Mar
6
2013

# GRE Math Trick: Ratio Tables

When a GRE quantitative problem features multiple ratios, many of you suffer headaches. This is because the “math” way of solving the problem is brutal, and students who don’t use logic will dive head-first into a morass of ugly substitutions, mistakenly assuming that the GRE is a math test. Here’s the kind of problem I’m talking about:

In a particular mixed candy bag, the ratio of Skittles to M&M’s is 4 to 5, while the ratio of Reese’s Pieces to M&M’s is 9 to 7. What is the ratio of Skittles to Reese’s Pieces?

The “math” way to do this problem is to set up two equations, solve one for M&M’s, and plug that value into the other one. If that sounds painful, that’s because it is. Don’t do this. Make a simple table instead:

```S | M | R 4 : 5     7 : 9```

Take a moment to confirm that you understand where the numbers above are coming from. They’re just a translation of the information in the word problem.

The question asks for the ratio of S to R. Can you just say it’s 4 to 9? No way. The value connecting them — the M — is different. It’s 5 in one ratio and 7 in the other. So, rewrite the ratios to make the M term the same in both, creating a kind of “bridge.”

```Multiply the first ratio by 7:  7×(4:5) = 28:35 Multiply the second ratio by 5: 5×(7:9) = 35:45```

Next, check out your new table:

```S  |  M |  R 28 : 35      35 : 45```

Now you can just “walk across the bridge,” as it were — the ratio of S to R is simply 28:45. Try this technique on your next multiple-ratios problem and let us know how it goes!

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#### About the Author: Boris Dvorkin

After picking up degrees in English and computer science from Case Western, Boris Dvorkin worked for six unfortunate months as a computer programmer before finding a home at Kaplan in May 2008. He is now a full-time GRE faculty member on-site and online, and he's worked on Kaplan's curriculum for the recent GRE revision. Boris was named Kaplan's Teacher of the Year for 2010. When he's not gushing about standardized test trivia, Boris enjoys playing obscure strategy board games, and is the proud owner of no less than three different board games about Portuguese spice merchants.

• Stephen

Sigh.

The author’s claim that his approach isn’t math is mind-boggling. His claim that there is one “math” approach–”the” math way–is simply wrong, as he goes on to demonstrate. Although I would rather not think of students relying on rote memorization of specific shortcuts for specific problems, at least the author provides some of the mathematical reasoning underlying his approach when he talks about a “bridge” between the two units. (Personally, I would teach this with a focus on what units tell us, set it up so that the units cancel, and get the answer more directly, but his “trick” works.)

I’ve read a number of the author’s columns now about how the GRE isn’t a math test, and the main conclusion I’m forced to reach is that this guy has an incredibly narrow idea of what “math” is. At least he isn’t letting that narrowness keep him from actually using and teaching the broader toolbox of math skills. But in this column I don’t see the rhetorical or pedagogical value of saying the GRE test doesn’t test math, as opposed to saying that it’s probably worth taking a breath before starting to do a problem through brute force and asking whether there’s a simpler way. (In this case, the simplest way would be to just divide the ratios already, but I can see that that approach is non-obvious and probably not the most pedagogically accessible approach. It can get deeply frustrating teaching graduate students who have just memorized these sorts of “shortcuts,” though, because then they have a very hard time adapting to even slightly different questions. “Cross-multiplying” is a training wheel that students should have left behind by the time they reach my graduate economics course.)

• Boris Dvorkin

Hi Stephen! I’m sorry that you’ve found some of my articles frustrating. I’ve enjoyed reading your comments and, even though we ultimately disagree, I appreciate the time and care you put into composing them.

One thing we do agree on is that your graduate-level economics students should be able to solve the problem I posted without any help from you or from me. I urge you to bear in mind that for most of my students, unlike for most of yours, math plays no role in their graduate ambitions, so if the training wheels help ‘em get a higher GRE score, then they can keep the training wheels on forever, so far as I’m concerned.