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Is xy>3?

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Is xy>3?

by fskilnik@GMATH » Tue Mar 05, 2019 12:52 pm

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GMATH practice exercise (Quant Class 14)

Is xy > 3 ?

(1) (7^x) > 729
(2) (9^y) = 7

Answer: [spoiler]_____(C)__[/spoiler]

P.S.: this IS in GMAT´s quant section scope.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Source: — Data Sufficiency |
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by fskilnik@GMATH » Tue Mar 05, 2019 1:11 pm

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fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 14)

Is xy > 3 ?

(1) (7^x) > 729
(2) (9^y) = 7
$$xy\,\,\mathop > \limits^? \,\,3$$
$$\left( 1 \right)\,\,{7^x} > 729\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {4,0} \right)\,\,\,\,\,\left[ {{7^4} = {{49}^2}} \right]\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {4,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,{9^y} = 7\,\,\,\, \Rightarrow \,\,\,y = {y_p}\,\,\,{\rm{unique}}\,\,{\rm{,}}\,\,\,{1 \over 2}\,\,{\rm{ < }}\,\,{y_p} < 1\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {0,{y_p}} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {6,{y_p}} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,{3^6} = 729\,\,\,\mathop < \limits^{\left( 1 \right)} \,\,\,{7^x}\,\,\mathop = \limits^{\left( 2 \right)} \,\,\,{\left( {{9^y}} \right)^x} = {3^{2xy}}\,\,\,\,\,\mathop \Rightarrow \limits^{3\,\, > \,\,1} \,\,\,2xy > 6\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$


The correct answer is (C).


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

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Fabio.
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by Brent@GMATPrepNow » Tue Mar 05, 2019 2:13 pm

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fskilnik@GMATH wrote:Is xy > 3 ?

(1) (7^x) > 729
(2) (9^y) = 7
Target question: Is xy > 3 ?

Statement 1: (7^x) > 729
Since there's no information about y, we cannot answer the target question with certainty.
Statement 1 is NOT SUFFICIENT

Statement 2: (9^y) = 7
Since there's no information about x, we cannot answer the target question with certainty.
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that (7^x) > 729
Statement 2 tells us that (9^y) = 7

Take the inequality (7^x) > 729, and replace 7 with 9^y to get: (9^y)^x > 729
Simplify to get: 9^xy > 729
Rewrite 729 as 9^3 to get: 9^xy > 9^3
From this, we can conclude that xy > 3
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
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