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What is the value of a+b?

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

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New post 26 Feb 2019, 10:58
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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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Re: What is the value of a+b?  [#permalink]

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

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New post 03 Mar 2019, 22:14
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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.
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Re: What is the value of a+b?   [#permalink] 03 Mar 2019, 22:14
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