The sign of a is positive. The sign of b is negative. If it can be determined, what is the sign of the mean of a and b?

Options

A. Positive
B. Negative
C. Both positive and negative
D. Cannot be determined from the given information.

Answer

To determine the sign of the mean of two numbers \( a \) and \( b \) where \( a \) is positive and \( b \) is negative, we can follow these steps:

### Step 1: Understand the values of \( a \) and \( b \)
We know that:
- \( a > 0 \) (positive)
- \( b < 0 \) (negative)

### Step 2: Write the mean formula
The mean of two numbers \( a \) and \( b \) is given by:
\[ \text{Mean} = \frac{a + b}{2} \]

### Step 3: Substitute the values of \( a \) and \( b \)
Substituting the values into the mean formula, we have:
\[ \text{Mean} = \frac{a + b}{2} \]

### Step 4: Analyze the expression \( a + b \)
Since \( a \) is positive and \( b \) is negative, the sum \( a + b \) can be either positive, negative, or zero depending on the relative magnitudes of \( a \) and \( b \):
- If \( |b| < a \) (the absolute value of \( b \) is less than \( a \)), then \( a + b > 0 \) (mean is positive).
- If \( |b| > a \) (the absolute value of \( b \) is greater than \( a \)), then \( a + b < 0 \) (mean is negative).
- If \( |b| = a \), then \( a + b = 0 \) (mean is zero).

### Step 5: Conclusion
Since we do not have information about the relative sizes of \( a \) and \( b \), we cannot definitively determine the sign of the mean. Therefore, the answer is:
\[ \text{Cannot be determined from the given information.} \]

### Final Answer
The correct option is D: cannot be determined from the given information.

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