted frequency for each class is multiplied by the midpoint of each class. We simplify this computation by using numbers that will make solving easier since we know (x) and (y).