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Does (a – 2)(b + 4) = 8?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by Brent@GMATPrepNow » Fri Jan 18, 2019 6:10 am
DivyaD wrote:Does (a - 2)(b + 4) = 8?
(1) ab = 2b - 4a
(2) a = 6
Target question: Does (a - 2)(b + 4) = 8?
This is a good candidate for rephrasing the target question.
When I SCAN the statements, I see that statement 1 has the term ab. This suggests to me that we might benefit from EXPANDING the expression, since that will result in an ab term.
Take: (a - 2)(b + 4) = 8
Use FOIL to expand left side: ab + 4a - 2b - 8 = 8
Add 8 to both sides to get: ab + 4a - 2b = 16
REPHRASED target question: Does ab + 4a - 2b = 16?

Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: ab = 2b - 4a
Hmm, this looks A LOT like our REPHRASED target question.
Take: ab = 2b - 4a
Add 4a to both sides: ab + 4a = 2b
Subtract 2b from both sides: ab + 4a - 2b = 0
So, the answer to the target question is NO, ab + 4a - 2b does NOT equal 16
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: a = 6
In order to answer the REPHRASED target question, we still need the value of b.
So, statement 2 is NOT SUFFICIENT

If you're not convinced, consider the following cases:
Case a: a = 6 and b = -2. So, ab + 4a - 2b = (6)(-2) + 4(6) - 2(-2) = 16. In this case, the answer to the REPHRASED target question is YES, ab + 4a - 2b equals 16
Case b: a = 6 and b = 0. So, ab + 4a - 2b = (6)(0) + 4(6) - 2(0) = 24. In this case, the answer to the REPHRASED target question is NO, ab + 4a - 2b does NOT equal 16
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
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by fskilnik@GMATH » Fri Jan 18, 2019 4:20 pm
DivyaD wrote:Does (a - 2)(b + 4) = 8?
(1) ab = 2b - 4a
(2) a = 6
$$\left( {a - 2} \right)\left( {b + 4} \right)\,\,\mathop = \limits^? \,\,8$$

$$\left( 1 \right)\,\,ab = 2b - 4a\,\,\,\,\mathop \Rightarrow \limits^{{\rm{focus}}!} \,\,\,\,a\left( {b + 4} \right) = 2b\,\,\,\,\mathop \Rightarrow \limits^{{\rm{focus}}!} \,\,\,\,\left( {a - \underline 2 } \right)\left( {b + 4} \right) + \underline {2\left( {b + 4} \right)} = 2b$$
$$\,\,\,\,\mathop \Rightarrow \limits^{{\rm{focus}}!} \,\,\,\,\left( {a - 2} \right)\left( {b + 4} \right) = 2b - 2\left( {b + 4} \right) = - 8 \ne 8\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$

$$\left( 2 \right)\,\,a = 6\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {6, - 4} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {6, - 2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$

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

Regards,
Fabio.
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