Problem Solving

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
(Exercise 18 of the Quant Class #04 included in our free test drive!)

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.