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What is a–b?

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Sun Sep 09, 2018 3:56 am
BTGmoderatorDC wrote:What is a-b?

(1) a^3 - b^3 = 117
(2) a^2 + ab + b^2 = 39

OA C

Source: Veritas Prep
Note that a^3 - b^3 = (a - b) (a^2 + ab + b^2).

=> a - b = (a^3 - b^3) / (a^2 + ab + b^2)

Since each statement alone cannot return the unique value of (a - b), we need the values of (a^3 - b^3) and (a^2 + ab + b^2).

Thus, a - b = (a^3 - b^3) / (a^2 + ab + b^2) = 117/39 = 3. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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by fskilnik@GMATH » Sun Sep 09, 2018 12:00 pm
BTGmoderatorDC wrote:What is a-b?

(1) a^3 - b^3 = 117
(2) a^2 + ab + b^2 = 39

Source: Veritas Prep
\[? = a - b\]
\[\left( 1 \right)\,\,{a^3} - {b^3} = 117\,\,\,\left\{ \begin{gathered}
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {\sqrt[{3\,}]{{117}}\,\,;\,\,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = \sqrt[{3\,}]{{117}}\,\,\, \hfill \\
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {1\,\,;\,\, - \sqrt[{3\,}]{{116}}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = 1 + \sqrt[{3\,}]{{116}}\,\,\, \ne \sqrt[{3\,}]{{117}}\,\, \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,{a^2} + ab + {b^2} = 39\]
\[\left\{ \begin{gathered}
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {\sqrt {39} \,\,;\,\,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = \sqrt {39} \,\,\, \hfill \\
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {0\,\,;\,\,\sqrt {39} } \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = - \sqrt {39} \,\,\, \ne \sqrt {39} \,\, \hfill \\
\end{gathered} \right.\,\,\,\]
\[\left( {1 + 2} \right)\,\,\,\left\{ \begin{gathered}
\,\,\left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) = {a^3} - {b^3} = 117 \hfill \\
\,\,{a^2} + ab + {b^2} = 39 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,39\left( {a - b} \right) = 117\,\,\,\,\, \Rightarrow \,\,? = a - b\,\,\,{\text{unique}}\,\,\]

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

Regards,
fskilnik.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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