Are positive integers j and k both greater than m?

Question

(1)  m + k - j  is negative.
(2)  k - j - m  is positive.

Can we solve it algebraicaly?

IanStewart · Accepted Answer

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.

siddharthmuzumdar · Answer

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.

metallicafan · Answer

Wonderful! I think it is a good approach.
Thank you. Kudos to you!

siddharthmuzumdar · Answer

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.

IanStewart · Answer

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.

Madhavi1990 · Answer

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.

ImAnkitKaushik · Answer

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.

bumpbot · Answer

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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?

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