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Common Math Errors on the GMAT  [#permalink]

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New post 22 Nov 2019, 15:00
FROM Manhattan GMAT Blog: Common Math Errors on the GMAT
[img]https://cdn2.manhattanprep.com/gmat/wp-content/uploads/sites/18/2019/11/mprep-blogimages-wave1-47-e1574458143860.png[/img]

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:

[list]
[*]When you’re under pressure, your memory becomes less reliable. [/*]
[*]Each person will find some things easier to remember than others. [/*]
[/list]

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?

[list]
[*]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. [/*]
[/list]
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.

[b]1. Decimals and exponents[/b]
Let’s go back to the example above.

0.00004 x 10-3

Instead of memorizing which way to move the decimal, [b]think about whether the decimal’s value should become larger or smaller. [/b]

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.

[b]2. Decimals and percents[/b]
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. [b]The literal meaning of the percent symbol is “/100”[/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 “/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. [b]Variables in fractions[/b]
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. [b]Factor out the same value from both the top and the bottom of the fraction[/b]. 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: [b]what am I factoring out of the top and bottom of this fraction?[/b] If you can’t factor it out, you don’t get to divide by it!

4. [b]Properties of 0[/b]
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!

[list]
[*]Zero is NOT positive or negative, it’s neither.[/*]
[*]Zero is EVEN, not odd. [/*]
[/list]
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. [b]Dividing by variables[/b]
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.

[list]
[*][b]Solve a quadratic like a quadratic[/b][/*]
[*][b]Don’t divide by 0[/b][/*]
[/list]
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. [b]Dividing by variables, with a twist[/b]
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

If you just divide by x, you get this:

3

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

You may already know a rule about dividing inequalities: [b]if you divide or multiply an inequality by a negative number, you have to flip the sign[/b]. 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 [b]never divide an inequality by a number unless you know for sure whether it’s positive or negative. [/b]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

0  – 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 3

[b]7. Negative variables[/b]
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, [b]imagine putting individual variables inside of parentheses[/b]. -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.

[b]You can attend the first session of any of our online or in-person GMAT courses absolutely free! We’re not kidding. [/b][url=https://www.manhattanprep.com/gmat/classes/][b]Check out our upcoming courses here[/b][/url][b].[/b][b]

[/b][b][/b]

[b][b][url=https://www.manhattanprep.com/instructors/chelsey-cooley/?utm_source=manhattanprep.com%2Fgre%2Fblog&utm_medium=blog&utm_content=CooleyBioGREBlog&utm_campaign=GRE%20Blog][img]https://cdn2.manhattanprep.com/gre/wp-content/uploads/sites/19/2015/11/chelsey-cooley-150x150.jpg[/img][/url]
 is a Manhattan Prep instructor based in Seattle, Washington.[/b] [/b]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. [url=https://www.manhattanprep.com/gmat/classes/#instructor/336]Check out Chelsey’s upcoming GMAT prep offerings here[/url].

The post [url=https://www.manhattanprep.com/gmat/blog/common-math-errors-on-the-gmat/]Common Math Errors on the GMAT[/url] appeared first on [url=https://www.manhattanprep.com/gmat]GMAT[/url].
ForumBlogs - GMAT Club’s latest feature blends timely Blog entries with forum discussions. Now GMAT Club Forums incorporate all relevant information from Student, Admissions blogs, Twitter, and other sources in one place. You no longer have to check and follow dozens of blogs, just subscribe to the relevant topics and forums on GMAT club or follow the posters and you will get email notifications when something new is posted. Add your blog to the list! and be featured to over 300,000 unique monthly visitors

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Quick GMAT Math Hacks  [#permalink]

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New post 04 Dec 2019, 15:00
FROM Manhattan GMAT Blog: Quick GMAT Math Hacks
[img]https://cdn2.manhattanprep.com/gmat/wp-content/uploads/sites/18/2019/12/mprep-blogimages-wave1-39-e1575494245444.png[/img]

Here are a few of the most useful quick GMAT math tricks I’ve learned over the years. They won’t show up on every problem, or even on every Quant section. But, if you happen to use one of these GMAT math hacks on test day, it could save you anywhere from a few seconds to a few minutes.

[b]Number Properties[/b]
[list]
[*]The product of two consecutive integers is always divisible by two, the product of three consecutive integers is always divisible by three, and so on. [/*]
[*]To check whether a complicated expression is even or odd, plug in 0 and 1. For instance, try the expression 2x3 + x2 + x. If you plug in 0, you get 0, which is even. If you plug in 1, you get 4, which is also even. So, this expression is always even. [/*]
[*]If you want to find all of the factors of a number by guessing and testing, you can stop when you reach the square root of that number. For instance, if you’re finding all of the factors of 228, you can stop checking numbers when you hit 15, since that’s approximately the square root of 228. [/*]
[/list]
[b]Geometry[/b]
[list]
[*]If you double the side length of a shape (such as a square or triangle), its area quadruples. If you halve the side length, its area is quartered. [/*]
[*]Learn the [url=https://www.manhattanprep.com/gre/blog/gre-geometry-three-ways-to-spot-similar-triangles/]three ways to spot similar triangles[/url], so you’ll instantly recognize that two triangles are similar without having to prove it from scratch. [/*]
[*]If a problem tells you a shape is a rectangle, don’t forget that the shape could be a square! In fact, a square is often a good case to test on [url=https://www.manhattanprep.com/gmat/blog/how-to-review-a-data-sufficiency-question/]Geometry Data Sufficiency[/url] problems. [/*]
[*]If a GMAT math problem asks you whether a point is on a line, plug the coordinates of the point into the equation for the line. If you get a valid result, then the point is on the line. For example, the point (2, 6) is on the line y = 2x + 2.[/*]
[/list]
[b]Word Problems[/b]
[list]
[*]The average of a set of numbers always has to be somewhere in the middle of that set. It can’t be larger than the largest number in the set, or smaller than the smallest number. This is useful for weighted average problems: if you average the weights of 6 cats that each weigh 10 pounds, and 8 dogs that each weigh 30 pounds, the result will be somewhere in the middle in between 10 and 30. The more evenly spread the numbers are, the closer the average will actually be to the middle. [/*]
[*]Only use a Venn diagram for rare “3-group” overlapping set problems. For almost all overlapping sets, the Overlapping Set Matrix is quicker and easier. [url=https://www.manhattanprep.com/gmat/blog/how-to-handle-3-group-overlapping-sets-on-the-gmat/]Both are described in this article[/url]. [/*]
[*]You’ll sometimes see rates problems that look like this: if it takes four people twelve days to sew eight jackets, how long does it take ten people to sew ten jackets? A quick trick for approaching these is to start with the original statement, and then “scale” it upwards or downwards. Here’s what that might look like: [/*]
[/list]
It takes 4 people 12 days to sew 8 jackets.

1 person will take 4 times as long to do the same amount of work, so it will take 1 person 48 days to sew 8 jackets.

If that 1 person sews ⅛ as many jackets, it will take ⅛ as many days. So, it takes 1 person 6 days to sew 1 jacket.

If a person takes 6 days to sew a jacket, then it will take 10 people 6 days to sew 10 jackets (one per person). The answer is 6.

[b]Fractions, Decimals, and Percents[/b]
[list]
[*]When a fraction has zeroes on the end of both the numerator and the denominator, chop off the same number of zeroes from each (just make sure you count carefully!). 1,000,000 / 5,000 simplifies to 1,000 / 5. [/*]
[*]Likewise, if a fraction has decimals in both the numerator and denominator, you can simplify by moving both decimal places by the same amount and in the same direction. For instance, 0.0007 / 0.14 = 0.007 / 1.4 = 0.07 / 14 = 0.7 / 140 = 7 / 1,400. [/*]
[*]Use [url=https://www.manhattanprep.com/gre/blog/heres-the-safest-way-to-handle-gre-percentage-problems/]this technique to directly translate percent problems from English into math[/url] without having to convert between decimals and percents. [/*]
[/list]
[b]Working with Numbers[/b]
[list]
[*]You can use a similar ‘scaling’ technique to calculate percents, fractions, or decimals. For instance, if you want to find 0.1% of 50,000, start like this: [/*]
[/list]
10% of 50,000 is 5,000.

So, 1% of 50,000 is a tenth of 5,000, or 500.

So, 0.1% of 50,000 is a tenth of 500, or 50. The answer is 50.

[list]
[*]To quickly divide a number by 5, divide it by 10 first, then multiply by 2. For example, 1,880/5 = 1,880/10 * 2 = 188 * 2 = 376. [/*]
[*]Arithmetic can be easier if you “split up” or rearrange the numbers before you do the math. Suppose that you need to calculate 117 – 98. Rewrite this as 117 – 100 + 2, or 17 + 2, which equals 19. [/*]
[*]Use a similar technique to quickly calculate the square of a number that’s close to an easy value.[/*]
[/list]
79² = (80 – 1)² = 80² – 2(80) + 1 = 6,400 – 160 + 1 = 6241

[list]
[*]To find a good common denominator, think of a value (if there is one) that both numbers are divisible by. Divide [b]one[/b] of the two numbers by that value. Then, multiply that by the other number.
[list]
[*]For example, to find a common denominator between 25 and 15, note that both are divisible by 5. So, divide 25 by 5, which gives you 5, then multiply that by 15, giving you 75. 75 would be a good common denominator.[/*]
[/list]
[/*]
[*]It can be useful to memorize the approximate square roots of 2 and 3: √2≈1.4 and√3≈1.7. To remember this, at least if you’re in the US, think of two dates: Valentine’s Day is on 2/14 and St. Patrick’s Day is on 3/17. [/*]
[*]To estimate other square roots, think of a perfect square that’s as close as possible to the value you’re dealing with. (You have your perfect squares memorized, right…?) Estimate based on that—so, for instance, √79 is a bit smaller than √81, which equals 9. [/*]
[/list]
[b]What Next?[/b]
Math isn’t the most important part of the GMAT math section! [url=https://www.manhattanprep.com/gmat/blog/what-the-gmat-really-tests/]Strong executive reasoning skills trump math knowledge[/url]. So, while these tips and tricks are useful, if you’re having a tough time with the math section, incorporate some work on [url=https://www.manhattanprep.com/gmat/blog/everything-know-gmat-time-management-part-3/]timing[/url], [url=https://www.manhattanprep.com/gmat/blog/which-gmat-problems-should-i-guess-on-part-3-making-great-guesses-on-quant-problems/]guessing[/url], [url=https://www.manhattanprep.com/gmat/blog/the-4-math-strategies-everyone-must-master-part-1/]problem-solving strategies[/url], and [url=https://www.manhattanprep.com/gmat/blog/how-to-handle-gmat-stress-without-freaking-out/]stress management[/url]. But keep some pages in your notes for these GMAT math tricks, plus any others you may come across while studying: you never know what may turn out to be useful.

[b]You can attend the first session of any of our online or in-person GMAT courses absolutely free! We’re not kidding. [/b][url=https://www.manhattanprep.com/gmat/classes/][b]Check out our upcoming courses here[/b][/url][b].[/b][b]

[/b][b][/b]

[b][b][url=https://www.manhattanprep.com/instructors/chelsey-cooley/?utm_source=manhattanprep.com%2Fgre%2Fblog&utm_medium=blog&utm_content=CooleyBioGREBlog&utm_campaign=GRE%20Blog][img]https://cdn2.manhattanprep.com/gre/wp-content/uploads/sites/19/2015/11/chelsey-cooley-150x150.jpg[/img][/url]
 is a Manhattan Prep instructor based in Seattle, Washington.[/b] [/b]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. [url=https://www.manhattanprep.com/gmat/classes/#instructor/336]Check out Chelsey’s upcoming GMAT prep offerings here[/url].

The post [url=https://www.manhattanprep.com/gmat/blog/quick-gmat-math-hacks/]Quick GMAT Math Hacks[/url] appeared first on [url=https://www.manhattanprep.com/gmat]GMAT[/url].
ForumBlogs - GMAT Club’s latest feature blends timely Blog entries with forum discussions. Now GMAT Club Forums incorporate all relevant information from Student, Admissions blogs, Twitter, and other sources in one place. You no longer have to check and follow dozens of blogs, just subscribe to the relevant topics and forums on GMAT club or follow the posters and you will get email notifications when something new is posted. Add your blog to the list! and be featured to over 300,000 unique monthly visitors

_________________
http://manhattangmat.com
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Quick GMAT Math Hacks   [#permalink] 04 Dec 2019, 15:00

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