CAT2019-2: 75

1 vote
145 views

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?

1. $4$
2. $3.5$
3. $4.5$
4. $5$

edited

1 vote

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 $4.5.$

498 points 1 2 5
selected by

Related questions

1
196 views
The salaries of Ramesh, Ganesh and Rajesh were in the ratio $6:5:7$ in $2010$, and in the ratio $3:4:3$ in $2015$. If Ramesh’s salary increased by $25$% during $2010-2015$, then the percentage increase in Rajesh’s salary during this period is closest to $8$ $7$ $9$ $10$
2
107 views
If x is a real number, then $\sqrt{\log _{e}\frac{4x-x^{2}}{3}}$ is a real number if and only if $1\leq x\leq 2$ $-3\leq x\leq 3$ $1\leq x\leq 3$ $-1\leq x\leq 3$
In an examination, Rama's score was one-twelfth of the sum of the scores of Mohan and Anjali. After a review, the score of each of them increased by $6$. The revised scores of Anjali, Mohan, and Rama were in the ratio $11:10:3$. Then Anjali's score exceeded Rama's score by $24$ $26$ $32$ $35$
How many pairs $(m,n)$ of positive integers satisfy the equation $m^{2}+105=n^{2}$?____
Anil alone can do a job in $20$ days while Sunil alone can do it in $40$ days. Anil starts the job, and after $3$ days, Sunil joins him. Again, after a few more days, Bimal joins them and they together finish the job. If Bimal has done $10$% of the job, then in how many days was the job done? $14$ $13$ $15$ $12$