Jan
22
2013

# GRE Arithmetic: Mixtures, Again!

Last night I TA-ed a “GRE Bootcamp” online event hosted by Lee Weiss. We won’t be running that particular event again for a while, so I figured I’d cover what a lot of the 400+ attendees considered the most challenging problem of the evening. Lee did an excellent job explaining it, but even so we got lots of requests in the Q&A to “go back to that mixture problem again!”

If you were at the event: hi! You might remember me as “Kaplan GRE Expert Boris.” It so happens that I’ve written before about how to solve this very type of problem, but concepts that the GRE repeats are always worth a second look.

Here was the question that confounded so many of you:

Liquid X is composed of 30% alcohol and 70% water, while Liquid Y is composed of 18% alcohol and 82% water. The two liquids are combined to form Mixture Z, which is composed of 21% alcohol and 79% water. What is the ratio in Mixture Z of Liquid X to Liquid Y?

Now let’s solve it together. We can start by cleaning up the problem a bit. If X has 30% alcohol, it’s redundant to say that it has 70% water — the GRE just puts that there to overwhelm you with extra numbers. Same goes for Y (18% alcohol) and mixture Z (21%) alcohol. You’ll find that you can usually summarize the information much better than the GRE does!

Liquid X: 30% alcohol

Liquid Y: 18% alcohol

Mixture Z (X + Y): 21% alcohol

The question is, what’s the ratio of X to Y? Note that for simplicity’s sake, I’ve left the choices out of the picture — you can find the answer straight-up.

To solve a problem like this, you can spare yourself a heck of a headache if you use the balance approach. Imagine you’re a mad scientist, adding “parts” of X and Y to a bubbling flask until you get just the right concentration of alcohol.

Every “part” of X you add gives 9% too much alcohol to the mixture. X is 30% alcohol, but the mixture is only 21%: 30 – 21 = 9.

Every “part” of Y you add gives you 3% too little alcohol. Y is 18% alcohol, and 21-18 = 3.

You need those 3′s and 9′s to cancel out if you want just the right percentage of alcohol. To do that, of course, you need three 3′s for every 9, since 3 × 3 = 9. This means that there’s one “part” of X for every three “parts” of Y — in other words, that the ratio of X to Y is 1:3.

(Sidenote: in the problem, one of the choices was 3:1, which is the ratio of Y:X, not X:Y. Always take care, when you finish solving a GRE quantitative problem, to confirm that you answered the question the test makers actually asked! Nothing is more heartbreaking than losing a GRE point because you got the right answer to the wrong question.)

Still got questions about mixtures? Let us know in the comments below!

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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.

• sharingjoy

First off- If you like board games about Portuguese spice merchants, then have you tried SmallWorld? Very fun boardgame, in all its permutations. You probably already own Agricola and Dominion by the sound of it. Both are also fun. Second- how does the new scoring system work when you have universities whose graduate programs recommend or require a minimum online score of 61 or 71 for example? I have found nothing online regarding Kaplan’s online scoring system and the appropriate bell curve accordingly.

• Boris Dvorkin

I have played Small World! In fact, I used to own it, but it wasn’t quite the right game for me, so I sold it. It was fiddly and a bit too random for my tastes. My roommate owns all of Dominion, and Agricola is one my favorite games ever.

As for your question, I’m afraid I must confess I am less helpful there. Where have you seen online scores of 61 or 71? The quant and verbal sections produce scores of 130 to 170, and the essay is from 1 to 6, so I don’t know where you would be getting 61 and 71.

• Linda Flinn

where did you get the 9 from in liquid mixture?

• Boris Dvorkin

Good question! Liquid X is 30% alcohol. The question wants a mixture that’s 21% alcohol. Note that 30 – 21 = 9: that’s where the 9 comes from. Liquid X has 9% too much alcohol.

• sharifulislam

We only calculate the ratio of alcohol.But why?The mixture is the combined compound of both alcohol and water.Why we only calculate the ratio of alcohol?

• Boris Dvorkin

Ahhhh, that is because once you know the percent of alcohol, you ALSO know the percent of water! For example, let’s say you have a mixture that’s 40% alcohol. I don’t need to tell you that the mixture is 60% water — you already know that.

You could just as easily solve this problem using the percentages of water. But because one percentage determines the other, you *never* need to use both.