![\"common](\"https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2019\/11\/mprep-blogimages-wave1-47-e1574458143860.png\")
<\/p>\n<\/p>\n
Do you ever make mistakes on GMAT math that just don\u2019t make sense when you review? That\u2019s not unusual, and in fact, it\u2019s probably one of the most common reasons to miss easy GMAT math problems. Here\u2019s why:\u00a0<\/span><\/p>\n <\/p>\n A lot of GMAT math errors are based on memorization. Suppose that you want to simplify the following expression:\u00a0<\/span><\/p>\n 0.00004 x 10-3<\/sup> Quick, which of the following rules is correct?<\/span><\/p>\n 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 <\/span>so<\/span><\/i> close to the wrong rule. Can you really be sure that if you memorize it now, you\u2019ll remember it flawlessly on test day? (By the way, the second rule is the correct one.)\u00a0<\/span><\/p>\n In this article, I\u2019ll list a handful of mistakes that people often make on GMAT math. Then, I\u2019ll 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.<\/span><\/p>\n Let\u2019s go back to the example above.<\/span><\/p>\n 0.00004 x 10-3<\/sup><\/span><\/p>\n Instead of memorizing which way to move the decimal, <\/span>think about whether the decimal\u2019s value should become larger or smaller.\u00a0<\/b><\/p>\n Ten raised to a negative power, like 10<\/span>-3<\/span>, is a fraction. In this case, it\u2019s equal to 1\/1,000. Multiplying something by a small fraction will definitely make it smaller.\u00a0<\/span><\/p>\n A small decimal has <\/span>more<\/span><\/i> zeroes in front of it. So, to simplify this expression, you want to add more zeroes in front of the 4.\u00a0<\/span><\/p>\n To remember<\/span> how many<\/span><\/i> zeroes to add, think about dividing by 10. Each time you divide a decimal by 10, you\u2019d add in one zero. Dividing by 10<\/span>3<\/span>, which is what we\u2019re doing in this problem, is the same as dividing by 10 three times. So, you need to add three zeroes.\u00a0<\/span><\/p>\n The right answer is 0.00000004.<\/span><\/p>\n When you want to find 0.05% of 13,000, what do you multiply 13,000 by? It\u2019s easy to lose a decimal place or two and end up with an answer that\u2019s off by a factor of 10.\u00a0<\/span><\/p>\n Here\u2019s the solution. <\/span>The literal meaning of the percent symbol is \u201c\/100\u201d<\/b>. 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 \u201c\/100\u201d instead.\u00a0<\/span><\/p>\n For this question, you\u2019d write the following on your paper:\u00a0<\/span><\/p>\n 0.05\/100 x 13,000\u00a0<\/span><\/p>\n This simplifies to 0.05 x 130, or 6.5.\u00a0<\/span><\/p>\n You can use this trick even when there are variables involved in the expression. For instance, a question might ask you \u201cIf y% of x equals 50, what is x% of y?\u201d\u00a0<\/span><\/p>\n Write this as follows:\u00a0<\/span><\/p>\n (y\/100)(x) = 50<\/span><\/p>\n (x\/100)(y) = ?\u00a0<\/span><\/p>\n In both cases, the left side of the expression simplifies to xy\/100. So, they\u2019re equal, and the answer to the question is 50.<\/span><\/p>\n 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.\u00a0<\/span><\/p>\n Here\u2019s an example of something you might have on your paper while doing a GMAT math problem:\u00a0<\/span><\/p>\n (x + 7y) \/ (y\u00b2<\/span>)<\/span><\/p>\n You may have memorized a rule that says \u201cyou can cancel common terms from the top and bottom of a fraction.\u201d But that rule comes with some fiddly little caveats, like the fact that you <\/span>aren\u2019t<\/span><\/i> allowed to do this:\u00a0<\/span><\/p>\n (x + 7y) \/ (y\u00b2<\/span>)<\/span><\/p>\n (x + 7) \/ (y)<\/span><\/p>\n Here\u2019s another way to think about it that\u2019s more reliable. <\/span>Factor out the same value from both the top and the bottom of the fraction<\/b>. Then, you can \u201ccancel\u201d (divide) both of those terms.\u00a0<\/span><\/p>\n In the example above, you can\u2019t factor a <\/span>y<\/span><\/i> out of the top of the fraction. So, you aren\u2019t allowed to cancel the <\/span>y<\/span><\/i>.\u00a0<\/span><\/p>\n But, in this example, you can:\u00a0<\/span><\/p>\n (y\u00b3<\/span>\u00a0+ 7y) \/ (y\u00b2<\/span>)\u00a0<\/span><\/p>\n y(y\u00b2<\/span>\u00a0+ 7) \/ y(y)<\/span><\/p>\n (y\u00b2<\/span>\u00a0+ 7) \/ y<\/span><\/p>\n If you\u2019ve made this mistake before, commit yourself to thinking each time: <\/span>what am I factoring out of the top and bottom of this fraction?<\/b> If you can\u2019t factor it out, you don\u2019t get to divide by it!<\/span><\/p>\n There are two common Number Properties rules in GMAT math that relate to the number 0. Unfortunately, they\u2019re almost identical to each other, and it\u2019s so easy to get them switched around!\u00a0<\/span><\/p>\n Let\u2019s dig into <\/span>why<\/span><\/i> this is the case.<\/span><\/p>\n All even numbers have one thing in common: if you divide them by 2, you don\u2019t 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.\u00a0<\/span><\/p>\n What happens when you divide zero by two? You get zero.\u00a0<\/span><\/p>\n 0\/2 = 0<\/span><\/p>\n Sure enough, there\u2019s no fraction or remainder. So, by our rules, zero is definitely even.\u00a0<\/span><\/p>\n Why isn\u2019t 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 <\/span>and<\/span><\/i> negative. However, on the GMAT, it\u2019s neither.\u00a0<\/span><\/p>\n 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\u2019t be positive or negative.\u00a0<\/span><\/p>\n How do you solve this equation?\u00a0<\/span><\/p>\n 3x = x\u00b2<\/span><\/p>\n The obvious first move is to divide both sides by x, giving you this answer:\u00a0<\/span><\/p>\n 3 = x<\/span><\/p>\n But, that\u2019s actually a big problem. Why? Because x <\/span>doesn\u2019t<\/span><\/i> necessarily equal 3. In fact, x could also equal 0. (Plug 0 into the equation 3x = x\u00b2<\/span>, and it works out just fine!)<\/span><\/p>\n You <\/span>could<\/span><\/i> memorize a rule: \u201cequations that have the same variable in every term also have 0 as a solution, on top of whatever solution you come up with.\u201d But, here are two alternatives.\u00a0<\/span><\/p>\n For the first alternative, notice that 3x = x\u00b2<\/span>\u00a0is a <\/span>quadratic<\/span><\/i> equation: it has a squared variable in it. The way to solve a quadratic isn\u2019t to divide out like terms! Instead, you move everything to the same side, and then factor. So, do this:\u00a0<\/span><\/p>\n x\u00b2<\/span>\u00a0– 3x = 0<\/span><\/p>\n x(x – 3) = 0<\/span><\/p>\n This gives you two solutions: x = 0, and x = 3.\u00a0<\/span><\/p>\n The other alternative is to be extra careful never to divide anything by zero. That includes variables! If a variable <\/span>might<\/span><\/i> equal zero, then you still can\u2019t divide by it. After all, you might be dividing by zero without realizing it.\u00a0<\/span><\/p>\n The right approach is the same one as shown above: instead of dividing out an x (don\u2019t 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.\u00a0<\/span><\/p>\n There\u2019s one other situation where it\u2019s dangerous to divide by a variable: when you\u2019re simplifying an inequality. This causes even bigger problems than the ones shown above.\u00a0<\/span><\/p>\n For example, suppose you\u2019re trying to simplify this inequality:\u00a0<\/span><\/p>\n 3x < x\u00b2<\/span><\/p>\n If you just divide by x, you get this:<\/span><\/p>\n 3 < x<\/span><\/p>\n That\u2019s perfect, except that it\u2019s the wrong answer. x definitely doesn\u2019t have to be bigger than 3! For instance, x could be -1:\u00a0<\/span><\/p>\n 3(-1) = -3<\/span><\/p>\n (-1)<\/span>2<\/span> = 1<\/span><\/p>\n -3 < 1<\/span><\/p>\n You may already know a rule about dividing inequalities: <\/span>if you divide or multiply an inequality by a negative number, you have to flip the sign<\/b>. That causes more problems when you\u2019re dividing or multiplying by a variable. You don\u2019t know the value of the variable, so you don\u2019t know whether it\u2019s negative or not! So, maybe you have to flip the sign, or maybe you don\u2019t. There\u2019s no way to tell. That\u2019s the issue.\u00a0<\/span><\/p>\n The solution is to <\/span>never divide an inequality by a number unless you know for sure whether it\u2019s positive or negative. <\/b>If you know that <\/span>x<\/span><\/i> is positive, you can go ahead and do the division above. If you know that <\/span>x<\/span><\/i> is negative, you can still do the division, you just have to flip the sign! But if you aren\u2019t sure, you can\u2019t divide by <\/span>x<\/span><\/i>.\u00a0<\/span><\/p>\n What can you do instead? It depends on what the overall GMAT problem looks like. On problems like these, it\u2019s often possible to solve more quickly and easily by testing numbers. Or, you can do something similar to the approach from the previous tip:\u00a0<\/span><\/p>\n 3x < x\u00b2<\/span><\/p>\n 0 < x\u00b2<\/span>\u00a0– 3x<\/span><\/p>\n In other words, x\u00b2<\/span>\u00a0– 3x is positive. Therefore, x(x-3) is positive.\u00a0<\/span><\/p>\n 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.<\/span><\/p>\n So, the correct answer is this:\u00a0<\/span><\/p>\n x < 0 or x > 3<\/span><\/p>\n This conversation about positive and negative numbers leads us to our final tip. Quick: is the following number positive or negative?\u00a0<\/span><\/p>\n -x<\/span><\/p>\n Especially in Number Properties problems, which often ask you whether a value is positive or negative, this can trip you up. It\u2019s easy to see the negative sign when you\u2019re working fast and assume that you definitely have a negative number. After all, -2 is negative, so why not -x?\u00a0<\/span><\/p>\n However, that\u2019s 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.\u00a0<\/span><\/p>\n To avoid mistakes, <\/span>\n
\n<\/span><\/p>\n\n
1. Decimals and exponents<\/b><\/h3>\n
2. Decimals and percents<\/b><\/h3>\n
3.\u00a0Variables in fractions<\/b><\/h3>\n
4.\u00a0Properties of 0<\/b><\/h3>\n
\n
5.\u00a0Dividing by variables<\/b><\/h3>\n
\n
6.\u00a0Dividing by variables, with a twist<\/b><\/h3>\n
7. Negative variables<\/b><\/h3>\n