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Expert replies
by prernamalhotra » Mon Jun 16, 2014 4:52 am
Hi,
Please help with the below question, can't figure where I am going wrong :

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 = ?

1) (A^2 + B^2)/ 2(A+B)

2) (A-B)^2/ 2 (A+B)

3) (A+B)/2

4) (A-B)/2

5) 0


My approach was to test this with numbers:

a=2, b=4 Therefore x= 3
and 1/y=3/4, y=4/3
x-y = 5/3, this matches with answer choice (1).

Thank you,
Prerna
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Source: — Problem Solving |

by Brent@GMATPrepNow » Mon Jun 16, 2014 2:11 pm
prernamalhotra wrote:Hi,
Please help with the below question, can't figure where I am going wrong :

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 = ?

1) (A^2 + B^2)/ 2(A+B)

2) (A-B)^2/ 2 (A+B)

3) (A+B)/2

4) (A-B)/2

5) 0


My approach was to test this with numbers:

a=2, b=4 Therefore x= 3
and 1/y=3/4, y=4/3
x-y = 5/3, this matches with answer choice (1).

Thank you,
Prerna
Your mistake is highlighted above.
The reciprocal of the number y is the average (arithmetic mean) of the reciprocals of a and b
If a = 2 and b = 4, then 1/y = (1/2 + 1/4)/2 [we're finding the mean of the reciprocals]
So, 1/y = (3/4)/2
1/y = 3/8
y = 8/3

So, x - y = 3 = 8/3 = 1/3

I'll let you take it from there.

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by tass@ » Tue Jun 17, 2014 8:41 am
a+b =2x ...... so x = a+b/2
2/y = 1/a + 1/b ...... so y = 2ab/a+b
Now x-y = a+b/2 - 2ab/a+b
by solving this x-y = (a-b)^2/ 2(a+b)
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