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An operation Ф is defined by the equation x Ф y

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by Brent@GMATPrepNow » Wed Aug 02, 2017 10:14 am
An operation Ф is defined by the equation x Ф y = x²/4 - xy + y² for all numbers x and y. What is the value of 186 Ф 91?

A) 1
B) 4
C) 8
D) 16
E) 24

Answer: B

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by Ian Stewart » Wed Aug 02, 2017 11:33 am
Brent@GMATPrepNow wrote:An operation Ф is defined by the equation x Ф y = x²/4 - xy + y² for all numbers x and y. What is the value of 186 Ф 91?
Great question, but I think there's a typo with the signs - I think it should read:

x Ф y = x²/ ( 4 + xy - y² )

and brackets would probably help to make clear that most of the right hand side is in a denominator.

edit: I was mistaken - the question is perfect as is, but I'll leave this post here because maybe this version of the question is also interesting to attempt (it has the same answer as the first question, though it's a bit harder to solve I think)
Last edited by Ian Stewart on Wed Aug 02, 2017 12:41 pm, edited 1 time in total.
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by Brent@GMATPrepNow » Wed Aug 02, 2017 12:12 pm
Ian Stewart wrote:
Brent@GMATPrepNow wrote:An operation Ф is defined by the equation x Ф y = x²/4 - xy + y² for all numbers x and y. What is the value of 186 Ф 91?
Great question, but I think there's a typo with the signs - I think it should read:

x Ф y = x²/ ( 4 + xy - y² )

and brackets would probably help to make clear that most of the right hand side is in a denominator.
I believe it's correct as shown.
If I were going to add brackets, I MIGHT add them as follows: x Ф y = (x²/4) - xy + y², but I don't think that's necessary.
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by Ian Stewart » Wed Aug 02, 2017 12:37 pm
Brent@GMATPrepNow wrote: I believe it's correct as shown.
Ha, I just did a quick mental estimate when I read the question and thought "that's way bigger than any of the answer choices", but I forgot to divide by the 4! So I thought most of the right hand side must be in the denominator, and interestingly if change the question as I did, flipping the signs where I did, you get exactly the same answer to the question!

You're completely right, the question was fine in its original form. Sorry about that. :)
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by Brent@GMATPrepNow » Thu Aug 03, 2017 6:38 am
Brent@GMATPrepNow wrote:An operation Ф is defined by the equation x Ф y = x²/4 - xy + y² for all numbers x and y. What is the value of 186 Ф 91?

A) 1
B) 4
C) 8
D) 16
E) 24

Answer: B

Source: www.gmatprepnow.com
Level of difficulty: 650 - 700
When we see that the expression x²/4 - xy + y² has two squares (x²/4 = (x/2)² and y² is obviously a square), we should start wondering whether the expression can be factored into the form (something)²

To make this factorization easier to see, let's factor out 1/4 first.
We get: x²/4 - xy + y² = (1/4)(x² - 4xy + 4y²)
= (1/4)(x - 2y)²

Perfect, we can now say that x Ф y = (1/4)(x - 2y)²
So, 186 Ф 91 = (1/4)[186 - 2(91)]²
= (1/4)[186 - 182]²
= (1/4)[4]²
= (1/4)[16]
= 4

Answer: B
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by Matt@VeritasPrep » Sun Aug 06, 2017 10:58 pm
x Ф y = x²/4 - xy + y²

4 * (x Ф y) = x² - 4xy + 4y²

4 * (x Ф y) = (x - 2y)²

x Ф y = (x - 2y)² / 4

From there, plug and chug, and we're set.
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by Matt@VeritasPrep » Sun Aug 06, 2017 11:01 pm
A much lamer approach:

x²/4 - xy + y² =>

186²/2² - 186*91 + 91² =>

93² - 186*91 + 91² =>

93² - 2*93*91 + 91² =>

(93 - 91)² =>

2²
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