Question

Take 6 distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers can be how much?

• A. 22.5
• B. 24.5
• C. 21.5
• D. 20
• E. 23.5

Hint:

To maximize the average under constraints, make the middle numbers as large as possible while satisfying the conditions on the extremes.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Let numbers be $x_1<x_2<x_3<x_4<x_5<x_6$.
Sum $= (x_1+x_2) + (x_3+x_4) + (x_5+x_6)$.
Step 2: Key Formula or Approach:
1. $x_1+x_2 = 28$.
2. $x_5+x_6 = 56$.
3. To maximize average, maximize $x_3$ and $x_4$.
Step 3: Detailed Explanation:
For max sum, we need max $x_5$. Since $x_5<x_6$ and sum is 56, max $x_5 = 27$ (leaving $x_6 = 29$).
Then max $x_4 = x_5 - 1 = 26$.
Then max $x_3 = x_4 - 1 = 25$.
Check if $x_2<x_3$: $x_2$ and $x_1$ must sum to 28. If $x_2 = 15$ and $x_1 = 13$, this is valid and $15<25$.
Max Sum $= 28 + 25 + 26 + 56 = 135$.
Max Average $= 135/6 = 22.5$.
Step 4: Final Answer:
The maximum average is 22.5.