All numbers in a list of six positive integers are less than 8. What is the smallest integer that if added to the list would guarantee that the average becomes greater than 8?
(A) 14
(B) 35
(C) 42
(D) 50
(E) 51
Here's a full solution:
In order to
guarantee that an average greater than 8, we must consider the scenario in which the original six integers are MINIMIZED.
Since the original numbers are "positive integers," their MINIMIZE values are: {1, 1, 1, 1, 1, 1}
From here, what is the smallest value we can add to this set so that the new average is greater than 8?
Let N = the number to be added to the set.
We get a new set of {1, 1, 1, 1, 1, 1, N}
If the average is greater than 8, we can write:
(1+1+1+1+1+1+N)/7 > 8
Simplify: (6+N)/7 > 8
Multiply both sides by 7 to get: 6 + N > 56
Subtract 6 from both sides to get N > 50
So, the number added must be greater than 50
Since we're told that the added number must be an INTEGER, the smallest possible value of N is
51
Answer:
E
Cheers,
Brent
