The mean of a certain number of observations is m. If each observation is divided by `x(ne 0)` and increased by y, then

Options

A. `mx+y`
B. `(mx+y)/(x)`
C. `(m+xy)/(x)`
D. `m+xy`

Answer

To solve the problem step by step, we will derive the new mean when each observation is divided by \( x \) (where \( x \neq 0 \)) and then increased by \( y \).

### Step 1: Understand the Mean
The mean \( m \) of a set of observations \( x_1, x_2, \ldots, x_n \) is given by the formula:
\[ m = \frac{\sum_{i=1}^{n} x_i}{n} \]
This implies:
\[ \sum_{i=1}^{n} x_i = m \cdot n \]

### Step 2: Transform the Observations
According to the problem, each observation \( x_i \) is transformed by dividing it by \( x \) and then adding \( y \). Thus, the new observations can be represented as:
\[ \text{New observation} = \frac{x_i}{x} + y \]

### Step 3: Calculate the New Mean
The new mean \( \text{New Mean} \) can be calculated as follows:
\[ \text{New Mean} = \frac{\sum_{i=1}^{n} \left(\frac{x_i}{x} + y\right)}{n} \]

### Step 4: Simplify the New Mean
We can separate the terms in the numerator:
\[ \text{New Mean} = \frac{\sum_{i=1}^{n} \frac{x_i}{x} + \sum_{i=1}^{n} y}{n} \]
Since \( y \) is a constant, we can simplify further:
\[ \text{New Mean} = \frac{\frac{1}{x} \sum_{i=1}^{n} x_i + n \cdot y}{n} \]

### Step 5: Substitute the Old Mean
Now, substituting \( \sum_{i=1}^{n} x_i = m \cdot n \):
\[ \text{New Mean} = \frac{\frac{1}{x} (m \cdot n) + n \cdot y}{n} \]
This simplifies to:
\[ \text{New Mean} = \frac{m}{x} + y \]

### Step 6: Combine the Terms
We can express this as:
\[ \text{New Mean} = \frac{m + xy}{x} \]

### Final Result
Thus, the new mean after transforming the observations is:
\[ \text{New Mean} = \frac{m + xy}{x} \]