Data Sufficiency

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?

Apologies, please see edit.

from question:
Total of 5 different numbers= 5*14=70

from statement -I :
Total of 2 small numbers=5*2=10
So total of 3 large numbers= (Total of 5 numbers) - (Total of 2 small numbers)=70-10=60
So Average of 3 large numbers=60/3=20
SUFFICIENT


from statement II :
Average of 3 large numbers= x(say)
Average of 2 small numbers= x/4

Total of 3 large numbers= 3x
Total of 2 small numbers=2*x/4

So 3x+2x/4=70
Solving x , we get x=20 (average of 3 large numbers)
SUFFICIENT

So answer is D

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.

Average of 5 different numbers is 14.

A+B+C+D+E=5*14=70 (A being the smallest up to E being the largest).

If we can find C+D+E then we can find the average of the 3 largest numbers.

Statement 1: The average (arithmetic mean) of the two smallest numbers is 5.

This tells us A+B=5*2=10 Substitute 10 for A+B and you can immediately see you can find the value of C+D+E. SUFFICIENT

Statement 2: The average (arithmetic mean) of the two smallest numbers is 1/4 of the average of the 3 largest numbers.

This tells us A+B/2=1/4(C+D+E)/3. Again, you can substitute (A+B=) into the A+B+C+D+E=70. You don't have to do the calculation since you know you will be left with only the three variables C,D and E. SUFFICIENT.

OA=
D