The average of $10$ consecutive odd numbers is $54$. Find the sum of the fifth and seventh numbers.

For any set of consecutive even or odd numbers, the average is located exactly at the middle position of the series.
Since there are $10$ terms (an even count), the average lies exactly midway between the $5\text{th}$ and $6\text{th}$ terms.

Let the $5\text{th}$ term be $x-1$ and the $6\text{th}$ term be $x+1$. The consecutive odd integers are:

$$\dots ,x-5,x-3,x-1,[x],x+1,x+3,x+5,\dots$$

The middle average value $[x]=54$.

Now we trace the values of individual terms around this point:

* $5\text{th}$ term = $54-1=53$
* $6\text{th}$ term = $54+1=55$
* $7\text{th}$ term = $54+3=57$

We need to find the sum of the fifth and seventh terms:

$$\text{Sum}=53+57=110$$