As Rich has noted, the problem at top has more than one correct answer choice and thus is flawed.
A better problem:
talaangoshtari wrote:The average age of the buildings on a certain city block is greater than 40 years old. Four of the buildings were built two years ago, and none of the buildings is more than 80 years old. If there are a total of x buildings on the block, what is the smallest possible value of x?
A. 5
B. 6
C. 7
D. 8
E. 10
We can PLUG IN THE ANSWERS, which represent the smallest possible value of x.
Four of the buildings were built two years ago, implying that the average age of these 4 buildings = 2.
To raise the average age of all of the buildings to a value greater than 40 -- while MINIMIZING THE TOTAL number of buildings -- the average age of the remaining buildings must be the MAXIMUM POSSIBLE VALUE (80).
When the correct answer choice is plugged in, the average age for all of the buildings will be greater than 40.
A: A total of 5 buildings, implying 4 buildings with an average age of 2 and 1 building with an average age of 80
Average for all 5 buildings = (4*2 + 80)/5 = 88/5 ≈ 18.
Too small.
B: A total of 6 buildings, implying 4 buildings with an average age of 2 and 2 buildings with an average age of 80
Average for all 6 buildings = (4*2 + 2*80)/6 = 168/6 = 28.
Too small.
C: A total of 7 buildings, implying 4 buildings with an average age of 2 and 3 buildings with an average age of 80
Average for all 7 buildings = (4*2 + 3*80)/7 = 248/7 ≈ 35.
Too small.
D: A total of 8 buildings, implying 4 buildings with an average age of 2 and 4 buildings with an average age of 80
Average for all 8 buildings = (4*2 + 4*80)/6 = 328/8 = 41.
Success!
The correct answer is
D.
Algebraic approach:
SUM = (NUMBER)(AVERAGE).
As noted above:
Four of the buildings were built two years ago, implying that the average age of these 4 buildings = 2.
To raise the average age of all of the buildings to a value greater than 40 -- while MINIMIZING THE TOTAL number of buildings -- the average age of the remaining buildings must be the MAXIMUM POSSIBLE VALUE (80).
Sum of the ages for the four 2-year old buildings = 4*2 = 8.
Let y = the remaining buildings on the block.
If each of these buildings is 80 years old -- the maximum possible age -- then the sum of the ages for these y buildings = 80y.
Since the average age of the 4+y buildings must be greater 40, we get:
(8 + 80y/(4 + y) > 40
8 + 80y > 160 + 40y
40y > 152
y > 3.8.
Since the least possible integer value for y is 4, the least possible value for the total number of buildings = 4+y = 4+4 = 8.
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