Final answer:
After calculating the total number of people in the complex and their combined ages, the largest possible average age of the people whose ages are below 51 years is found to be 28 years. This is determined by using the average age of the entire population and subtracting the minimum total age of individuals aged 51 and above. So the correct answer is Option B.
Step-by-step explanation:
In an apartment complex, there are 30 people aged 51 years and above and at most 39 people whose ages are below 51 years. The average age of all the people in the complex is 38 years, and we're looking for the largest possible average age of people below 51 years.
To solve this, let X be the total number of people in the apartment complex and Y be the total number of people below 51 years. We are given that Y ≤ 39. The total age for all X people is 38X, and the total age for the 30 people aged 51 and above is at least 51 × 30 = 1530. To maximize the average age for people below 51, we need to minimize the total number of people, thus we take Y = 39.
The total age for the Y people below 51 can be found by subtracting the total age of the 30 older people from the total age of X people:
38X - 1530 = Total age for Y people below 51
To find X, we need to understand that the maximum total of people is when we have the most people below 51 possible, so:
X = 30 people over 51 + 39 people below 51 = 69
Now we calculate the total age for the 39 people below 51:
38 × 69 - 1530 = Total age for 39 people below 51
2622 - 1530 = 1092
Finally, to find the average age for the 39 people whose ages are below 51:
1092 / 39 = 28 years
Therefore, the largest possible average age of the people whose ages are below 51 years is 28 years, which corresponds to option B.