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Common GMAT Math Mistakes

 

Learn the cluster of most common math mistakes on the GMAT, mistakes that the GMAT regularly exploits.

 

Pattern-Matching,
Good and Bad

The human brain is, far and away, the best pattern-matching machine we have ever seen.  No computer comes close.  We match patterns for all kinds of things: facial recognition, movies, neighborhood, assessing whether someone is loony, assessing whether someone would make a good friend, etc.  Needless to say, because we do so much pattern-matching, our choices are not spot-on correct 100% of the time.

 

Mathematical Patterns

What’s germane to GMAT Math is that we also learn and understand math via patterns.  Like all things in mathematics, what’s mathematically true is absolutely precise, and even a slight change from that may be completely false.  That simply does in our pattern-matching right-brain, which likes to fudge to fit things.  Hence, there are a some very predictably mathematical patterns that people think are true, or suspect are true, even though they are blatantly false.  And, of course, dozens of GMAT Math questions are designed to probe exactly those areas!

 

Some True Patterns

First of all, let’s talk about two true patterns.  True pattern #1 is the Distributive Law, which says that multiplication distributes over addition & subtraction.

a*(b + c) = ab + ac

 

Incidentally, when you go from left to right, the action is called “distributing”, and when you go right to left, the action is called “factoring out.”  Those two are a complement pair of actions.  Factoring out is a move that is often helpful in solving the more algebraic GMAT problems.

The second doesn’t have a universally recognized name like the first, although it’s sometimes called the “Distributive Law for Exponents,” which says that exponents distribute over multiplication& division.

(ab)^n = (a^n)(b^n)

(a/b)^n = (a^n)/(b^n)

 

Two special cases of this are square-roots and reciprocals, which are really specific cases of exponents.  They both distribute over multiplication.

 

sqrt{ab} = (sqrt{a})*(sqrt{b})                 1/{ab} = {1/a}*{1/b}

 

All of these are 100% percent true.

 

The Not-So-True Patterns

You can almost see where this is going. It’s absolutely true that multiplication distributes over addition and subtraction.  It’s absolutely true that exponents (including square-roots and reciprocals) distribute over multiplication and division.   BUT, it’s 100% false that exponents, square-roots, or reciprocals distribute over addition and subtraction at any time in any way.  The following red equations are examples of false extensions of the pattern.

 

Take a good look at these.  I encourage you to plug in numbers to verify that each one is false.  They are close to the correct laws above, but in characteristic mathematical fashion, “close” here means “dead wrong.” Questions that encourage you to make one of these mistakes simply litter the GMAT math sections.

 

Robust Errors

The pattern-matching function of the brain runs deep, as does our trust in it.  Just because you can identify the faulty pattern consciously in a lucid moment doesn’t mean you won’t fall into the same mistake again.

In math pedagogy, these are called “robust errors”: even when you understand clearly why these are false, it’s as if the pattern-matching machinery of the mind draws you back to making the same mistake.  If you less than 100% crystal clear about these mistakes, you will make them when you are tired, when you are less focused, and when you are stressed – for example, during the GMAT.  I highly recommend that you bookmark this page and reread this article, each time reacquainting yourself with why each mistake is mathematically unsound.  I highly recommend you start making a log of the GMAT practice questions that, in some way, solicited one of these mistakes – and whether you fell for it or not.

Taken as a group, these must be the single greatest source of algebra errors on the GMAT.  If, through practice, you can learn to catch yourself and prevent yourself from making one of these mistakes every time, this will phenomenally improve your GMAT Quantitative Score.

 

Practice Questions

1) Find the value of a + b

(1) a^2 + b^2 = 50

(2) a - b = 8

  1. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
  2. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
  3. Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
  4. Each statement alone is sufficient to answer the question.
  5. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

 

2) Is x<sqrt{34}?

(1) x<sqrt{33}+1

(2) x< 5 + 3

  1. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
  2. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
  3. Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
  4. Each statement alone is sufficient to answer the question.
  5. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

 

Practice Question Answers and Explanations

(1) E; (2) E

In both of those questions, I was bending over backwards trying to lead you into making one of those errors.

 

1)  Clear prompt: find the value of a + b.

Statement #1: We cannot take the square root of a^2 + b^2 to get a + b.  There’s no way to simplify a^2 + b^2, and the equation a^2 + b^2 = 50 has an infinite number of pairs (a, b) that solve it, each with a different sum.  This statement does not allow us to answer the question.  Statement #1 is insufficient.

Statement #2: Again, there are an infinite number of pairs (a, b) that satisfy the equation a - b = 8.  No conclusion can be drawn about a + b.  Statement #2 is insufficient.

Statements #1 & #2 Combined:  Now, we have two equations for two unknowns, so we can solve.

a-b=8 right a=b+8

a^2+b^2=50 right (b+8)^2+b^2=50

b^2+16b+64+b^2=50

2*b^2+16b+14=0

b^2+8b+7=0

(b+1)(b+7)=0

b=-1
b=-7

Two solutions: {a = +7, b = -1} or {a = +1, b = -7}.

These two have different sums, so we cannot uniquely determine the value of a + b.  Even combined, the statements are insufficient.

Answer = E.

 

2) The prompt: Is x<sqrt{34}?

Statement #1: We cannot say that sqrt{34} is equal to 1 + sqrt{33}.  Like sqrt{34}, sqrt{33} is somewhere between 5 and 6.  When we add 1, it’s somewhere between 6 and 7.  If we know x is less than a number between 6 and 7, that’s not a guarantee that it’s less than a number between 5 and 6.  Statement #1 is insufficient.

Statement #2: We cannot say that sqrt{34} is equal to sqrt{25} + sqrt{9} = 5 + 3.  Knowing that x<8 doesn’t tell us conclusively whether or not x is less than sqrt{34}, a number between 5 and 6.  Statement #2 is insufficient.

Statements #1 & #2 Combined: Now, the combined conditions just amount to Statement #1, which was more restrictive than Statement #2.  We already know that Statement #1 is insufficient, so this means that the combined statements are also insufficient.

Answer = E

 

For more practice questions, sign up for Magoosh GMAT Prep, which offers hundreds of lessons and practice questions, all with video explanations!

 

On Tuesday, March 6th, our GMAT experts Chris Lele and Mike McGarry will be doing free one-on-one tutoring sessions with students to answer any and all questions you have about how you can do your best on the GMAT by avoiding mistakes like the ones above. During each fifteen-minute session, you can ask for general help or a demonstration of a question type you’ve been having trouble with, math or verbal!

E-mail gmat@magoosh.com to schedule a time slot for yourself!

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