The mean income of a group of workers is `bar(X)` and that of another group is `bar(Y)`. If the number of workers in the second group is 10 times the number of workers in the first group, then the mean income of the combined group is

Options

A. `(bar(X)+10bar(Y))/(3)`
B. `(bar(X)+10bar(Y))/(11)`
C. `(10bar(X)+bar(Y))/(Y)`
D. `(bar(X)+10bar(Y))/(9)`

Answer

To find the mean income of the combined group of workers, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( n \) = number of workers in the first group
- \( m \) = number of workers in the second group
- \( \bar{X} \) = mean income of the first group
- \( \bar{Y} \) = mean income of the second group

### Step 2: Express the Total Incomes
The total income of the first group can be expressed as:
\[ \Sigma X_i = n \bar{X} \]
And the total income of the second group can be expressed as:
\[ \Sigma Y_i = m \bar{Y} \]

### Step 3: Find the Total Number of Workers
The total number of workers in the combined group is:
\[ n + m \]

### Step 4: Substitute the Relationship Between Groups
According to the problem, the number of workers in the second group is 10 times the number of workers in the first group:
\[ m = 10n \]

### Step 5: Substitute \( m \) in the Total Workers Expression
Now, substituting \( m \) in the total number of workers:
\[ n + m = n + 10n = 11n \]

### Step 6: Calculate the Mean Income of the Combined Group
The mean income of the combined group can be calculated as:
\[ \text{Mean} = \frac{\Sigma X_i + \Sigma Y_i}{n + m} = \frac{n \bar{X} + m \bar{Y}}{n + m} \]
Substituting \( m = 10n \):
\[ \text{Mean} = \frac{n \bar{X} + 10n \bar{Y}}{11n} \]

### Step 7: Simplify the Expression
We can factor out \( n \) from the numerator:
\[ \text{Mean} = \frac{n(\bar{X} + 10 \bar{Y})}{11n} \]
The \( n \) cancels out:
\[ \text{Mean} = \frac{\bar{X} + 10 \bar{Y}}{11} \]

### Final Result
Thus, the mean income of the combined group is:
\[ \text{Mean} = \frac{\bar{X} + 10 \bar{Y}}{11} \]

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