Concept:
English contains many fixed expressions and idioms in which a particular preposition must be used. Such expressions cannot be changed without making the sentence incorrect.
Step 1: Identify the meaning intended by the sentence.
The sentence describes an elderly person who has difficulty hearing sounds clearly.
The intended meaning is:
\[
\text{unable to hear properly}
\]
Step 2: Recall the standard English expression.
The accepted phrase is:
\[
\text{hard of hearing}
\]
This expression means that a person's hearing ability is weak.
Step 3: Check the remaining options.
The expressions
\[
\text{hard on hearing}
\]
\[
\text{hard in hearing}
\]
\[
\text{hard at hearing}
\]
are grammatically incorrect.
Only “hard of hearing” is a recognized English expression.
Step 4: Complete the sentence.
The sentence becomes:
\[
\text{The old man is hard of hearing.}
\]
Therefore,
\[
\boxed{\text{of}}
\]