Let's solve the problem step by step:

1. Calculate the difference between 28 and 10:

\[
28 - 10 = 18
\]

2. Find half of this difference:

\[
\frac{18}{2} = 9
\]

3. Let the number be \( x \). According to the problem, one third of this number is less than 9:

\[
\frac{x}{3} < 9
\]

4. Solve for \( x \):

Multiply both sides of the inequality by 3 to isolate \( x \):

\[
x < 9 \times 3
\]
\[
x < 27
\]

5. Among the options provided, find which number is less than 27:

- 13 is less than 27.
- 7 is less than 27.
- 27 is not less than 27.
- 22 is less than 27.

To match the conditions given, we need to ensure that one third of the number is strictly less than 9.

- For 13:
\[
\frac{13}{3} \approx 4.33 < 9
\]

- For 7:
\[
\frac{7}{3} \approx 2.33 < 9
\]

- For 22:
\[
\frac{22}{3} \approx 7.33 < 9
\]

Thus, all options except 27 satisfy the condition.

However, if the problem explicitly asks for a specific number among the provided choices, 13 meets the condition.

Thus, the number is 13, which corresponds to option A.