There are a handful of ways to solve this problem. Let's look at two of them. First, we could solve this with just the raw algebra. The question tells us that the original ratio of

blue to

red pens is

5:

7. Thus,

\(B = 5x\) and

\(R = 7x\) (where \(x\) is the scaling factor of the ratio.) We don't know what "\(x\)" is yet, but thinking of the ratio in these terms allows us to quickly simplify down to one variable.

The problem then tells us that if we were to add 3 to

\(B\) (in other words,

\(5x +3\)) and subtract 9 from

\(R\) (in other words,

\(7x-9\)), the new ratio would be

3:

2. Mathematically, it would look like this:

\(\frac{5x+3}{7x-9} = \frac{3}{2}\)

Cross-multiplying the fractions gives us:

\(2(5x+3) = 3(7x-9)\)

\(10x+6 = 21x - 27\)

\(11x = 33\)

\(x = 3\)

Note how "3" is a possible answer. However, this is a classic trap of the GMAT: including the "right answer to the wrong question" as one of the answer choices. We were not asked to solve for the original scaling factor prior to the change, \(x\). (Incidentally, if we solve for the number of

blue pens after the change, we arrive at 15, another trap answer!) Make sure you focus on the actual target of the question: the number of

red pens after the change. Thus, we need

\(7x - 9\). Plugging \(x\) into our equation gives us 12.

The answer is (D).There is a totally different way to solve this problem as well, using the answer choices as leverage. (I call this tactic "Look Out Below!") The nature of ratios sometimes allows us to compare the answer choices against each other and eliminate any answer choice that doesn't "follow the rules." For example, the problem tells us that after the change, ratio of

blue pens to

red pens should be

3:

2 (or

3x:

2x). Since you can't have fractional pens, this means that the number of

red pens must be a

multiple of 2. And, since answer choices (A), (B), and (C) are all

odd, we can eliminate these. We are down to only (D) and (E).

Before the change (in other words, before we added 3 blue pens and subtracted 9 pens), the ratio was

5x:

7x.Thus, adding +9 back to answer choices (D) and (E) would give us the theoretical "original" value for

\(R\), which should be a

multiple of 7.

Adding 9 to answer choice (D) gives us 21 -- a multiple of 7 and therefore a possible candidate.

Adding 9 to answer choice (E) gives us 27 -- not a multiple of 7. We can eliminate this, leaving only one answer, (D).

Any way you solve it, the answer is still (D).
_________________

Aaron J. Pond

Veritas Prep Elite-Level Instructor

Hit "+1 Kudos" if my post helped you understand the GMAT better.

Look me up at https://www.veritasprep.com/gmat/aaron-pond/ if you want to learn more GMAT Jujitsu.