Let the number of people below $51 \; (<51) \; \text {years}$ be $x.$
The total number of people in an apartment complex $ =$ The number of people whose ages $51 \; \text{years}$ and above $( \geq 51) + $ the number of people whose ages below $ 51 \; (<51) \; \text{years} = 30 + x $
The average age of all the people in the apartment complex $ = 38 \; \text {years}.$
The total age of people in apartment complex $ = (30+x) \times 38 $
The smallest possible average age of people above $51 \; \text{years}$ is $51,$ and it gives the largest value for the other group.
The total age of people above $51 \; \text{years} = 30 \times 51 = 1530$
Now, the total age of people below $51 \; \text{years} = (30+x) \times 38 – 1530$
$\quad = 1140 + 38x – 1530 = 38x – 390 $
The average age people below $51 \; \text{years} = \frac{38x – 390}{x} \quad \longrightarrow (1) $
The number of people whose ages are below $51 \; \text{years}$ is almost $39 \; (\leqslant 39).$
For getting the largest possible average age, the number of people should be $ x = 39 .$
$\therefore$ The average age people below $51 \; \text{years} = \frac{(38 \times 39) – 390}{39} = \frac{39(38-10)}{39} = 28 \; \text{years}.$
Correct Answer $: \; \text{B}$