sdotgarcia wrote:The average of 5 different numbers is 14. What is the average (arithmetic mean) of the 3 largest numbers?
(1) The average (arithmetic mean) of the two smallest numbers is 5.
(2) The average (arithmetic mean) of the two smallest numbers is 1/4 of the average of the 3 largest numbers.
OA is D
I understand how Statement 1 is sufficient, but can someone explain how statement 2 is?
Sum = (number)(average).
When given an average, calculate THE SUM.
Here, the sum of the 5 numbers = 5*14 = 70.
Statement 1: The average (arithmetic mean) of the two smallest numbers is 5.
Thus:
Sum of the 2 smallest numbers = 2*5 = 10.
Sum of the 3 largest numbers = (sum of all 5 numbers) - (sum of the 2 smallest numbers) = 70-10 = 60.
Average of the 3 largest numbers = 60/3 = 20.
SUFFICIENT.
Statement 2: The average (arithmetic mean) of the two smallest numbers is 1/4 of the average of the 3 largest numbers.
Statement 1 yields the following averages:
Average of the 2 smallest numbers = 5.
Average of the 3 largest numbers = 20.
This case satisfies statement 2:
(average of the 2 smallest numbers)/(average of the 3 largest numbers) = 5/20 = 1/4.
To determine whether this case is the ONLY case that will satisfy statement 2, change the information given in statement 1.
Let the average of the 2 smallest numbers = 2.
Then:
Sum of the 2 smallest numbers = 2*2 = 4.
Sum of the 3 largest numbers = (sum of all 5 numbers) - (sum of the 2 smallest numbers) = 70-4 = 66.
Average of the 3 largest numbers = 66/3 = 22.
(average of the 2 smallest numbers)/(average of the 3 largest numbers) = 2/22 = 1/11.
Doesn't work: the resulting fraction is NOT 1/4.
Implication:
The ONLY case that will satisfy statement 2 is the case implied by statement 1.
Thus, the average of the 2 smallest numbers = 5, while the average of the 3 largest numbers = 20.
SUFFICIENT.
The correct answer is
D.
Algebraic approach for statement 2:
Let x = the average of the 2 smallest numbers and y = the average of the 3 largest numbers.
Sum of the 2 smallest numbers = 2x.
Sum of the 3 largest numbers = 3y.
Since the sum of all 5 numbers is 70, we get:
2x + 3y = 70.
Statement 2 indicates that x = (1/4)y.
Since we have 2 variables (x and y) and 2 distinct linear equations -- 2x+3y=70 and x=(1/4)y -- we can solve for each variable.
Thus, the value of y can be determined.
SUFFICIENT.
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