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For all values of x and y, let x y be defined by x y = xy - x + 1. If

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GMATH Teacher
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Joined: 12 Oct 2010
Posts: 935
For all values of x and y, let x y be defined by x y = xy - x + 1. If  [#permalink]

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01 Oct 2018, 15:39
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74% (02:03) correct 26% (01:54) wrong based on 26 sessions

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For all values of $$x$$ and $$y$$, let $$x\blacktriangle y$$ be defined by $$x\blacktriangle y = xy - x + 1$$. If $$\left( {a - 2} \right)\blacktriangle \,a\,\, = \,\,a\,\blacktriangle \left( {a + 1} \right)$$, 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: http://www.GMATH.net
(Exercise 18 of the Quant Class #04 included in our free test drive!)

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Re: For all values of x and y, let x y be defined by x y = xy - x + 1. If  [#permalink]

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01 Oct 2018, 18:35
(a−2)▲a = a▲(a+1)

(a−2)a-(a−2)+1 = a(a+1)-a+1

$$a^2$$-2a-a+2+1 = $$a^2$$+a-a+1

-3a = -2

Or

a = 2/3 = 0.66

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Re: For all values of x and y, let x y be defined by x y = xy - x + 1. If  [#permalink]

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02 Oct 2018, 11:40
fskilnik wrote:
For all values of $$x$$ and $$y$$, let $$x\blacktriangle y$$ be defined by $$x\blacktriangle y = xy - x + 1$$. If $$\left( {a - 2} \right)\blacktriangle \,a\,\, = \,\,a\,\blacktriangle \left( {a + 1} \right)$$, 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: http://www.GMATH.net

Thank you for your contribution, Afc0892!
The solution I present below is the same, just to follow my method´s style (for the interested readers).

$$?\,\,\,:\,\,a\,\,\left( {{\rm{approx}}.} \right)\,$$

$${\text{LHS}} = \left( {a - 2} \right)\blacktriangle \,a = \left( {a - 2} \right)a - \left( {a - 2} \right) + 1 = {a^2} - 3a + 3$$

$${\text{RHS}} = a\,\blacktriangle \left( {a + 1} \right) = a\left( {a + 1} \right) - a + 1 = {a^2} + 1$$

$${\rm{LHS}} = {\rm{RHS}}\,\,\,\,\, \Rightarrow \,\,\,\,\, - 3a + 3 = 1\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,?\,\,\,:\,\,\,a = {2 \over 3} \cong \,\,0.66$$

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

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: For all values of x and y, let x y be defined by x y = xy - x + 1. If   [#permalink] 02 Oct 2018, 11:40
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