Simplifying Algebraic Equations on Data Sufficiency GMAT Questions

GMAT QuantIn the past few weeks, I’ve written a couple of posts extolling the virtues of using strategies in lieu of doing difficult algebra. But over the course of the quant section, there’s no getting around it: at times, algebra will be an effective tool that you’ll want to deploy. The key is for us to use this tool judiciously.

Because the GMAT is largely a test of pattern recognition, it’s worthwhile to first discuss the structural clues that we’ll want to be on the lookout for when determining whether algebra will be the most effective approach. My older posts discussed two scenarios when algebra would be problematic: the first was problem-solving questions involving difficult quadratic simplification, and the second was problem-solving percent questions that involved variables. In both cases, we’re better off either picking numbers or back-solving. Alternatively, when we see Data Sufficiency word problems, algebra serves a much more useful function, allowing us to distill complex information in simpler, more concrete form.

Once we recognize that we’ll be attacking a question algebraically, the next step is to consider how we can make our equations and expressions as simple as possible. Say, for example, that we’re told that the ratio of men to women to children in a park is 6 to 5 to 4. One way to depict this information is to write M:W:C = 6:5:4. The problem with this approach is that it leaves us with three variables. Hardly the simplicity and elegance that we’re looking for if we’re dealing with a time constraint. The alternative is to use only one variable and depict the information in terms of x:

Men:  6x

Women: 5x

Children: 4x

Now when we receive additional information about how these values are related, the equations we can assemble will be far more straightforward. Let’s try a GMATPrep* question to see this in action.

A certain company divides its total advertising budget into television, radio, newspaper, and magazine budgets in the ratio of 8:7:3:2 respectively. How many dollars are in the radio budget?

(1) The television budget is $18,750 more than the newspaper budget

(2) The magazine budget is $7,500.

We’ve got a Data Sufficiency word problem, so let’s start by putting all of the relevant information into algebraic form. Rather than using four different variables, we’ll organize our information like so:

Television: 8x

Radio: 7x

Newspaper: 3x

Magazine: 2x

Our ultimate goal is find the radio budget, which is 7x. Clearly, if we have the value of x, we can find 7x, so we can rephrase the question as: ‘What is the value of x?’

Statement 1 tells us that the television budget, 8x, is 18,750 more than the newspaper budget, 3x. In algebraic form, that will be: 8x = 18750 + 3x. Obviously, we can solve for x here, so SUFFICIENT.

Statement 2 tells us that the magazine budget, or 2x, is 7500. So 2x = 7500. Again, we can clearly solve for x, so SUFFICIENT.

And the answer is D; either statement alone is sufficient to answer the question.

Let’s try another.

Of the shares of stock owned by a certain investor, 30 percent are shares of Company X stock and 1/7 of the remaining shares are shares of Company Y stock. How many shares of Company X stock does the investor own?

(1) The investor owns 100 shares of Company Y stock.
(2) The investor owns 200 more shares of Company X stock than of Company Y stock.

Same drill: we recognize that we’re dealing with a Data Sufficiency word problem, so let’s convert the initial into algebraic form.

If we designate our total shares of stock ‘T,’ and we know that 30% of those are Company X, we’ll have .3T shares of company X. We’re told that 1/7 of the remaining shares are Company Y. If .3T shares are company X, we’ll have .7T shares left over. If 1/7 of those .7T shares belong to Company Y, we can designate Company Y’s shares as (1/7) * .7T =   .1T.

Summarized, we have the following information:

Company X: .3T

Company Y: .1T

We’re asked about Company X, so we want .3T. Clearly, if we have T, we can solve for .3T, so our rephrased question is just: “What is the value of T?”

Statement 1 tells us there are 100 share of Y, so .1T = 100. We can solve for T, so SUFFICIENT.

Statement 2 tells us that the investor has 200 more shares of X than Y. Algebraically: .3T = 200 + .1T. Again, we can solve for T, but no need to actually do the math. SUFFICIENT.

The answer is D; either alone is sufficient to answer the question.

Takeaway: preparation for the GMAT is not about learning which strategies are ‘best.’ Different strategies will work well in different scenarios, and for some test-takers, it will be a matter of taste to determine which they prefer. If you do decide to approach a question algebraically – and again, in Data Sufficiency word problems, this will often work nicely – try to diminish the complexity of the problem by minimizing the number of variables you use to depict the relevant information.

*GMATPrep questions courtesy of the Graduate Management Admissions Council.

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By David Goldstein, a Veritas Prep GMAT instructor based in Boston. You can find more articles by him here