The mean of a set of numbers is `overline(X)`. If each number is divided by 3, then the new mean is

Options

A. `overline(X)`
B. `overline(X)+3`
C. `3 overline(X)`
D. `(overline(X))/(3)`

Explanation

To solve the problem, we need to find the new mean when each number in a set is divided by 3. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Mean**:
The mean (average) of a set of numbers \( X_1, X_2, X_3, \ldots, X_n \) is given by the formula:
\[ \overline{X} = \frac{X_1 + X_2 + X_3 + \ldots + X_n}{n} \]
where \( n \) is the total number of values in the set.

2. **Dividing Each Number by 3**:
If we divide each number by 3, the new set of numbers becomes:
\[ \frac{X_1}{3}, \frac{X_2}{3}, \frac{X_3}{3}, \ldots, \frac{X_n}{3} \]

3. **Calculating the New Mean**:
The new mean, denoted as \( \overline{Y} \), for the new set of numbers can be calculated as:
\[ \overline{Y} = \frac{\frac{X_1}{3} + \frac{X_2}{3} + \frac{X_3}{3} + \ldots + \frac{X_n}{3}}{n} \]

4. **Simplifying the New Mean**:
We can factor out \( \frac{1}{3} \) from the numerator:
\[ \overline{Y} = \frac{1}{3} \cdot \frac{X_1 + X_2 + X_3 + \ldots + X_n}{n} \]
This simplifies to:
\[ \overline{Y} = \frac{1}{3} \cdot \overline{X} \]

5. **Final Result**:
Thus, the new mean when each number is divided by 3 is:
\[ \overline{Y} = \frac{\overline{X}}{3} \]

### Conclusion:
The new mean is one-third of the original mean, which can be expressed as:
\[ \text{New Mean} = \frac{\overline{X}}{3} \]