The average of the numbers 23 and $x$ is 27. What is the positive difference between 23 and $x$? (Recall that the average of two numbers is their sum divided by 2).
To calculate the average of two numbers, we add them and divide the sum by $2$.  From the given information, we have the equation  \[\frac{23+x}{2}=27.\]  Multiplying both sides by $2$ yields \[23+x=54.\]  Thus, $x=54-23=31$.  Finally, the positive difference between $31$ and $23$ is \[31-23=\boxed{8}.\]
8
Level 2
Prealgebra
Problem: The average of the numbers 23 and $x$ is 27. What is the positive difference between 23 and $x$? (Recall that the average of two numbers is their sum divided by 2).
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the average of the numbers 23 and $x$, we use the formula for the average: (23 + $x$) / 2. We know from the problem that this average equals 27. Therefore, we can set up the equation:

(23 + $x$) / 2 = 27.

To eliminate the fraction, we multiply both sides of the equation by 2:

23 + $x$ = 54.

Next, we isolate $x$ by subtracting 23 from both sides:

$x$ = 54 - 23.

Calculating the right side gives us:

$x$ = 31.

Now, we need to find the positive difference between 23 and $x$. The positive difference is calculated as |23 - $x$|. Substituting the value of $x$ we found:

|23 - 31| = |-8| = 8.

Thus, the positive difference between 23 and $x$ is 8. The final answer is 8.<|END_OF_LATENT|> The final answer is $8$.