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For all values of x and y , let x&y be defined by x&

by fskilnik@GMATH » Mon Oct 01, 2018 2:43 pm
For all values of x and y , let x&y be defined by x&y = xy−x+1 . If (a−2)&a = a&(a+1) , which of the following numbers is closest to the value of a ?

(A) 0.66
(B) 0.83
(C) 0.95
(D) 1.14
(E) 1.43

Answer: [spoiler]__(A)____[/spoiler]
Difficulty Level: 600 - 650
Source: https://www.GMATH.net
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by Brent@GMATPrepNow » Tue Oct 02, 2018 7:23 am
fskilnik@GMATH wrote:For all values of x and y , let x&y be defined by x&y = xy−x+1 . If (a − 2)&a = a&(a + 1) , which of the following numbers is closest to the value of a ?

(A) 0.66
(B) 0.83
(C) 0.95
(D) 1.14
(E) 1.43
Nice question!

GIVEN: x&y = xy − x + 1

So, take the equation: (a − 2)&a = a&(a + 1)
Apply rules to get the equation: (a - 2)(a) - (a - 2) + 1 = (a)(a + 1) - a + 1
Expand: a² - 2a - a + 2 + 1 = a² + a - a + 1
Simplify: a² - 3a + 3 = a² + 1
Subtract a² from both sides to get: -3a + 3 = 1
Subtract 3 from both sides to get: -3a = -2
Solve: a = (-2)/(-3) = 2/3 = 0.6666.....

Answer: A

Cheers,
Brent
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by fskilnik@GMATH » Tue Oct 02, 2018 10:28 am
fskilnik@GMATH wrote:For all values of x and y , let x&y be defined by x&y = xy−x+1 . If (a−2)&a = a&(a+1) , which of the following numbers is closest to the value of a ?

(A) 0.66
(B) 0.83
(C) 0.95
(D) 1.14
(E) 1.43

Source: https://www.GMATH.net
Thank you for your compliment and contribution, Brent!

The "alternate" solution I present below does not deserve the "alternate" name, really. It´s just an alternate "style", so to speak!

$$?\,\,\,:\,\,a\,\,\left( {{\rm{approx}}.} \right)\,$$
$${\rm{LHS}} = \left( {a - 2} \right)\& \,a = \left( {a - 2} \right)a - \left( {a - 2} \right) + 1 = {a^2} - 3a + 3$$
$${\rm{RHS}} = a\,\& \left( {a + 1} \right) = a\left( {a + 1} \right) - a + 1 = {a^2} + 1$$
$${\rm{LHS}} = {\rm{RHS}}\,\,\,\,\, \Rightarrow \,\,\,\,\, - 3a + 3 = 1\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,?\,\,\,:\,\,\,a = {2 \over 3} \cong \,\,0.66$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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