The average of three numbers is 20. If the two numbers are 16 and 22, the third number is:

1. ### Break Down the Problem

We need to find the third number when the average of three numbers is given, along with two of those numbers.

2. ### Relevant Concepts

The formula for the average of a set of numbers is given by:

$\text{Average} = \frac{\text{Sum of all numbers}}{\text{Total count of numbers}}$

Since we know the average and two of the three numbers, we can rearrange the formula to find the sum of the numbers.

3. ### Analysis and Detail

1. Let the three numbers be $x_1, x_2, x_3$.
2. The average of the three numbers is given as 20, which translates to:

$\frac{x_1 + x_2 + x_3}{3} = 20$

3. Multiplying both sides by 3 gives:

$x_1 + x_2 + x_3 = 60$

4. We are given $x_1 = 16$ and $x_2 = 22$.
5. Therefore, we can substitute these values into our equation:

$16 + 22 + x_3 = 60$

6. Simplifying, we have:

$38 + x_3 = 60$

7. Now solve for $x_3$:

$x_3 = 60 - 38 = 22$

4. ### Verify and Summarize

We found the third number to be 22. To verify, we can recalculate the average:

$\text{Average} = \frac{16 + 22 + 22}{3} = \frac{60}{3} = 20$

This confirms our solution is correct.

Final Answer

The third number is $\mathbf{22}$.