Is k + m < 0 ? (1) k < 0 (2) km > 0

1) K <0, but this does not say anything about m. When k = -2 and m = 3, k +m > 0 and when k = -2 and m = -1, K + m < 0.
Not Sufficient

2) km > 0. Both k and m are positive (k +m > 0 ) or both k,m are negative (k+m < 0) . Not sufficient

When you combine both statements, you can conclude that k < 0 (statement 1) and m <0 (from statement 2, km > 0, so if k <0, then m <0). So, k + m < 0.

My answer would be C.

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Since we have 2 variables (\(k\) and \(m\)) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since \(k<0\) and \(km>0\), we have \(m<0\).
Thus we have \(k + m < 0\) and the answer is 'yes'.
Since both conditions together yield a unique answer, they are sufficient.

Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since we don't have any information about \(m\), condition 1) is not sufficient obviously.

Condition 2)
If we have \(k = -1\) and \(m = -1\), then we have \(k + m = -2 < 0\) and the answer is 'yes'.
If we have \(k = 1\) and \(m = 1\), then we have \(k + m = 2 > 0\) and the answer is 'no'.
Since condition 2) does not yield a unique answer, it not sufficient.

Therefore, C is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

Solution:

Question Stem Analysis:

We need to determine whether k + m < 0.

Statement One Alone:

We are given that the value of k is negative. However, without knowing anything about the value of m, we can’t determine whether k + m < 0. Statement one alone is not sufficient.

Statement Two Alone:

We are given that the value of km is positive, which means either k and m are both positive or they are both negative. If they are both positive, then k + m is not less than 0. On the other hand, if they are both negative, then k + m is less than 0. Statement two is not sufficient.

Statements One and Two Together:

From statement one, we know that k is negative. From statement two, we know that km > 0. Thus, m must also be negative. Since both k and m are negative, we see that k + m is indeed less than 0. Both statements are sufficient.

Answer: C
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Here instead of taking cases we can only consider what sign will hold the equation :
Is k + m < 0 ?
this means is
1) k and m both are -ve
2) one is neg and its magnitude is greater than the +ve one

(1) k < 0 : this means k is -ve but we don't know about m which can be -ve or +ve - Not sufficient.
(2) km > 0 : this means both are having same sign either +ve or -ve - Not sufficient.

(1) &(2) : combine give me k is -ve and km > 0 means m has to be -ve
so both are -ve -- Sufficient as per 1) condition.

And C

Bunuel

Can this question be solved algebraically?

why can't I assume from the question that k < 0-m and B would be sufficient?