If the arithmetic mean of the numbers `x_1 , x_2, x_3,……., x_n` is `barx`, then the arithmetic mean of the numbers `ax_1 + b, ax_2 + b, ax_3 + b, ……, ax_n + b`, where a and b are two constants, would be:

Options

A. `bar(x)`
B. `nabar(x) + nb`
C. `abar(x)`
D. `abar(x) + b`

Answer

To find the arithmetic mean of the transformed numbers \( ax_1 + b, ax_2 + b, ax_3 + b, \ldots, ax_n + b \), we can follow these steps:

### Step 1: Understand the Arithmetic Mean
The arithmetic mean (average) of a set of numbers \( x_1, x_2, x_3, \ldots, x_n \) is given by the formula:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
where \( n \) is the total number of terms.

**Hint for Step 1:** Remember that the arithmetic mean is simply the sum of the numbers divided by the count of those numbers.

### Step 2: Write the Expression for the New Set
We need to find the arithmetic mean of the new set of numbers:
\[ ax_1 + b, ax_2 + b, ax_3 + b, \ldots, ax_n + b \]
The sum of these numbers can be expressed as:
\[ (ax_1 + b) + (ax_2 + b) + (ax_3 + b) + \ldots + (ax_n + b) \]

**Hint for Step 2:** Break down the sum into two parts: the sum of the \( ax_i \) terms and the sum of the constant \( b \).

### Step 3: Factor Out Common Terms
We can factor out \( a \) from the first part of the sum:
\[ = a(x_1 + x_2 + x_3 + \ldots + x_n) + nb \]
Here, \( nb \) represents the contribution of the constant \( b \) from each of the \( n \) terms.

**Hint for Step 3:** Look for common factors in the expression to simplify your calculations.

### Step 4: Substitute the Arithmetic Mean
We know from Step 1 that:
\[ x_1 + x_2 + x_3 + \ldots + x_n = n\bar{x} \]
Substituting this into our expression gives:
\[ = a(n\bar{x}) + nb \]

**Hint for Step 4:** Use the definition of the arithmetic mean to substitute for the sum of \( x_i \).

### Step 5: Combine and Simplify
Now, we can simplify the expression:
\[ = an\bar{x} + nb \]
To find the arithmetic mean, we divide this entire sum by \( n \):
\[ \text{Arithmetic Mean} = \frac{an\bar{x} + nb}{n} \]
This simplifies to:
\[ = a\bar{x} + b \]

**Hint for Step 5:** Remember to divide the entire sum by the number of terms to find the mean.

### Final Answer
Thus, the arithmetic mean of the numbers \( ax_1 + b, ax_2 + b, ax_3 + b, \ldots, ax_n + b \) is:
\[ \boxed{a\bar{x} + b} \]