Question

The median of 11 different positive integers is 15 and seven of those 11 integers are 8,12, 20, 6, 14, 22 and 13.
I. The difference between the averages of four largest integers and four smallest integers is 13.25.
II. The averages of all 11 integers is 16.
Which of the following statements would be sufficient to find the largest possible integer of these numbers?

• A. Only Statement I
• B. Only Statement II
• C. Both Statements I and II are required
• D. Neither Statement I nor Statement II is sufficient
• E. Either Statement I or Statement II is sufficient

Answer 1

The correct option is E Either Statement I or Statement II is sufficient

If these 7 numbers are arranged in ascending order, they will be 6, 8, 12, 13, 14, 20 and 22. If the median of 11 numbers is 15, then 15 is the 6th number. Hence, all 3 remaining numbers are above 15.
From Statement I,
Average of 4 smallest integers = 9.75
Thus, average of 4 largest integers
= 9.75+13.25 = 23
Summation of 4 largest integers = 92
To find the largest possible integer, 7th and 8th numbers should be minimum, i.e. 16 and 17
Thus, 17 + 20 + 22 + x = 92
$\Rightarrow$ 59 + x = 92 $\Rightarrow$ x = 33
Hence, statement I alone is sufficient to find the solution.
From Statement II,
Average is 16. Then, summation of 11 numbers
= 16$\times$11 = 176
Now, sum of 8 available numbers = 110
And sum of 3 unknown numbers = 66
To find the largest possible integer, 7th and 8th numbers should be minimum, i.e. 16 and 17.
Thus, 16 + 17 + x = 66
$\Rightarrow$ 33 + x = 66
$\Rightarrow$ x = 33
Hence, Statement II alone is sufficient to find the solution.