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A better explanation please!

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A better explanation please!

by Rastis » Mon Sep 15, 2014 12:39 pm
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 was confused as how to derive the equation for the reciprocal for Y. I did 1/y = (1/a + 1/b)/2 and couldn't understand why it's 1/y = 1/2(1/a + 1/b)
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by [email protected] » Mon Sep 15, 2014 12:51 pm
Hi Rastis,

While this is a "thick" question, it can be solved by TESTing Values.

We're told that...
X = average of A and B
The reciprocal of Y = the average of the reciprocals of A and B

A = 2
B = 4

X = (2+4)/2 = 3

1/Y = (1/2 + 1/4)/2 = (3/4)/2 = 3/8
1/Y = 3/8
Y = 8/3

We're asked for the value of X - Y:
X - Y =
3 - 8/3 =
9/3 - 8/3 =
1/3

So we're looking for an answer that equals 1/3 when we use X = 2 and Y = 4

From the answer choices, we can quickly eliminate C, D and E (none of them have a denominator that is (or could become) a "3"; so we're down to the first 2 answers.

Answer A: (4 + 16)/12 = 20/12 = 5/3
Answer B: (-2)^2/12 = 4/12 = 1/3

Final Answer: B

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Rich
Contact Rich at [email protected]
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by Brent@GMATPrepNow » Mon Sep 15, 2014 1:08 pm
Rastis 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) (a² + b²)/2(a + b)

B) (a - b)²/ 2(a + b)

C) (a + b)/2

D) (a - b)/2

E) 0
Rich's input-output is the best (i.e., fastest) approach, but we can also solve this algebraically (although it does get a bit messy)

x is the average (arithmetic mean) of the positive numbers a and b
So, x = (a + b)/2

The reciprocal of the number y is the average (arithmetic mean) of the reciprocals of a and b.
So, 1/y = the mean of the reciprocals 1/a and 1/b
In other words, 1/y = (1/a + 1/b)/2
Rewrite parts in bracket with common denominator: 1/y = (b/ab + a/ab)/2
Simplify: 1/y = [(b+a)/ab]/2
Simplify: 1/y = (b+a)/2ab
Flip both sides to get: y = 2ab/(a+b)

So, x - y = (a + b)/2 - 2ab/(a+b)
= (a+b)(a+b)/2(a+b) - 2(2ab)/2(a+b)
= (a² + 2ab + b²)/2(a+b) - 4ab/2(a+b)
= (a² - 2ab + b²)/2(a+b)
= (a - b)²/ 2(a + b)
= B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by gmatcracker0123 » Tue Sep 16, 2014 11:23 am
I was confused as how to derive the equation for the reciprocal for Y. I did 1/y = (1/a + 1/b)/2 and couldn't understand why it's 1/y = 1/2(1/a + 1/b)
Hi Rastis,

Both the equations you mentioned
1. 1/y = (1/a + 1/b)/2
2. 1/y = 1/2(1/a + 1/b)
mean the same.

I assume the mistake that you made was in the first case was that after solving the bracket you multiplied 2 to the numerator.
(1/a + 1/b)/2
= (a + b /ab) /2
= (a + b /ab) / (2/1)
= (a + b /ab) * (1/2)
= a + b / 2ab
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