The artithmetic mean of k numbers `y_(1),y_(2), . . ., y_(k)` is A. If `y_(k)` replaced by `x_(k)`, then the new arithmetic mean will be

Options

A. `A-y_(k)+x_(k)`
B. `(KA-y_(k)+x_(k))/(k)`
C. `A-(y_(k)-x_(k))`
D. `((k-1)A)/(k)-y_(k)+x_(k)`

Answer

To solve the problem step by step, we will derive the new arithmetic mean after replacing the last number \( y_k \) with \( x_k \).

### Step 1: Understand the Arithmetic Mean
The arithmetic mean \( A \) of \( k \) numbers \( y_1, y_2, \ldots, y_k \) is given by the formula:
\[ A = \frac{y_1 + y_2 + \ldots + y_k}{k} \]

### Step 2: Express the Sum of the Numbers
From the formula for the arithmetic mean, we can express the sum of the numbers as:
\[ y_1 + y_2 + \ldots + y_k = kA \]

### Step 3: Replace \( y_k \) with \( x_k \)
Now, if we replace \( y_k \) with \( x_k \), the new set of numbers becomes \( y_1, y_2, \ldots, y_{k-1}, x_k \). The new sum of these numbers will be:
\[ y_1 + y_2 + \ldots + y_{k-1} + x_k \]

### Step 4: Find the New Sum
Using the previous expression for the sum, we can express \( y_1 + y_2 + \ldots + y_{k-1} \) as:
\[ y_1 + y_2 + \ldots + y_{k-1} = kA - y_k \]
Thus, the new sum becomes:
\[ (kA - y_k) + x_k \]

### Step 5: Calculate the New Arithmetic Mean
The new arithmetic mean \( A' \) after replacing \( y_k \) with \( x_k \) is given by:
\[ A' = \frac{(kA - y_k) + x_k}{k} \]

### Step 6: Simplify the Expression
Now, we can simplify the expression for the new mean:
\[ A' = \frac{kA - y_k + x_k}{k} \]
This can be rearranged to:
\[ A' = \frac{kA - y_k + x_k}{k} = A - \frac{y_k}{k} + \frac{x_k}{k} \]

### Final Result
Thus, the new arithmetic mean after replacing \( y_k \) with \( x_k \) is:
\[ A' = A - \frac{y_k}{k} + \frac{x_k}{k} \]