1 vote

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?

- $4$
- $3.5$
- $4.5$
- $5$

1 vote

Best answer

Given that, the average of $30$ integers is $5 \Rightarrow$ Sum of $30$ integers $= 30\times5=150$

Among these $30$ integers, there are exactly $20$ which do not exceed $5$.$\Rightarrow 20$ $|\leq5|$

But $10$ integers might exceed $5$. We need the maximum possible average for $20$ integers, so for the remaining $10$ integers, we can take the minimum values so that the average get balanced.

So, minimum possible sum of remaining $10$ integers $= 10$ $|>$ $5|$ $ \Rightarrow 10 \times6 = 60$

So, sum of remaining $20$ integers $\Rightarrow$ $150-60=90 $

Let the average of $20$ integers be $x.$

Integers Average Sum of integers

$30$ $5$ $150$

$20$ $x$ $20x$

$10$ $6$ $60$

$\Rightarrow 20x=90$

$\Rightarrow x= \frac{90}{20}=4.5$

$\therefore$ The highest possible value of the average of these $20$ integers are $4.5.$

Correct Answer: C

Among these $30$ integers, there are exactly $20$ which do not exceed $5$.$\Rightarrow 20$ $|\leq5|$

But $10$ integers might exceed $5$. We need the maximum possible average for $20$ integers, so for the remaining $10$ integers, we can take the minimum values so that the average get balanced.

So, minimum possible sum of remaining $10$ integers $= 10$ $|>$ $5|$ $ \Rightarrow 10 \times6 = 60$

So, sum of remaining $20$ integers $\Rightarrow$ $150-60=90 $

Let the average of $20$ integers be $x.$

Integers Average Sum of integers

$30$ $5$ $150$

$20$ $x$ $20x$

$10$ $6$ $60$

$\Rightarrow 20x=90$

$\Rightarrow x= \frac{90}{20}=4.5$

$\therefore$ The highest possible value of the average of these $20$ integers are $4.5.$

Correct Answer: C