OG - DS - #89

[dk08]

If . represents one of the operations +, -, and x, is k. (l+m)=(k.l) + (k.m) for all numbers k, l, and m?

(1) k.1 is not equal to 1.k for some numbers k.

(2) . represents subtraction

I am not sure how to work through this?

[esledge]

First, you should try to rephrase the question. You know there are 3 possibilities for the operator: addition, subtraction, multiplication.

If the operator is addition, the question really is:
Is k + (l + m) = (k + l) + (k + m) for all numbers k, l, and m?
Is k + l + m = 2k + l + m for all numbers k, l, and m?
Is k = 2k for all values of k?
We can see that the answer is NO when the operator is addition.

If the operator is subtraction, the question really is:
Is k - (l + m) = (k - l) + (k - m) for all numbers k, l, and m?
Is k - l - m = 2k - l - m for all numbers k, l, and m?
Is k = 2k for all values of k?
We can see that the answer is NO when the operator is subtraction.

If the operator is multiplication, the question really is:
Is k * (l + m) = (k * l) + (k * m) for all numbers k, l, and m?
Is kl + km = kl + km for all numbers k, l, and m?
We can see that the answer is YES when the operator is multiplication.

So, if the answer must be NO when the operator is addition or subtraction, but the answer must be YES when the operator is multiplication, the ultimate question here is: "Is the operator multiplication, or one of the other two choices?"

Statement (1) SUFFICIENT:
Try out the three operators to see which one matches what the statement tells us.
Addition: (k + 1) = (1 + k) for all values. Therefore, the operator is not addition.
Subtraction: (k - 1) = (1 - k) when k = 1, but not for other values of k. Therefore, the operator could be subtraction.
Multiplication: (k * 1) = (1 * k) for all values. Therefore, the operator is not multiplication.

Thus, statement (1) tells us the operator is subtraction, sufficiently answering the rephrased question.

Statement (2) SUFFICIENT:
This is the easier statement, as it answers the rephrased question outright.

{By the way, there is an error in the OG explanation. In the discussion of statement (1), it says "k + 1 = 1 + k, and also k - 1 = 1 - k." They meant "...and also k * 1 = 1 * k."}

[dk08]

thank you

[v pat]

If "." is "-" then k. (l+m)=(k.l) + (k.m) reduces to k = 2k which can be true if k=0 and nothing else. Hence st1 could be yes or no depending on k.
St2 tells us the same thing. Hence I guess the ans is E. If not please explain.

First, you should try to rephrase the question. You know there are 3 possibilities for the operator: addition, subtraction, multiplication.

If the operator is addition, the question really is:
Is k + (l + m) = (k + l) + (k + m) for all numbers k, l, and m?
Is k + l + m = 2k + l + m for all numbers k, l, and m?
Is k = 2k for all values of k?
We can see that the answer is NO when the operator is addition.

If the operator is subtraction, the question really is:
Is k - (l + m) = (k - l) + (k - m) for all numbers k, l, and m?
Is k - l - m = 2k - l - m for all numbers k, l, and m?
Is k = 2k for all values of k?
We can see that the answer is NO when the operator is subtraction.

If the operator is multiplication, the question really is:
Is k * (l + m) = (k * l) + (k * m) for all numbers k, l, and m?
Is kl + km = kl + km for all numbers k, l, and m?
We can see that the answer is YES when the operator is multiplication.

So, if the answer must be NO when the operator is addition or subtraction, but the answer must be YES when the operator is multiplication, the ultimate question here is: "Is the operator multiplication, or one of the other two choices?"

Statement (1) SUFFICIENT:
Try out the three operators to see which one matches what the statement tells us.
Addition: (k + 1) = (1 + k) for all values. Therefore, the operator is not addition.
Subtraction: (k - 1) = (1 - k) when k = 1, but not for other values of k. Therefore, the operator could be subtraction.
Multiplication: (k * 1) = (1 * k) for all values. Therefore, the operator is not multiplication.

Thus, statement (1) tells us the operator is subtraction, sufficiently answering the rephrased question.

Statement (2) SUFFICIENT:
This is the easier statement, as it answers the rephrased question outright.

{By the way, there is an error in the OG explanation. In the discussion of statement (1), it says "k + 1 = 1 + k, and also k - 1 = 1 - k." They meant "...and also k * 1 = 1 * k."}

[dbernst]

v. pat,

Your confusion with this question is not surprising. At MGMAT, we have a saying about this dastardly test: "The Math is the Verbal, and the Verbal is the Math." The first portion of this mantra suggests that the Quant section of the GMAT is often confusing not because the mathematical operations are overly complex, but because the WORDING of the questions is so confounding (the second half of the mantra refers to the necessity of adopting a structured, logical, "mathematical" approach to each verbal question type, processes we discuss on the Verbal sections of the Forum).

In the problem at hand, the question asks, "If . represents one of the operations +, -, and x, is k. (l+m)=(k.l) + (k.m) for all numbersk, l, and m?" As Emily eloquently explained in her response, each statement indicates that the dot must be subtraction. Thus, you are exactly correct in your mathematical assessment: If k = 0, k=2k; If k = anything else, k does not = 2k. However, let's look more closely at the wording of the question. The question asked whether the equation holds true "for all numbers k, l, and m?" You just proved to me that the equation DOES NOT hold true for all numbers k, l, and m. Since you can definitively answer the question (the answer is NO!), each statement is sufficient to answer the question, and the correct answer is D.

Hope that makes sense
-dan

If "." is "-" then k. (l+m)=(k.l) + (k.m) reduces to k = 2k which can be true if k=0 and nothing else. Hence st1 could be yes or no depending on k.
St2 tells us the same thing. Hence I guess the ans is E. If not please explain.