Common Math Errors on the GMAT
Do you ever make mistakes on GMAT math that just don’t make sense when you review? That’s not unusual, and in fact, it’s probably one of the most common reasons to miss easy GMAT math problems. Here’s why:
- When you’re under pressure, your memory becomes less reliable.
- Each person will find some things easier to remember than others.
A lot of GMAT math errors are based on memorization. Suppose that you want to simplify the following expression:
0.00004 x 10-3
Quick, which of the following rules is correct?
- To multiply a decimal by ten raised to a negative power, move the decimal place to the right that many times.
- To multiply a decimal by ten raised to a negative power, move the decimal place to the left that many times.
Only one of these rules is right. But look how similar they are! The right one may be obvious to you right now, but the right rule is so close to the wrong rule. Can you really be sure that if you memorize it now, you’ll remember it flawlessly on test day? (By the way, the second rule is the correct one.)
In this article, I’ll list a handful of mistakes that people often make on GMAT math. Then, I’ll share a self-check you can use to avoid each one. Because everyone is different, some of these mistakes may be easy for you to avoid. For others, you might decide to double-check every single time.
1. Decimals and exponents
Let’s go back to the example above.
0.00004 x 10-3
Instead of memorizing which way to move the decimal, think about whether the decimal’s value should become larger or smaller.
Ten raised to a negative power, like 10-3, is a fraction. In this case, it’s equal to 1/1,000. Multiplying something by a small fraction will definitely make it smaller.
A small decimal has more zeroes in front of it. So, to simplify this expression, you want to add more zeroes in front of the 4.
To remember how many zeroes to add, think about dividing by 10. Each time you divide a decimal by 10, you’d add in one zero. Dividing by 103, which is what we’re doing in this problem, is the same as dividing by 10 three times. So, you need to add three zeroes.
The right answer is 0.00000004.
2. Decimals and percents
When you want to find 0.05% of 13,000, what do you multiply 13,000 by? It’s easy to lose a decimal place or two and end up with an answer that’s off by a factor of 10.
Here’s the solution. The literal meaning of the percent symbol is “/100”. In fact, the percent symbol sort of looks like a division sign with two zeroes, symbolizing a 100. Any time you see a math expression including a percent, write it on your paper as if the percent sign said “/100” instead.
For this question, you’d write the following on your paper:
0.05/100 x 13,000
This simplifies to 0.05 x 130, or 6.5.
You can use this trick even when there are variables involved in the expression. For instance, a question might ask you “If y% of x equals 50, what is x% of y?”
Write this as follows:
(y/100)(x) = 50
(x/100)(y) = ?
In both cases, the left side of the expression simplifies to xy/100. So, they’re equal, and the answer to the question is 50.
3. Variables in fractions
Simplifying a fraction that only includes numbers is relatively straightforward, although the math might be tedious. But, when the fraction includes variables, the math gets less obvious.
Here’s an example of something you might have on your paper while doing a GMAT math problem:
(x + 7y) / (y²)
You may have memorized a rule that says “you can cancel common terms from the top and bottom of a fraction.” But that rule comes with some fiddly little caveats, like the fact that you aren’t allowed to do this:
(x + 7y) / (y²)
(x + 7) / (y)
Here’s another way to think about it that’s more reliable. Factor out the same value from both the top and the bottom of the fraction. Then, you can “cancel” (divide) both of those terms.
In the example above, you can’t factor a y out of the top of the fraction. So, you aren’t allowed to cancel the y.
But, in this example, you can:
(y³ + 7y) / (y²)
y(y² + 7) / y(y)
(y² + 7) / y
If you’ve made this mistake before, commit yourself to thinking each time: what am I factoring out of the top and bottom of this fraction? If you can’t factor it out, you don’t get to divide by it!
4. Properties of 0
There are two common Number Properties rules in GMAT math that relate to the number 0. Unfortunately, they’re almost identical to each other, and it’s so easy to get them switched around!
- Zero is NOT positive or negative, it’s neither.
- Zero is EVEN, not odd.
Let’s dig into why this is the case.
All even numbers have one thing in common: if you divide them by 2, you don’t end up with a fraction or a remainder. For instance, 2,476 is even, because if you divide it by 2, you get a round number with nothing left over. The same is true of, say, -18. This rule of thumb will always accurately tell you whether a number is even.
What happens when you divide zero by two? You get zero.
0/2 = 0
Sure enough, there’s no fraction or remainder. So, by our rules, zero is definitely even.
Why isn’t zero positive or negative? This is a trickier one, because it depends, in part, on language. In some languages other than English, zero is actually said to be both positive and negative. However, on the GMAT, it’s neither.
On the GMAT, a good general strategy is to visualize a number line. Numbers to the left are smaller than numbers to the right. Anything to the left of zero is negative, and anything to the right of zero is positive. And because zero itself is neither to the left nor to the right of zero, it can’t be positive or negative.
5. Dividing by variables
How do you solve this equation?
3x = x²
The obvious first move is to divide both sides by x, giving you this answer:
3 = x
But, that’s actually a big problem. Why? Because x doesn’t necessarily equal 3. In fact, x could also equal 0. (Plug 0 into the equation 3x = x², and it works out just fine!)
You could memorize a rule: “equations that have the same variable in every term also have 0 as a solution, on top of whatever solution you come up with.” But, here are two alternatives.
- Solve a quadratic like a quadratic
- Don’t divide by 0
For the first alternative, notice that 3x = x² is a quadratic equation: it has a squared variable in it. The way to solve a quadratic isn’t to divide out like terms! Instead, you move everything to the same side, and then factor. So, do this:
x² – 3x = 0
x(x – 3) = 0
This gives you two solutions: x = 0, and x = 3.
The other alternative is to be extra careful never to divide anything by zero. That includes variables! If a variable might equal zero, then you still can’t divide by it. After all, you might be dividing by zero without realizing it.
The right approach is the same one as shown above: instead of dividing out an x (don’t do it, since it might equal zero!), focus on factoring it out without dividing. To do that, put both terms on the same side of the equation, then factor out the x that they have in common.
6. Dividing by variables, with a twist
There’s one other situation where it’s dangerous to divide by a variable: when you’re simplifying an inequality. This causes even bigger problems than the ones shown above.
For example, suppose you’re trying to simplify this inequality:
3x < x²
If you just divide by x, you get this:
3 < x
That’s perfect, except that it’s the wrong answer. x definitely doesn’t have to be bigger than 3! For instance, x could be -1:
3(-1) = -3
(-1)2 = 1
-3 < 1
You may already know a rule about dividing inequalities: if you divide or multiply an inequality by a negative number, you have to flip the sign. That causes more problems when you’re dividing or multiplying by a variable. You don’t know the value of the variable, so you don’t know whether it’s negative or not! So, maybe you have to flip the sign, or maybe you don’t. There’s no way to tell. That’s the issue.
The solution is to never divide an inequality by a number unless you know for sure whether it’s positive or negative. If you know that x is positive, you can go ahead and do the division above. If you know that x is negative, you can still do the division, you just have to flip the sign! But if you aren’t sure, you can’t divide by x.
What can you do instead? It depends on what the overall GMAT problem looks like. On problems like these, it’s often possible to solve more quickly and easily by testing numbers. Or, you can do something similar to the approach from the previous tip:
3x < x²
0 < x² – 3x
In other words, x² – 3x is positive. Therefore, x(x-3) is positive.
Next, use some Number Properties facts. The product of x and x-3 is only positive if x and x-3 are both positive, or x and x-3 are both negative. That will happen in exactly two situations. If x is greater than 3, then x and x-3 are both positive, so their product is positive. Or, if x is less than 0, then x and x-3 are both negative, so their product is positive.
So, the correct answer is this:
x < 0 or x > 3
7. Negative variables
This conversation about positive and negative numbers leads us to our final tip. Quick: is the following number positive or negative?
-x
Especially in Number Properties problems, which often ask you whether a value is positive or negative, this can trip you up. It’s easy to see the negative sign when you’re working fast and assume that you definitely have a negative number. After all, -2 is negative, so why not -x?
However, that’s only true if x itself is positive. If x is negative, then the number above is actually positive. For instance, if x = -5, then -x = 5.
To avoid mistakes, imagine putting individual variables inside of parentheses. -x is really -(x). Therefore, if x = -5, then -x = -(x) = -(-5) = 5. After all, two negatives make a positive.
This can also help you remember what to do when you raise a variable to a power. x² really equals (x)², so if x = -5, then x² = (-5)² = 25. Just don’t accidentally include anything else inside of the parentheses! If you do this, you’ll be able to simplify expressions including negative variables correctly.
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is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170Q/170V on the GRE. Check out Chelsey’s upcoming GMAT prep offerings here.