CAT #1 700-800 Level Question Recipriocal Averages

EMAN wrote:The number x is the average (arithmetic mean) of the positive numbers a and b, and the reciprocal of the number y is the average (arithmetic mean) of the reciprocals of a and b. In terms of a and b, x – y = ?

(A)A^2 + b^2 / 2(A+B)
(B)(A-B)^2 / 2(A+B)
(C)A+B / 2
(D)A-B / 2
(E)0


I answered this question incorrectly on the test. I understand how to put this in equation format to a certain point, but after that my algebra skills are insufficient. The explanation involves a lot of tedious algebra. Someone please let me know if there is an easy way to do this without a lot of complicated factoring and other manipulative BS.

JeffB wrote:

EMAN wrote:The number x is the average (arithmetic mean) of the positive numbers a and b, and the reciprocal of the number y is the average (arithmetic mean) of the reciprocals of a and b. In terms of a and b, x – y = ?

(A)A^2 + b^2 / 2(A+B)
(B)(A-B)^2 / 2(A+B)
(C)A+B / 2
(D)A-B / 2
(E)0


I answered this question incorrectly on the test. I understand how to put this in equation format to a certain point, but after that my algebra skills are insufficient. The explanation involves a lot of tedious algebra. Someone please let me know if there is an easy way to do this without a lot of complicated factoring and other manipulative BS.

Pick numbers:

A = 2
B = 4

A+B/2 = X
2+4/2 = 3

1/a + 1/b = 1/y
1/2 + 1/4 = 3/4

y = 4/3

x-y
3- 4/3 = 5/3

Can eliminate E,D,C

Try A:

A^2 + b^2 / 2(A+B)
4 + 16 / 2(2+4)
20/12 = 5/3