Count the Variables, Count the Equations

December 29, 2008

One of the advantages of the GMAT Data Sufficiency format is that you often don't have to solve algebraic equations. Simply recognizing that the equations could be solved is enough.

This is especially useful when looking at information that addresses 2, 3, or more variables. It's important to remember that, no matter how many variables and equations you're working with:

To solve for x variables, you need x distinct, linear equations containing those variables.

For instance, given the equations x + y = 4 and x - y = 2, you can solve for x and y. That example you can probably handle easily. But the principle extends to much more complex problems.

A Realistic GMAT-like Example

Here's a problem from my Word Problems: Fundamentals set to show you what I mean:

A sum of $300,000 from a certain estate was divided among the executor and three children. How much of the estate did the oldest child receive?

(1) The executor received 1/10 of the sum from the estate, and the youngest child received 1/3 of the remainder.

(2) Each of the two older children received the same amount of money.

As it turns out, the dollar figures in this problem are simple, so you don't have to treat it abstractly. However, it's good practice to do so.

Notice in the question that the estate will be split four ways. Call the executor e, the oldest child a, the middle child b, and the youngest child c. Thus, the question tells us that e + a + b + c = $300,000.

Algebraically speaking, we have four variables and one equation. In order to solve for the four variables, we'll need four distinct, linear equations. Let's look to the statements for more.

Statement (1) gives us two more equations. e = (1/10)($300,000), and since the remainder is (9/10)($300,000), c = (1/3)(9/10)($300,000). Before you run away from all that algebra, keep in mind that you don't even have to do this much. If you recognize that the statement gives you two equations, that's plenty. The question contains one equation, (1) has two more, but we need four equations, so the statement is insufficient.

Statement (2) has one equation of its own: a = b. Combined with the one equation in the question, that's not enough -- it's a total of two equations, not four. However, when we take statements (1) and (2) together, we finally have the four distinct, linear equations we need:

• e + a + b + c = $300,000
• e = (1/10)($300,000)
• c = (1/3)(9/10)($300,000)
• a = b

Do you want to solve for the value of a? I don't think so. Combined, the statements are sufficient, regardless of what a works out to be.

Distinct and Linear

A couple of times now, I've said that these equations must be "distinct" and "linear." Both are important caveats.

In order for two equations to be distinct, they must not simplify to the same equation. For instance, x + y = 5 and 2x + 2y = 10 are not distinct. If you divide both sides of the second equation by 2, you'll be left with x + y = 5. That's just the same equation a second time.

Thus, if a Data Sufficiency statement gives you an equation that could be simplified, simplify it! If you don't, you won't know if it's distinct from the other equations in the question and statements.

"Linear" is a more involved concept. The word is used literally: A linear equation is one that, when graphed on the coordinate plane, is a straight line. If you recall from high school geometry, pre-calculus, and the like, when squares, cubes, square roots, and other exponents appear in equations, the graphs are curved.

You don't need to know anything about graphing curves and ellipses for the GMAT, just that in order for an equation to be linear, no variables can be raised to exponents (which includes square roots and other fractional exponents).

Count 'em, Don't Solve 'em

The only way you can use this tip to your advantage is to practice it. The next time you see a Data Sufficiency problem with multiple equations, give it a try. It won't save you time on every single such question, but the only way you'll know when to apply it is to experiment. Good luck!

About the author: Jeff Sackmann is a GMAT tutor based in New York City. He has created many resources for GMAT preparation, including the popular Total GMAT Math and Total GMAT Verbal, as well as 1,800 practice GMAT math questions.

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