Step 1: Read off what each mean tells us.
The arithmetic mean of $a$ and $b$ is the average $\frac{a+b}{2}$, and we are told it equals $5$. So the sum is $a+b=10$. We will hold on to this.
Step 2: Turn the harmonic mean into the product.
The harmonic mean is $\frac{2ab}{a+b}=3.2$. Since the bottom is already $a+b=10$, this becomes $\frac{2ab}{10}=3.2$, so $ab=16$. Now we know the sum and product.
Step 3: Guess-and-check with the sum.
Instead of building a quadratic, list pairs of whole numbers adding to $10$: $(1,9),(2,8),(3,7),(4,6),(5,5)$. We just need the pair whose product is $16$.
Step 4: Test the products quickly.
$1\times 9=9$, $2\times 8=16$, $3\times 7=21$, $4\times 6=24$. The pair $(2,8)$ hits $16$ exactly.
Step 5: Match against the options.
The pair $a=2,\;b=8$ is exactly option 4, so this is our candidate.
Step 6: Verify both means.
A.M. $=\frac{2+8}{2}=5$ and H.M. $=\frac{2(2)(8)}{10}=\frac{32}{10}=3.2$. Both conditions hold perfectly.
\[ \boxed{a=2,\; b=8} \]