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# What is the value of a+b?

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GMATH Teacher
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What is the value of a+b?  [#permalink]

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26 Feb 2019, 10:58
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95% (hard)

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40% (02:22) correct 60% (02:05) wrong based on 80 sessions

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GMATH practice exercise (Quant Class 14)

What is the value of $$a+b$$ ?

(1) $$a^2+ab-2a = 91$$
(2) $$b^2+ab-2b = -28$$

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: What is the value of a+b?  [#permalink]

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26 Feb 2019, 14:39
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of $$a+b$$ ?

(1) $$a^2+ab-2a = 91$$
(2) $$b^2+ab-2b = -28$$

Statements combined:

Adding the two equations, we get:
$$a^2+b^2+2ab-2a-2b=63$$
$$(a+b)^2 - 2(a+b)=63$$
$$(a+b)(a+b-2)=63$$

Let $$x=a+b$$.
Substituting $$x=a+b$$ into $$(a+b)(a+b-2) = 63$$, we get:
$$(x)(x-2)=63$$
$$x^2-2x-63=0$$
$$(x-9)(x+7)=0$$
$$x=9$$ or $$x=-7$$

Since $$x=a+b$$, it is possible that $$a+b=9$$ or that $$a+b=-7$$.
Thus, the two statements combined are INSUFFICIENT.

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Re: What is the value of a+b?  [#permalink]

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27 Feb 2019, 07:29
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of $$a+b$$ ?

(1) $$a^2+ab-2a = 91$$
(2) $$b^2+ab-2b = -28$$

$$? = a + b$$

$$\left( 1 \right)\,\,a\left( {a + b - 2} \right) = 91\,\,\,\,:\,:\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {91,\, - 88} \right)\,\,\,\, \Rightarrow \,\,\,? = 3\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {1,92} \right)\,\,\,\, \Rightarrow \,\,\,? = 93\,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,b\left( {a + b - 2} \right) = - 28\,\,\,:\,:\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {31,\, - 28} \right)\,\,\,\, \Rightarrow \,\,\,? = 3\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 27,1} \right)\,\,\,\, \Rightarrow \,\,\,? = - 26\,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,\,\,\left( 1 \right)\left( + \right)\left( 2 \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left( {a + b - 2} \right)\left( {a + b} \right) = 63\,\,\,\, \Rightarrow \,\,\,\,{\left( {a + b} \right)^2} - 2\left( {a + b} \right) - 63 = 0$$

$$\left. \matrix{ {\rm{Sum}}:2 \hfill \cr {\rm{Product}}: - 63\,\, \hfill \cr} \right\}\,\,\,\, \Rightarrow \,\,\,?\,\,\,\,:\,\,\,\,\,\left( {\rm{i}} \right)a + b = \,9\,\,\,\,\,{\rm{or}}\,\,\,\,\,\,\left( {{\rm{ii}}} \right)a + b\, = - 7\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left( {\rm{E}} \right)$$

$$\left( * \right)\,\,{\rm{viability}}\,\,\,{\rm{:}}\,\,\,\,\left\{ \matrix{ \,\left( {\rm{i}} \right)\,\,a + b = \,9\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,7a = 91\,\,\,\, \Rightarrow \,\,\,\left( {a,b} \right) = \left( {13, - 4} \right)\,\,\,\,{\rm{viable}}!\,\,\,\,\,\,\left[ {\left( 1 \right),\left( 2 \right)\,\,{\rm{ok!}}} \right] \hfill \cr \,\left( {{\rm{ii}}} \right)\,\,a + b = \, - 7\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\, - 9a = 91\,\,\,\, \Rightarrow \,\,\,\left( {a,b} \right) = \left( { - {{91} \over 9},{{28} \over 9}} \right)\,\,\,\,{\rm{viable}}!\,\,\,\,\,\,\left[ {\left( 1 \right),\left( 2 \right)\,\,{\rm{ok!}}} \right] \hfill \cr} \right.$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: What is the value of a+b?  [#permalink]

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02 Mar 2019, 13:45
GMATGuruNY wrote:
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of $$a+b$$ ?

(1) $$a^2+ab-2a = 91$$
(2) $$b^2+ab-2b = -28$$

Statements combined:

Adding the two equations, we get:
$$a^2+b^2+2ab-2a-2b=63$$
$$(a+b)^2 - 2(a+b)=63$$
$$(a+b)(a+b-2)=63$$

Let $$x=a+b$$.

Substituting $$x=a+b$$ into $$(a+b)(a+b-2) = 63$$, we get:
$$(x)(x-2)=63$$
$$x^2-2x-63=0$$
$$(x-9)(x+7)=0$$
$$x=9$$ or $$x=-7$$

Since $$x=a+b$$, it is possible that $$a+b=9$$ or that $$a+b=-7$$.
Thus, the two statements combined are INSUFFICIENT.

Hi! Can you please explain How (a+b)^2 - 2(a+b)=63 became (a+b)(a+b-2)=63 ?
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Re: What is the value of a+b?  [#permalink]

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03 Mar 2019, 22:09
anmolgmat14 wrote:
GMATGuruNY wrote:
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of $$a+b$$ ?

(1) $$a^2+ab-2a = 91$$
(2) $$b^2+ab-2b = -28$$

Statements combined:

Adding the two equations, we get:
$$a^2+b^2+2ab-2a-2b=63$$
$$(a+b)^2 - 2(a+b)=63$$
$$(a+b)(a+b-2)=63$$

Let $$x=a+b$$.

Substituting $$x=a+b$$ into $$(a+b)(a+b-2) = 63$$, we get:
$$(x)(x-2)=63$$
$$x^2-2x-63=0$$
$$(x-9)(x+7)=0$$
$$x=9$$ or $$x=-7$$

Since $$x=a+b$$, it is possible that $$a+b=9$$ or that $$a+b=-7$$.
Thus, the two statements combined are INSUFFICIENT.

Hi! Can you please explain How (a+b)^2 - 2(a+b)=63 became (a+b)(a+b-2)=63 ?

Take (a+b) common..
Manager
Joined: 24 Jul 2018
Posts: 55
Location: India
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Re: What is the value of a+b?  [#permalink]

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03 Mar 2019, 22:14
1
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

What is the value of $$a+b$$ ?

(1) $$a^2+ab-2a = 91$$
(2) $$b^2+ab-2b = -28$$

1 more way I could deduce...Correct me if I am wrong...

(1) $$a^2+ab-2a=91$$
=> $$a(a+b-2)=91$$......................i

(2)$$b^2+ba-2b=-28$$
=>$$b(b+a-2)=-28$$.....................ii

Dividing i by ii,
$$\frac{a}{b}=\frac{-91}{28}$$

From here, we can't take out the value of a+b, hence option E.
Re: What is the value of a+b?   [#permalink] 03 Mar 2019, 22:14
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