## Common Sense on GMAT Data Sufficiency

*Rich is one of Knewton’s expert GMAT prep teachers, and he loves thinking of ways to crack the Data Sufficiency section*.

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Data Sufficiency questions are often difficult to get used to, because they require an adjustment in your approach to math problems. When you went through math classes growing up, the end goal was always “Find the value of x” or “Find the area of this circle.” You were asked to give hard responses to these questions, and nothing mattered more than finding a definite value.

With Data Sufficiency, answering the question does not matter as much as *the ability to answer the question*. You are not primarily concerned with the final answer, but rather whether you have enough information to get you to that answer. For example, if you’re asked to find the value of x, and a statement tells you that 300x + 257 = 1345, you know that this statement is sufficient, because you can perform arithmetic on that equation to isolate x. Are you going to perform it? No, because it’s too complicated and you don’t need to! All you’re concerned with is whether you *can* find the answer.

It might strike you as odd, but because of this principle, you can tackle some supposedly difficult DS questions without writing down a single equation or calculation! Sounds too good to be true, but in actuality, it makes a lot of sense. Remember, in business school you’ll be given data in case studies, and you’ll be expected to determine relatively quickly what information is relevant. DS questions are perfect for testing this ability because you have to look at the information given to you and cut to the heart of what is most important about that information.

As an example, let’s look at this rather wordy DS problem:

*Martha bought an armchair and a coffee table at an auction and sold both items at her store. Her gross profit from the purchase and sale of the armchair was what percent greater than her gross profit from the purchase and sale of the coffee table?*

*(1) Martha paid 10 percent more for the armchair than for the coffee table.*

*(2) Martha sold the armchair for 20 percent more than she sold the coffee table.*

First, let’s approach this algebraically to show how cumbersome it ends up being:

In general, Gross Profit (P) is the Selling Price (S) minus the Buying Price (B):

P = S – B

We want to know what percent greater the profit of the armchair (P_armchair) is than the profit of the coffeetable (P_coffeetable). If we represent the missing percent as x, then the equation would be

P_coffeetable*(1 + x/100) = P_armchair

Rearranging, we would get:

x = 100*(P_armchair / P_coffeetable – 1)

We know that P = S – B, so we can substitute:

x = 100*[(S_armchair – B_armchair)/ (S_coffeetable – B_coffeetable) – 1]

Confused yet?? I sure am!!

But try to look at things from a sufficiency point of view. Notice that you need absolute values for the gross profits in order to solve for x.Â You could also find the ratio between the two gross profits.

Now that we’ve seen how ugly this looks when all the algebra is written out, let’s take a more common-sense approach.

The prompt:

*Martha bought an armchair and a coffee table at an auction and sold both items at her store. Her gross profit from the purchase and sale of the armchair was what percent greater than her gross profit from the purchase and sale of the coffee table?*

What we need in order to determine sufficiency:

In essence, this question asks you to compare the values of two profits. You need the value of each profit OR the ratio between the two profits. Notice that you can figure out what information you will need without writing down a single number or algebraic expression.

What each statement tells us:

*(1)Â Martha paid 10 percent more for the armchair than for the coffee table.*

Without writing any math, you can deduce that this is insufficient, because buying prices are mentioned, but no selling prices.Â And with no selling prices, we certainly can’t determine anything about profit.

*(2) Martha sold the armchair for 20 percent more than she sold the coffee table.*

Now, we’ve got information about selling prices, but nothing about buying prices. Again, Insufficient because there is no way to determine profits.

So far, nothing too difficult. It’s pretty simple to narrow this down to C and E.Â But how do we determine whether the statements together are sufficient?

You could test numbers here, but really all you need to do is realize that the statements only give you percentages to work with.Â For the sake of illustration, let’s pick numbers to see what this means:

According to Statement 1, Marta could have spent $100 on the coffee table and $110 on the armchair, or it could have been $10 on the coffee table and $11 on the armchair. (Unlikely prices, maybe, but remember, the real world doesn’t apply here!) There are infinite possibilities for what the buying prices could have been.

Likewise, Statement 2 tells us that Martha could have sold the coffee table for $100 and the armchair for $120. Or it could have been $10 for the coffee table, $12 for the armchair.

You’ll notice that because the absolute numbers for selling and buying prices vary so much, so too do the gross profits! And if the gross profits can fluctuate that drastically, there is no way on Earth you can nail down one specific percentage increase from one profit to the next!

And thus, without writing a single equation, you can determine that the answer must be E.

It’s very very tricky to get your mind to think this way, especially since you’ve been trained all your life to hack away at a problem until you come up with a definite answer. But it is absolutely imperative that you begin to look past the math of DS questions and ask yourself what information is *necessary to solve the problem*.