Welcome back to this week’s Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!
If ab ≠ 0 and a ≠ -3b, what is the value of (4a + 6b)/(a + 3b)?
(1) a – 3b = 6
(2) 2a/(a + 3b) = 4
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.
We cannot simplify the given expression very much, because the denominator (which is a sum, a + 3b) is not a factor of the numerator. If we really wanted to, we could split the numerator and write the expression as a sum:
Or we could leave the question as is. Either way, be sure not to cancel any of the coefficients, because the denominator is a sum – we can’t simply cancel the 6 in the numerator with the 3 in the denominator, for instance.
Statement 1: INSUFFICIENT. This gives us a relationship between a and b. However, if we use it to solve for one of the variables and then we substitute that expression into the question, we’ll quickly see that we will not get a single number:
From the statement: a = 6 + 3b
Substitute into the original question:
We can stop here if we see that the denominator is 6 + 6b, which will not cancel with the numerator of the combined fraction (which equals 24 + 15b).
Statement 2: SUFFICIENT. We can get a constant ratio between a and b, which will actually cancel in the question.
From the statement:
2a = 4a + 12b
-2a = 12b
a = -6b
Substitute into the question:
Note that it is okay to cancel out the b’s, since ab ≠ 0 and thus neither variable equals 0.
As long as we have a constant ratio between a and b, we will get a number out of an expression such as .
The correct answer is (B): Statement 2 is sufficient, but Statement 1 is not sufficient.
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