Question

In an apartment complex, the number of people aged 51 years and above is 30 and there are at most 39 people whose ages are below 51 years. The average age of all the people in the apartment complex is 38 years. What is the largest possible average age, in years, of the people whose ages are below 51 years?

• A. 27
• B. 28
• C. 26
• D. 25

Answer 1

The Correct Option is B

To solve this problem, we need to find the largest possible average age of people under 51 years when the average age of all individuals in the apartment complex is 38 years. Here are the steps: 

1. Let \( n \) be the number of people whose ages are below 51 years. We know \( n \leq 39 \).
2. The total number of people in the complex is \( n + 30 \) because there are 30 people aged 51 or above.
3. Let the average age of those under 51 be \( x \). We need to maximize \( x \).
4. The average age of all the people is 38, so the total age sum is \((n + 30) \times 38\).
5. The equation for the sum of ages is:
   \((n \cdot x) + (30 \cdot 51) = (n + 30) \cdot 38\)
6. Rearrange to solve for \( x \):
   \(nx = (n+30) \cdot 38 - 30 \cdot 51\)
7. Simplify the right side:
   \(nx = 38n + 1140 - 1530\)
   \(nx = 38n - 390\)
8. Thus, \(x = 38 - \frac{390}{n}\).
9. To maximize \( x \), \( \frac{390}{n} \) should be minimized, which happens when \( n \) is maximized.
10. Given \( n \leq 39 \), we'll use \( n = 39 \).
11. Substituting, we get:
    \(x = 38 - \frac{390}{39} = 38 - 10 = 28\).

Therefore, the largest possible average age of the people whose ages are below 51 years is 28 years.