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If x and y are positive integers, is x/y < (x+5)/(y+5)?

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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Sun Sep 30, 2018 6:34 am
BTGmoderatorDC wrote:If x and y are positive integers, is x/y < (x+5)/(y+5)?

(1) y = 5
(2) x > y
Given: x and y are positive integers

Target question: Is x/y < (x+5)/(y+5)?
This is a good candidate for rephrasing the target question.

Since y is POSITIVE, we can safely take the inequality x/y < (x+5)/(y+5) and multiply both sides by y.
When we do this, we get: x < (y)(x+5)/(y+5)

Also, since y is POSITIVE, we also know that (y + 5) is POSITIVE
So, let's take the inequalityx < (y)(x+5)/(y+5) and multiply both sides by (y + 5).
We get: (x)(y+5) < (y)(x+5)
Expand both sides to get: xy + 5x < xy + 5y
Subtract xy from both sides to get: 5x < 5y
Divide both sides by 5 to get: x < y
We now have a very simple, REPHRASED target question. . . . .
REPHRASED target question: Is x < y?

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

Statement 1: y = 5
Since we have no information about x, there's no way to determine whether x < y
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x > y
Perfect.
With information, we know (with certainty) that the answer to the REPHRASED target question is NO, x is NOT less than y
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by fskilnik@GMATH » Sun Sep 30, 2018 1:48 pm
BTGmoderatorDC wrote:If x and y are positive integers, is x/y < (x+5)/(y+5)?

(1) y = 5
(2) x > y
Source: Magoosh
$$x,y\,\, \ge 1\,\,\,{\rm{ints}}$$
$${x \over y}\,\,\mathop < \limits^? \,\,{{x + 5} \over {y + 5}}$$
$$\left( 1 \right)\,\,y = 5\,\,\,\,\, \Rightarrow \,\,\,\,\,{x \over 5}\,\,\mathop < \limits^? \,\,{{x + 5} \over {10}}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,10} \,\,\,\,\,\,\,\,2x\,\,\mathop < \limits^? \,\,x + 5$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,x = 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,x = 5\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,{x \over y}\,\,\mathop < \limits^? \,\,{{x + 5} \over {y + 5}}\,\,\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{y\,\,\left( {y + 5} \right)\,\, > \,\,0} \,\,\,\,\,\,\,x\,\left( {y + 5} \right)\,\,\mathop < \limits^? \,\,y\left( {x + 5} \right)\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\,\mathop < \limits^? \,y\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,$$

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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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