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Etiqueta: GRE Math Misconceptions

More GRE Math Misconceptions

Manhattan Prep GRE Blog - More GRE Math Misconceptions by Chelsey Cooley

Did you enjoy our last set of GRE Math Misconceptions? Here are four more to watch out for.

Mistake: If you raise a negative number to a negative exponent, the answer will definitely be negative.

Fact: Weirdly enough, a negative number with a negative exponent can come out positive. Check out this expression:

Manhattan Prep GRE Blog - More GRE Math Misconceptions by Chelsey Cooley

To simplify it, remember that a negative exponent is “shorthand” for an exponent in the bottom of a fraction. This is what the expression really means:

Manhattan Prep GRE Blog - More GRE Math Misconceptions by Chelsey Cooley

What happens when you raise -2 to the 4th power?

Manhattan Prep GRE Blog - More GRE Math Misconceptions by Chelsey Cooley

The answer comes out positive.  

Why?: All that matters is whether the exponent is even or odd. If the exponent is even, the result will always be positive. That’s true even if the exponent is a negative even number. When you see a negative number and a negative exponent, don’t automatically assume that the answer will be negative too. Check whether the exponent is odd or even first.

 

Mistake: -1/3 is smaller than -1/2.

Fact: -1/3 is actually greater than -1/2.

Why?: Draw a number line:

Manhattan Prep GRE Blog - More GRE Math Misconceptions by Chelsey Cooley

On a number line, numbers to the left are always smaller than numbers to the right. Counterintuitively, since -1/2 is to the left of -1/3, -1/2 is smaller. (A quick way to check this is to decide which one should be further away from zero.) So, if you’re writing an inequality, it should look like this:

Manhattan Prep GRE Blog - More GRE Math Misconceptions by Chelsey Cooley

This is a good example of something that might seem clear now, but that can easily trick you if you’re working quickly. If you’re comparing negative numbers on the GRE, especially negative fractions, consider drawing a quick number line. At the very least, slow down for a moment!

 

Mistake: There are 12 numbers between 8 and 20, inclusive. (By the way, ‘inclusive’ just means ‘including the numbers at the ends,’ which are 8 and 20 themselves.)  

Fact: There are actually 13 numbers in that range. To be completely certain, count them:

8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Why?: The issue arises when you just subtract 8 from 20 and get 12. Just subtracting doesn’t give you the right number of terms. You need to add 1 to whatever you got.

To see why this happens, go back to the number line. Place a token on every integer from 1 to 20:

Manhattan Prep GRE Blog - More GRE Math Misconceptions by Chelsey Cooley

When you subtract 8 from 20, that’s the equivalent of removing the first 8 tokens:

Manhattan Prep GRE Blog - More GRE Math Misconceptions by Chelsey Cooley

When you do that, though, you don’t have tokens on the numbers from 8 to 20, inclusive! You’re actually missing one token: the one that’s supposed to be on 8. That’s why your result will turn out 1 too low. When you subtracted, you ‘removed’ one too many tokens.

 

Mistake: To find the solution to this equation:

x (3x – 1) = 8x

Start by dividing both sides of the equation by x.

Fact: Hold on! The equation actually has two solutions: x = 3, and x = 0. But if you start by dividing both sides by x, you only find the first solution. That could cause you to miss the right answer.

Why?: On the GRE, you’re never allowed to divide by 0. Since x is a variable, you don’t know what it equals: it could be 5, 100, -3, or even 0. If you divide by x, you could be dividing by 0 accidentally. That’s an ‘illegal move’ and will cause your math to come out wrong.

To avoid accidentally dividing by 0, don’t divide both sides of an equation by a variable unless you’re sure it doesn’t equal 0. Instead, simplify using addition, subtraction, and multiplication:

x (3x – 1) = 8x

3x² x = 8x

3x² – 9x = 0

x (3x – 9) = 0

Now, either x equals 0, or 3x – 9 equals 0 (in which case x is 3). 📝


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Chelsey CooleyChelsey Cooley Manhattan Prep GRE Instructor 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 GRE prep offerings here.

The post More GRE Math Misconceptions appeared first on GRE.

Fuente https://www.manhattanprep.com

GRE Math Misconceptions

Manhattan Prep GRE Blog - GRE Math Misconceptions by Chelsey Cooley

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Math can be counterintuitive. There are a few GRE Math misconceptions that really seem like they should be true—but actually aren’t. Being prepared for them will keep you aware on test day.

Mistake: 1 is prime.

Fact: 1 isn’t prime. In fact, the smallest prime number is 2.

Why?: It seems like 1 should be prime, because you can’t divide it by any other integers. However, mathematicians have agreed to say that 1 isn’t a prime. This makes certain mathematical theorems much simpler and more intuitive. Even though you won’t use those theorems on the GRE (phew!), you have to deal with their consequences by remembering that 1 isn’t prime.

Mistake: 3-4-5 and 30-60-90 triangles are the same thing.

Fact: A right triangle can be 3-4-5 or 30-60-90, but not both.

Why?: Here’s a couple of 3-4-5 triangles next to a couple of 30-60-90 triangles. Even if the triangles get bigger or smaller, the triangles on the left all have different proportions from the triangles on the right. So, if the sides of a right triangle have the ratio 3-4-5, you know the angles aren’t 30-60-90, and vice versa.

Manhattan Prep GRE Blog - GRE Math Conceptions by Chelsey Cooley

Mistake: If the ratio of teachers to students at a school is 1 to 4, then 1/4 of the people at the school are teachers.

Fact: In this scenario, only 1/5 of the people at the school are teachers!

Why?: A fraction always represents a part of a particular whole. In this case, the part is the number of teachers, and the whole is all of the people at the school. So, the denominator of the fraction has to be the sum of the teachers and the students, not just the students alone.

Try it out with numbers to confirm. If there are 10 teachers and  40 students, then 10 out of the 50 people at the school, or 1/5, are teachers.

Mistake: The average of the numbers from 1 to 10 is 5.

Fact: The average of the numbers from 1 to 10 is 5.5.

Why?: Intuition tells you that 5 is halfway from 1 to 10. However, to find the average of a bunch of consecutive numbers, you need to average the smallest and largest numbers together. The right answer will be the average of 1 and 10, which is (1+10)/2 = 11/2 = 5.5.

Confirm this by actually averaging the numbers from 1 to 10. Here’s the sum:

1+2+3+4+5+6+7+8+9+10 = 55

There are 10 terms, so the average is 55/10, which equals  5.5.

Mistake: If x is 25% greater than y, then y is 25% less than x.

Fact: If x is 25% greater than y, then y is only 20% less than x.

Why?: This is one of the most counterintuitive math facts out there, but the numbers back it up. Suppose that a coat costs 25% more than a sweater. If the sweater costs $100, the coat would cost 1.25($100), or $125.

However, if a sweater costs 25% less than a coat, and the coat costs $125, the sweater only costs 0.75($125) = $93.75.

‘Percent more than’ and ‘percent less than’ aren’t interchangeable. Pay close attention to which term the problem actually uses. If it says ‘percent more’ or ‘percent greater,’ then use a decimal greater than 1, such as the 1.25 figure from the example above. If it says ‘percent less’ or ‘percent smaller,’ then use a decimal lower than 1, such as 0.75.

You can also prove this specific example using fractions. If x is 25% greater than y, then x is 5/4 of y. Use algebra rules to get y by itself:

x = 5/4 y

4x = 5y

4/5 x = y

y is fourth-fifths as large as x. Since the missing 1/5 is equivalent to 20%, y is only 20% smaller than x. 📝


See that “SUBSCRIBE” button in the top right corner? Click on it to receive all our GRE blog updates straight to your inbox!


Chelsey CooleyChelsey Cooley Manhattan Prep GRE Instructor 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 GRE prep offerings here.

The post GRE Math Misconceptions appeared first on GRE.

Fuente https://www.manhattanprep.com