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

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Math Revolution GMAT Instructor
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m and n are two different numbers selected from the integers between  [#permalink]

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New post 30 Oct 2018, 17:35
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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 \(\frac{mn}{(m-n)}\)?

\(A. 320\)
\(B. 340\)
\(C. 360\)
\(D. 380\)
\(E. 400\)

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Re: m and n are two different numbers selected from the integers between  [#permalink]

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New post 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.
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Re: m and n are two different numbers selected from the integers between  [#permalink]

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New post 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
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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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