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Are positive integers j and k both greater than m?

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Are positive integers j and k both greater than m?  [#permalink]

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New post 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?  [#permalink]

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New post 14 Dec 2011, 12:20
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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?  [#permalink]

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New post 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?  [#permalink]

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New post 14 Dec 2011, 18:21
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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?  [#permalink]

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New post 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?  [#permalink]

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New post 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?  [#permalink]

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New post 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- 1-k
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?  [#permalink]

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New post 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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Re: Are positive integers j and k both greater than m? &nbs [#permalink] 13 Nov 2017, 06:53
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