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Welcome to the 2nd installment of our dive into Number Properties. If you haven’t yet tried the first problem, start with the first article in the series.
Let’s dive right into our second problem from the GMATPrep® free exams: Read more
If you have two equations, you can solve for two variables.
This rule is a cornerstone of algebra. It’s how we solve for values when we’re given a relationship between two unknowns:
If I can buy 2 kumquats and 3 rutabagas for $16, and 3 kumquats and 1 rutabaga for $9, how much does 1 kumquat cost?
We set up two equations:
2k + 4r = 16
3k + r = 9
Then we can use either substitution or elimination to solve. (Try it out yourself; answer* below).
On the GMAT, you’ll be using the “2 equations à 2 variables” rule to solve for a lot of word problems like the one above, especially in Problem Solving. Be careful, though! On the GMAT this rule doesn’t always apply, especially in Data Sufficiency. Here are some sneaky exceptions to the rule…
2 Equations aren’t always 2 equations
Memorize what? I’m not going to tell you yet. Try this problem from the GMATPrep® free practice tests first and see whether you can spot the most efficient solution.
All right, have you got an answer? How satisfied are you with your solution? If you did get an answer but you don’t feel as though you found an elegant solution, take some time to review the problem yourself before you keep reading.
Step 1: Glance Read Jot
Take a quick glance; what have you got? PS. A given equation, xy = 1. A seriously ugly-looking equation. Some fairly “nice” numbers in the answers. Hmm, maybe you should work backwards from the answers?
Jot the given info on the scrap paper.
Step 2: Reflect Organize
Oh, wait. Working backwards isn’t going to work—the answers don’t stand for just a simple variable.
Okay, what’s plan B? Does anything else jump out from the question stem?
Hey, those ugly exponents…there is one way in which they’re kind of nice. They’re both one of the three common special products. In general, when you see a special product, try rewriting the problem usually the other form of the special product.
Step 3: Work
Here’s the original expression again:
Interesting. I like that for two reasons. First of all, a couple of those terms incorporate xy and the question stem told me that xy = 1, so maybe I’m heading in the right direction. Here’s what I’ve got now:
And that takes me to the second reason I like this: the two sets of exponents look awfully similar now, and they gave me a fraction to start. In general, we’re supposed to try to simplify fractions, and we do that by dividing stuff out.
How else can I write this to try to divide the similar stuff out? Wait, I’ve got it:
They’re almost identical! Both of the terms cancel out, as do the terms, leaving me with:
I like that a lot better than the crazy thing they started me with. Okay, how do I deal with this last step?
First, be really careful. Fractions + negative exponents = messy. In order to get rid of the negative exponent, take the reciprocal of the base:
Next, dividing by 1/2 is the same as multiplying by 2:
That multiplies to 16, so the correct answer is (D).
Key Takeaways: Special Products
(1) Your math skills have to be solid. If you don’t know how to manipulate exponents or how to simplify fractions, you’re going to get this problem wrong. If you struggle to remember any of the rules, start building and drilling flash cards. If you know the rules but make careless mistakes as you work, start writing down every step and pausing to think about where you’re going before you go there. Don’t just run through everything without thinking!
(2) You need to memorize the special products and you also need to know when and how to use them. The test writers LOVE to use special products to create a seemingly impossible question with a very elegant solution. Whenever you spot any form of a special product, write the problem down using both the original form and the other form. If you’re not sure which one will lead to the answer, try the other form first, the one they didn’t give you; this is more likely to lead to the correct answer (though not always).
(3) You may not see your way to the end after just the first step. That’s okay. Look for clues that indicate that you may be on the right track, such as xy being part of the other form. If you take a few steps and come up with something totally crazy or ridiculously hard, go back to the beginning and try the other path. Often, though, you’ll find the problem simplifying itself as you get several steps in.
* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.
I have never met anyone who is better at algebra than he or she is at arithmetic. As good as a person may be with algebra, that person’s going to be even better with real numbers (arithmetic). How can we use that to our advantage on the test?
Algebra and arithmetic are very similar, but algebra uses variables where arithmetic would use real numbers. On certain GMAT problems, we can taken a problem in which we were given variables and use real numbers instead “ we’re turning algebra into arithmetic!
Note: a lot of my students will complain that this method takes too much time. Of course it does when you first start studying it. You’ve been doing algebra for years, but most of you are just learning how to turn algebra into arithmetic. Think how slow you were when you first started learning algebra. Put in the practice and you’ll pick up the speed!