Common mistake - Not recognizing “hidden” algebra questions

Many students (and some teachers for that matter!) don’t see the connection between algebra and numbers. For example, when I ask students what it means to say that 3x + 7x = 10x, the most common response is that three x’s plus seven x’s is ten x’s. While that response is kind of true, it certainly doesn’t tell us what the equation means.

These students are missing one of the main points about algebra: Algebra helps generalize the ways in which numbers behave.

In this case, 3x + 7x = 10x tells us that, for ANY value of x, the value of 3x + 7x will be exactly the same as the value of 10x.

For example, if x = 2.6, we know that, if we evaluate 3(2.6) + 7(2.6), we’ll get the same outcome we get by evaluating 10(2.6).

Students who don’t see the connection between algebra and how numbers behave often fail to see that some GMAT questions are really just algebra questions in disguise.

For example, upon seeing the algebraic expression x² - y², many students will instinctively recognize that this difference of squares can be factored to get (x+y)(x-y)

However, if I ask those same students to evaluate 444² - 443², they don’t see a difference of squares, because they feel that algebra is some collection of mysterious rules involving variables. So, if there aren’t any variables, then it’s not an algebra question.

If we don’t see the difference of squares in this question, then we’re forced to evaluate each part to get: 444² - 443² = 197,136 – 196,249 = 887 (uggh!!!)

The faster strategy is to treat 444² - 443² as the difference of squares it most definitely is.

When we do so, we get: 444² - 443² = (444 + 443)(444 – 443) = (887)(1) = 887

There are many other examples of algebraic concepts “hiding” within GMAT questions.

For example, I know that most students will automatically simplify x + x to get 2x, but those same students won’t see that we can simplify 2^n + 2^n using the exact same logic:

2^n + 2^n = 2(2^n) = (2^1)(2^n) = 2^(n+1)

So, always be on the lookout for algebraic concepts hiding within questions that don’t necessarily look like the algebra questions you’re accustomed to.

Here are some more official GMAT questions related to this topic:

Question #1 | Question #2 | Question #3

Cheers,

Brent