The average of $30$ integers is $5$ . Among these $30$ integers, there are exactly $20$ which do not exceed $5$ . What is the highest possible value of the average of these $20$ integers?

$5$

$3.5$

$4.5$

$4$

✅ Correct Option: 3

We have 30 integers with an average of 5. This tells us:

Total sum of all 30 integers = Average × Number of integers

= 30 × 5

= 150

We know that exactly 20 integers do not exceed 5 (meaning they are ≤ 5), and we want to find the highest possible average of these 20 integers.

The crucial insight: To maximize the average of the 20 integers, we need to minimize the sum of the remaining 10 integers.

We have a fixed total sum of 150. If we can make the 10 remaining integers as small as possible, we'll have more "leftover sum" for our 20 integers of interest.

The remaining 10 integers have an important constraint: they must be greater than 5 (since exactly 20 integers do not exceed 5, the other 10 must exceed 5).

Since we're dealing with integers, the smallest possible value for each of these 10 integers is 6.

Minimum sum of the 10 integers = 10 × 6 = 60

Maximum sum of the 20 integers = Total sum - Minimum sum of 10 integers

Maximum sum of the 20 integers = 150 - 60 = 90

Therefore, the highest possible average of these 20 integers is:

Maximum average = $\frac{90}{20} = 4.5$

This strategy works because we're constrained by the total sum (150), we want to maximize one part (the 20 integers), so we minimize the other part (the 10 integers). The constraint that the 10 integers must be > 5 gives us their minimum possible value (6 each).

Answer: 4.5

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The average of $30$ integers is $5$ . Among these $30$ integers, there are exactly $20$ which do not exceed $5$ . What is the highest possible value of the average of these $20$ integers?