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Are positive integers j and k both greater than m?
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14 Dec 2011, 09:31
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Are positive integers j and k both greater than m? (1) \(m + k  j\) is negative. (2) \(k  j  m\) is positive. Can we solve it algebraicaly?
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Re: Are positive integers j and k both greater than m?
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14 Dec 2011, 12:20
I do not know if this is the right approach but this is how I solved it (after quite some time though).
Given: j > 0; k > 0
Statement 1: m + k  j < 0 Thus m + k < j or j > (k + m) Now, j and k are both positive. So this inequality shows that j is greater than the sum of a positive number k and m(the sign of m does not matter). Thus, j > m
Statement 2: k  j  m > 0 Thus, k > (j + m) Using similar logic as above, we can understand that k is also greater than the sum of a positive number j and m. Thus k > m
Thus the individual statements answer about each j and m, and together they answer the question. That is why the correct answer should be C.
Please suggest a more efficient method to solve.




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Re: Are positive integers j and k both greater than m?
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14 Dec 2011, 14:03
siddharthmuzumdar wrote: I do not know if this is the right approach but this is how I solved it (after quite some time though).
Given: j > 0; k > 0
Statement 1: m + k  j < 0 Thus m + k < j or j > (k + m) Now, j and k are both positive. So this inequality shows that j is greater than the sum of a positive number k and m(the sign of m does not matter). Thus, j > m
Statement 2: k  j  m > 0 Thus, k > (j + m) Using similar logic as above, we can understand that k is also greater than the sum of a positive number j and m. Thus k > m
Thus the individual statements answer about each j and m, and together they answer the question. That is why the correct answer should be C.
Please suggest a more efficient method to solve. Wonderful! I think it is a good approach. Thank you. Kudos to you!
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Re: Are positive integers j and k both greater than m?
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14 Dec 2011, 18:21
metallicafan wrote: Are positive integers j and k both greater than m?
(1) \(m + k  j\) is negative. (2) \(k  j  m\) is positive.
Can we solve it algebraicaly? Neither statement is sufficient alone here (there's no way to say which of m or k is larger in statement 1, and no way to say which of m and j is larger in statement 2). We can combine the statements directly: if two inequalities face the same way, we can *add* them (but note that we cannot subtract them) just as we do with equations. So writing the inequalities so they face the same way and adding them we find: \(\begin{align*} 0 &> m + k  j \\ k j  m &> 0 \\ k  j  m &> m + k  j \\ 0 &> 2m \\ 0 &> m \end{align*}\) If m is negative, it's definitely smaller than any positive number, so it's smaller than j and k, and the answer is C.
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Re: Are positive integers j and k both greater than m?
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15 Dec 2011, 02:52
IanStewart wrote: Neither statement is sufficient alone here (there's no way to say which of m or k is larger in statement 1, and no way to say which of m and j is larger in statement 2). We can combine the statements directly: if two inequalities face the same way, we can *add* them (but note that we cannot subtract them) just as we do with equations. So writing the inequalities so they face the same way and adding them we find:
\(\begin{align*} 0 &> m + k  j \\ k j  m &> 0 \\ k  j  m &> m + k  j \\ 0 &> 2m \\ 0 &> m \end{align*}\)
If m is negative, it's definitely smaller than any positive number, so it's smaller than j and k, and the answer is C. I guess this is the best approach. metallicafan, I guess you found your best answer now. My method also works but is than the one mentioned here.
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Re: Are positive integers j and k both greater than m?
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15 Dec 2011, 06:18
siddharthmuzumdar wrote: I guess this is the best approach.
metallicafan, I guess you found your best answer now. My method also works but is than the one mentioned here. Your approach is also very good  if the question asked something different, you might need to use an approach similar to yours. For example, if the question had asked if j > m, then you would need to analyze each statement in the way you've done. As with many higher level questions, there are often several ways to get to an answer.
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Re: Are positive integers j and k both greater than m?
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13 Nov 2017, 01:33
Thought of this way. Kudos, if you felt it was useful 1) m + k  j = 1 (as answer is negative) m = j 1k m = j  k 1 m = j 1(k+1) Thus j > m (cause j minus values = m)K we don't know about > m + 1=k +1 (if we remove j), so m = K. But we don't know for sure. 2) k  j  m = 1 k = 1 + j + m k  1(j +1) = m. Thus K > m. J we are not sure. 1) + 2) j, k > m. Thus C.



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Re: Are positive integers j and k both greater than m?
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13 Nov 2017, 06:53
metallicafan wrote: Are positive integers j and k both greater than m?
(1) \(m + k  j\) is negative. (2) \(k  j  m\) is positive.
Can we solve it algebraicaly? As we can not solve it with statements 1 &2 individually: Just subtract the statement 1 from statement 2 (Positive(Negative)) will always be positive.After subtraction, we get 2m >0, so m is negative. As J & K are mentioned as positives, they will always be greater than m. Hence C is the answer.




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