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# m and n are two different numbers selected from the integers between

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6529
GMAT 1: 760 Q51 V42
GPA: 3.82
m and n are two different numbers selected from the integers between  [#permalink]

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30 Oct 2018, 17:35
1
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Difficulty:

5% (low)

Question Stats:

87% (00:48) correct 13% (02:02) wrong based on 31 sessions

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[Math Revolution GMAT math practice question]

$$m$$ and $$n$$ are two different numbers selected from the integers between $$11$$ and $$20$$, inclusive. What is the maximum value of $$\frac{mn}{(m-n)}$$?

$$A. 320$$
$$B. 340$$
$$C. 360$$
$$D. 380$$
$$E. 400$$

_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 484 Re: m and n are two different numbers selected from the integers between [#permalink] ### Show Tags 30 Oct 2018, 18:00 MathRevolution wrote: [Math Revolution GMAT math practice question] $$m$$ and $$n$$ are two different numbers selected from the integers between $$11$$ and $$20$$, inclusive. What is the maximum value of $$\frac{mn}{(m-n)}$$? $$A. 320$$ $$B. 340$$ $$C. 360$$ $$D. 380$$ $$E. 400$$ Beautiful problem, Max. Congrats! $$m,n\,\,{\rm{distinct}}\,\, \in \left\{ {11,12, \ldots ,20} \right\}$$ $$? = \max \left( {{{mn} \over {m - n}}} \right)$$ $$\left( {\rm{I}} \right)\,\,m > n\,\,\,\,\left( {{\rm{otherwise}}\,\,{{mn} \over {m - n}} < 0\,\,,\,\,{\rm{not}}\,\,\max } \right)$$ $$\left( {{\rm{II}}} \right)\,\,0 < {{mn} \over {m - n}} \le \max \left( {{{mn} \over {m - n}}} \right)\,\,\,\,\, \Leftrightarrow \,\,\,\,\,{1 \over n} - {1 \over m} = {{m - n} \over {mn}} \ge {\left[ {\max \left( {{{mn} \over {m - n}}} \right)} \right]^{ - 1}}$$ $$\left( {{\rm{III}}} \right)\,\,\,?\,\,\,\, \Leftrightarrow \,\,\,\,\min \left( {{1 \over n} - {1 \over m}} \right)\,\,\,\,\, \Leftrightarrow \,\,\,\left( {m,n} \right) = \left( {20,19} \right)$$ $$\left( {{\rm{IV}}} \right)\,\,\,? = {\left( {{{20 - 19} \over {20 \cdot 19}}} \right)^{ - 1}} = 20 \cdot 19 = 380$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount! Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6529 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: m and n are two different numbers selected from the integers between [#permalink] ### Show Tags 31 Oct 2018, 23:36 => To obtain the maximum value for $$\frac{mn}{(m-n)}$$, $$m-n$$ must be the smallest possible positive number and $$mn$$ must be the largest possible product. $$mn$$ is largest when both $$m$$ and $$n$$ are largest, that is, when $$m = 20$$ and $$n – 19$$. In this case, $$m – n = 1$$ is as small as possible and $$mn = 20*19 = 380$$ is as large as possible. Thus, the maximum value of $$\frac{mn}{(m-n)}$$ is $$\frac{20*19}{(20-19)} = \frac{380}{1} = 380.$$ Therefore, D is the answer. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
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Re: m and n are two different numbers selected from the integers between &nbs [#permalink] 31 Oct 2018, 23:36
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