If ° represents one of the operations…
We are given that ° represents one of the following operations: addition, subtraction, or multiplication, and we must determine whether k ° (l + m) = (k ° l) + (k ° m).
Statement One Alone:
k ° 1 is not equal to 1 ° k for some numbers k.
Using the information in statement one, we see that ° can’t represent multiplication or addition. If ° represents multiplication or addition, then we would have k ° 1 = 1 ° k (notice that k x 1 = 1 x k and k + 1 = 1 + k are true for any values of k). However, we are given that k ° 1 ≠ 1 ° k.
Thus, we know that ° must represent subtraction and that is enough information to answer the question of whether k ° (l + m) = (k ° l) + (k ° m). We could stop at this point because we have a definitive answer to the question, but let’s clarify by using an example.
We can illustrate this by substituting the subtraction sign for the ° sign and using numbers. Let’s have k be 10, l be 4, and m be 3.
k ° (l + m) = (k ° l) + (k ° m) ?
k – (l + m) = (k – l) + (k – m) ?
10 – (4 + 3) = (10 – 4) + (10 – 3) ?
10 – 7 = 6 + 7 ?
3 ≠ 13
We see that, when we carry out the calculations with ° representing subtraction, k ° (l + m) is NOT EQUAL to (k ° l) + (k ° m), and therefore, the answer to the question is NO.
Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.
Statement Two Alone:
° represents subtraction.
Since we’ve already determined that ° represents subtraction from statement one, that is enough information to answer the question of whether k ° (l + m) = (k ° l) + (k ° m). Statement two alone is sufficient to answer the question.