The average of 18 numbers is 37.5. If six numbers of average x are added to them,then the average of all the numbers increases by one. The value of x is

Options

A. `41.5`
B. `40`
C. `42`
D. `38.5`

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Calculate the sum of the 18 numbers
Given that the average of 18 numbers is 37.5, we can find the sum of these numbers using the formula for average:
\[ \text{Sum} = \text{Average} \times \text{Number of items} \]
So, the sum of the 18 numbers is:
\[ \text{Sum} = 37.5 \times 18 = 675 \]

### Step 2: Define the sum of the six new numbers
Let the average of the six new numbers be \( x \). Therefore, the sum of these six numbers can be expressed as:
\[ \text{Sum of six numbers} = 6x \]

### Step 3: Calculate the total number of numbers
When we add the six new numbers to the original 18 numbers, the total number of numbers becomes:
\[ \text{Total numbers} = 18 + 6 = 24 \]

### Step 4: Set up the equation for the new average
According to the problem, the average of all 24 numbers increases by 1, which means the new average is:
\[ \text{New average} = 37.5 + 1 = 38.5 \]
The formula for the new average is:
\[ \text{New average} = \frac{\text{Sum of 18 numbers} + \text{Sum of six numbers}}{\text{Total numbers}} \]
Substituting the known values, we have:
\[ 38.5 = \frac{675 + 6x}{24} \]

### Step 5: Solve for \( x \)
To eliminate the fraction, multiply both sides by 24:
\[ 38.5 \times 24 = 675 + 6x \]
Calculating the left side:
\[ 924 = 675 + 6x \]
Now, isolate \( 6x \):
\[ 6x = 924 - 675 \]
Calculating the right side:
\[ 6x = 249 \]
Now, divide by 6 to find \( x \):
\[ x = \frac{249}{6} = 41.5 \]

### Final Answer
The value of \( x \) is:
\[ \boxed{41.5} \]

---