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Average of P and its reciprocal

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Average of P and its reciprocal [#permalink] New post 24 Jun 2019, 08:45
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Question Stats:

77% (00:45) correct 22% (00:52) wrong based on 27 sessions
\(0<P<1\) where P is a fraction.

Quantity A
Quantity B
Average of P and its reciprocal
1



A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

I already know by substituting the values we can get the answer. I want someone to solve this question using Algebra and not substitution.
[Reveal] Spoiler: OA

Last edited by Carcass on 25 Jun 2019, 16:34, edited 3 times in total.
Edited by Carcass
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Re: Sum of Fraction and its Reciprocal [#permalink] New post 24 Jun 2019, 09:17
Expert's post
Please follow the rule how to format a question in the proper manner

https://greprepclub.com/forum/how-to-po ... 12752.html
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Re: Sum of Fraction and its Reciprocal [#permalink] New post 24 Jun 2019, 09:31
Expert's post
\(\frac{\frac{1}{P} + P}{2}\)

\(\frac{\frac{1 + P^2}{P}}{\frac{2}{1}}\)

\(\frac{1+P^2}{2P}\)

Clearly, 1 + a certain quantity \(P^2\), even though is a fraction as a number is > of 1

Hope this helps
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Re: Sum of Fraction and its Reciprocal [#permalink] New post 25 Jun 2019, 05:43
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skjoshi267 wrote:
\(0<P<1\) where P is a fraction.

Quantity A
Quantity B
Average of P and its reciprocal
1



Average of P and its reciprocal = \(\frac{\frac{1}{P} + P}{2}\)

\(=\frac{\frac{1 + P^2}{P}}{\frac{2}{1}}\)

\(=\frac{1+P^2}{2P}\)

So, we have:
QUANTITY A: \(\frac{1+P^2}{2P}\)

QUANTITY B: \(1\)

We can complete this solution using matching operations

Since \(2P\) is POSITIVE, we can safely multiply both quantities by \(2P\) to get:
QUANTITY A: \(1+P^2\)
QUANTITY B: \(2P\)

If you notice that we have a PERFECT SQUARE hiding among the two quantities, then we can first subtract \(2P\) from both quantities to get:
QUANTITY A: \(1+P^2-2P\)
QUANTITY B: \(0\)

Rearrange to get:
QUANTITY A: \(P^2-2P+1\)
QUANTITY B: \(0\)

Factor to get:
QUANTITY A: \((P-1)^2\)
QUANTITY B: \(0\)

Since we know that P-1 does not equal 0, we know that \((P-1)^2\) is POSITIVE

Answer: A

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Re: Sum of Fraction and its Reciprocal   [#permalink] 25 Jun 2019, 05:43
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