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How to Create a GRE Problem Log for Quant

Manhattan Prep GRE Blog - How to Create a GRE Problem Log for Quant by Chelsey Cooley

Having a GRE problem log is like having a budget: sort of a pain sometimes, but much smarter than the alternative. Skeptical? Check out this article first—then come back here when you’re ready to roll.

1. Choose a format that inspires you.

Are you a gel-pen-loving bullet-journal enthusiast? Or would you rather something plain but practical, like a nice Excel spreadsheet? Your GRE problem log won’t work at all if you don’t write in it or look at it. A GRE problem log can be in any format that lets you record information in an organized way.

2. Light, heavy, or in between?

Some of us are natural self-analyzers. Some of us would rather just skip straight to the action. It’s okay if your GRE problem log is very simple. An elaborate problem log is great too. What matters is that you choose something that won’t feel like a burden.

3. The world’s simplest GRE problem log…

At the heart of it, the point of a GRE problem log is to remember what you’ve learned and to help you learn in the future. There are all kinds of little ‘aha’ moments that come from doing GRE problems: keeping a problem log makes sure those moments are recorded rather than vanishing.

With that in mind, here’s the world’s simplest GRE problem log:

Manhattan Prep GRE Blog - How to Create a GRE Problem Log for Quant by Chelsey Cooley

4. Not all takeaways are created equal.

The best takeaways are general. When you do a problem, you’re not going to see that same problem on your actual GRE. So, recording exactly how you did that specific problem is a waste of time. Your goal is to glean ideas from that problem that you could use on other problems.

The best takeaways remind you of not only what to do, but when to do it. Try to record not only which actions you took during a problem, but also how you knew what to do.

Here’s a problem from the 5 lb. Book of GRE Practice Problems:

If y≠0, what percent of y percent of 50 is 40 percent of y?

Here’s a quick solution:

  • Since y isn’t part of the answer, choose a number for y. We’ll choose 100, so we can read the problem like this:

What percent of 100 percent of 50 is 40 percent of 100?

  • Start by writing the equation as follows:

Manhattan Prep GRE Blog - How to Create a GRE Problem Log for Quant by Chelsey Cooley

  • Then, simplify the equation to solve for p:

Manhattan Prep GRE Blog - How to Create a GRE Problem Log for Quant by Chelsey Cooley

And here are some great takeaways:

  • If you see a variable percent (y%), but you aren’t solving for y, just choose 100 for y!
  • Translate ‘what percent’ as ‘p/100’ and translate ‘of’ as multiplication
  • If you see ‘100 percent’, you can just ignore it while doing math
  • Complicated percent problems can be easier to solve with fractions, rather than decimals

5. ‘Do it again?’

You don’t have to redo every single problem, or even every problem you missed. The best problems to redo are the ones that were right at the edge of your ability level. Don’t bother with the ones that were ridiculously hard, or the ones that you missed for a silly reason (although you should still write those ‘silly reasons’ down.) Redo the ones that you know you could get right with just a bit more studying.

6. Taking it up a notch…

Here’s some more information you may want to include in your GRE problem log:

  • What topic was the problem testing? This way, you can quickly skim your log to find all of the Algebra problems, or all of the Geometry problems, and so on.
  • What answer did you pick? This is useful when you redo a problem: compare your answer now to the one you got originally.
  • How long did it take? Log problems that take you a long time as well as the ones you got wrong. When you do them again, try to beat your previous time.

7. Take it up two notches.

You might not include this information for every single problem you do, but it can be useful for some problems!

  • If you got it wrong: what type of error was it? I like to think in terms of conceptual errors (you didn’t know how to do something), process errors (you knew how to do it, but didn’t choose the right approach), and careless errors (you added 2 plus 3 and got 7).
  • For Quantitative Comparison problems: what cases did you test? If you didn’t test cases… that may be a clue to why you missed the problem! What cases should you have tried?
  • Were there any interesting trap answers or easy mistakes to make in the problem – regardless of whether you fell for them yourself? What would be the easiest ways to get this problem wrong?
  • Is there a better or faster way to solve it – even if your approach worked?

8. Now what?

So, you’ve created this GRE problem log, and you’ve started filling it up with Quant problems. Now what? Twice per week, on the same days every week, read over your GRE problem log. On one of those two days, just reread it and look at your takeaways. On the other day, redo some or all of the problems you’ve decided to redo, and record whether you got them right this time. With time, you’ll find yourself thinking about lessons learned from old problems when you’re doing new ones—and that’s exactly what you need for test day. 📝


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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 How to Create a GRE Problem Log for Quant appeared first on GRE.

Fuente https://www.manhattanprep.com

Using Smart Numbers for GRE Quant

Manhattan Prep GRE Blog - Using Smart Numbers for GRE Quant by Chelsey Cooley

Here’s a quick cheat sheet on how, when, and why to use Smart Numbers to solve GRE Quant problems.

What is Smart Numbers?

Smart Numbers is a strategy for certain GRE Quant problems, usually word problems. It’s not a guessing method—in other words, using Smart Numbers will give you the exact right answer, just like doing algebra will.

When can you use Smart Numbers on GRE Quant?

You’ll decide whether to use Smart Numbers by looking at the answer choices (so, it’s most often useful on Discrete Quant problems, which have answer choices!).

If you see the following in the answer choices, you can definitely use Smart Numbers:

  • Expressions with variables in them, such as 3x or 4y + z.

You can also usually use Smart Numbers if you see the following in the answer choices:

  • Percents
  • Ratios
  • Fractions

If you see percents, ratios, or fractions, here’s how to make the decision. Read the whole problem, and decide whether you’re dealing with specific numbers, or just with relationships between numbers.

For instance, does the problem say that x equals 12, or that Beryl has sixteen cats? Those are specific numbers, and you probably can’t use Smart Numbers.

On the other hand, if x is 50% more than y, or if Beryl has twice as many cats as Jane, those are relationships—and you probably can use Smart Numbers.

There are a few other special situations, so I’ll also give you a rule that covers everything—although it takes a little bit more thinking to apply it. If a GRE Discrete Quant problem doesn’t tell you the numbers, but just tells you how they relate to each other, you can use Smart Numbers. If it does tell you specific numbers, you can’t.

How does Smart Numbers work?

Suppose you’ve decided to use Smart Numbers because there are variable expressions in your answer choices. For instance, the problem looks like this:

If a, b, c, and d are consecutive integers and a < b < c < d, what is the average (arithmetic mean) of a, b, c, and d in terms of d?

A) d – 5/2
B) d – 2
C) d – 3/2
D) d + 3/2
E) (4d – 6)/7

In this situation, start by choosing numbers that fit all of the facts the problem gives you. In this one, the four numbers you choose have to be consecutive, with a being the smallest, and d being the largest.

As long as the numbers fit the facts, you should use the easiest numbers you can think of. For this problem, let’s go for 1, 2, 3, and 4.

The next step is probably the most important one: everywhere you see a variable in the problem—including the answer choices!—replace it with the number you chose. You can use a combination of mental math and scratch work to do this, depending on how complex the problem looks.

By the way, during this step, you should forget about the phrase “in terms of d.” “In terms of” only matters when you’re using variables. Since we’re replacing our variables with numbers, we can just drop it.

Here’s what that problem would look like, once we’re finished with this step:

If 1, 2, 3, and 4 are consecutive integers and 1<2<3<4, what is the average of 1, 2, 3, and 4?

A) 4 – 5/2
B) 4 – 2
C) 4 – 3/2
D) 4 + 3/2
E) (4*4 – 6)/7

Next, answer the question. What is the average of 1, 2, 3, and 4? It’s 2.5.

Which of the answer choices equals 2.5? Only (C) does. (By the way, you can often figure this out without doing too much math—for instance, you should eliminate (B) quickly, since it won’t result in a decimal.)

Let’s try another one. This time, suppose you’re using Smart Numbers because you noticed percents in the answer choices. Your problem might look like this:

Aloysius spends 50% of his income on rent, utilities, and insurance, and 20% on food. If he spends 30% of the remainder on video games and has no other expenditures, what percent of his income is left after all of the expenditures?

A) 30%
B) 21%
C) 20%
D) 9%
E) 0%

Pick a number that fits everything you’re told in the problem. This problem doesn’t really give us any constraints on the number—except that it’s a dollar amount, so it shouldn’t be negative—so we can pick more or less any number we want. Let’s say that Aloysius’s income is $100.

You don’t have to replace the variables with numbers in this scenario, because there aren’t any variables! If the problem only has percents or ratios, not variables, you can skip that step. Go right ahead and solve the problem.

50% of $100 is $50, and 20% of $100 is $20. That leaves $30 remaining. Aloysius spends 30% of that $30, or $9, on video games. His total expenditures are $50+$20+$9, or $79, with $21 left over. Since $21 is 21% of his original income, the right answer is (B).

Why should you use Smart Numbers?

In some situations, using Smart Numbers takes more time than just doing the algebra. If you’re fast and confident with algebra, there will be problems where you’ll save time by “just doing the math.” However, there are other advantages to using Smart Numbers:

  • It’s easier to check your work with numbers than with variables.
  • It makes it easier to convert between different units. It’s much easier to convert 100 pennies to dollars than to convert 4x pennies to dollars.
  • It makes it easier to work with percentages. I know that 3 is 50% of 6, but it’s not nearly as obvious that 3xy is 50y% of 6x.
  • It’s often an easier way to solve a very tough word problem. If you’re having a hard time setting up equations based on a word problem, it may become clearer when you try using specific numbers.

However, I do have one warning: don’t think of Smart Numbers as a last resort! If you wait until you’ve already spent two minutes on the GRE Quant problem, using Smart Numbers isn’t going to help you. Try using it first—after all, there’s no rule saying you have to try algebra before you can do something else. On the GRE, you’re free to use whichever approach works, even if your middle school algebra teacher would disapprove! 📝


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 Using Smart Numbers for GRE Quant appeared first on GRE.

Fuente https://www.manhattanprep.com

Can You Ace GRE Quant if You’re Bad at Math? (Part 3)

Manhattan Prep GRE Blog - Can You Ace GRE Quant if You're Bad at Math? (Part 3) by Chelsey Cooley

If you have a complicated relationship with math, you need to be especially careful about how you study. Some GRE Quant study techniques might seem to make perfect sense, but can actually leave you frustrated and demoralized in the long run. For painless studying, try these next few ideas instead.

(If you’re just joining us now, check out the previous two articles in this series before you keep reading. In the first one, we dispel the “bad at math” myth. In the second, we go over some simple approaches to gain momentum and learn the basics.)  

The When and Why of GRE Quant Rules

Part of the “bad at math” mindset is the feeling that math is sort of like magic. When you watch an expert solve a math problem, it’s like watching someone pull a rabbit out of a hat: you can see what’s happening, but you don’t know what they actually did.   

That’s compounded by the way that a lot of us learn math in school. Unless you had great elementary school math teachers, you probably learned math as a long list of rules and operations. You probably spent a lot of time learning to apply each rule correctly, and much less time learning when to use each rule.

So, if you took a test on multiplication in elementary school, you’d pass as long as you multiplied the numbers correctly. That doesn’t work on GRE Quant. To ‘pass’ the GRE, you have to not only multiply correctly, you have to decide whether to multiply in the first place.

That’s a skill that you won’t get from memorizing rules. You also won’t get there by drilling one problem type over and over until you can perform it perfectly, then moving on to the next one. If you don’t also know the “when and why,” the real test will seem much harder than your practice sessions.

So, what can you do? My first piece of advice is to create “when I see this, do this” flashcards. Those are discussed in detail here. Every time you do a GRE Quant problem, try to spot clues that you could use in other problems. Then, identify what you’re supposed to do when you notice one of those clues. Put those two things on the front and back of a flashcard, and keep it handy. Periodically, go through all of these flashcards and test your “what to do next” knowledge.

Second, regularly set aside time to do random sets of actual GRE Quant problems. This is more and more important the closer you get to test day. It forces you to not only solve the problems, but also figure out what they’re testing in the first place, and what approach to take. Instead of just skimming through your mental cheat sheet on a single topic, you have to choose from among everything you know about GRE Quant. That’s not something that comes naturally, but it will improve if you start practicing it!

Take GRE Quant Step by Step

Think of your GRE Quant knowledge as a jigsaw puzzle. Each time you learn a new fact or skill, someone hands you a new puzzle piece. If you already have the surrounding pieces in place, it’ll be easy to fit the new one in. But if you’re just getting started, and someone hands you a random piece from the middle of the puzzle, it’s almost impossible to decide where it goes.

Don’t start your GRE Quant studies by picking random pieces from the middle of the puzzle. Start with the corners and the edges: the math foundations. Check out the previous article for a list of starting places and some ideas on how to approach them.   

From there, aim to “push your GRE Quant score up from below,” rather than “dragging it up from above.” You’ll gain more points by really mastering the easy or moderate problems than you will by conquering the very hardest problems—and this will take less of your limited study time and build your confidence as well. Spend a little more of your time on the problems that are just a bit too hard for you—the ones where you have all of the surrounding puzzle pieces in place, but you haven’t quite placed the very last one. And avoid wasting time on the very toughest problems, unless those are really the only ones that are challenging for you.

It may seem satisfying to continue drilling one topic until you’re comfortable with it, but this can also lead to frustration when it doesn’t work out. Worse, it’s a poor strategy for memory formation. You’re better off moving around the jigsaw puzzle, changing which bit you’re working on in order to stay fresh. (This means that even if you’re spending almost all of your study time on GRE Quant, a little work on Verbal can be good for both your morale and your score.)

It’s fine to not understand things, to make mistakes, and to get problems wrong, even all the way up until test day. Focus on learning the material that’s most within your grasp right now, and learning it in the most efficient and effective way you can. Why not check out GRE Interact to get started? 📝


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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 Can You Ace GRE Quant if You’re Bad at Math? (Part 3) appeared first on GRE.

Fuente https://www.manhattanprep.com

GRE Sentence Equivalence: Charge Traps

Manhattan Prep GRE Blog - GRE Sentence Equivalence: Charge Traps by Chelsey Cooley

In this article, GRE instructor Tom Anderson asks a smart question: is it better to sort of know a lot of GRE words, or to really know a few GRE words? It turns out that you’re better off if you learn fewer words, but really learn them well. If you don’t, here’s one way the GRE could trick you.

Most of the toughest GRE words are adjectives: descriptions of people, situations, or things. Adjectives are a bit like cupcakes. A sweet cupcake is good, but a cupcake that’s too sweet can make your teeth ache. They’re both sweet, but one is tasty, and the other is, well, gross.

Likewise, lots of GRE adjectives have “evil twins.” One word is sweet, but the other is too sweet. Here are some examples.

thrifty – miserly

sentimental – mawkish

respectful – obsequious

devout – priggish

ornate – ostentatious

All of these pairs share the same relationship. Someone who’s miserly is too thrifty. If a poem is mawkish, it’s too sentimental. If an employee is obsequious, she’s not just respectful, she’s so respectful that it’s kind of weird. And so on. The second word is a “too sweet” version of the first word.

If you only sort of know these words, you can see how you might assume they mean the same thing. After all, thrifty and miserly both mean “cheap,” and ornate and ostentatious both mean “fancy.” But do they mean the same thing on the GRE? Nope.

So, what if you see both of them in the answer choices? It depends.

Suppose you’re doing a GRE Sentence Equivalence problem—the type of problem where the two right answers will be synonyms. Here’s one possible set of answer choices:

crafty

gawky

hardy

miserly

stingy

thrifty

The first three answer choices are right out, since none of them has a twin. That leaves us with miserly, stingy, and thrifty. Let’s call this ‘situation number 1’—where you have three answer choices that sort of mean the same thing.

This is what we call a charge trap. The three words have similar meanings, but one of them has a different “charge”—thrifty is a neutral word, while miserly and stingy are much more extreme, and therefore bad. Since only two of the words really match each other, you should choose miserly and stingy, regardless of what your fill-in was.

When you learn a new word, take note of whether it has a strong charge, either good or bad. This is especially true if it’s a more extreme version of some other word you already know. If you’re not sure what the charge of a word is, search for it online and check out how people are using it!

Okay, here’s situation number 2, with a different set of answer choices:

elaborate

gaudy

ornate

ostentatious

pragmatic

rustic

Two answer choices—pragmatic and rustic—are definitely out, since they have no twins. That leaves four possibilities, of which you need to choose two. Take a moment and divide those four words into two pairs, based on their charge.

Ready? Here we go. Elaborate and ornate have the same (neutral) charge, while gaudy and ostentatious share a negative charge. Other than that, they basically mean the same thing: fancy.

To choose a pair, let’s go back to the golden rule of GRE Verbal: Find the Proof. Every GRE Verbal problem has one and only one right answer, and you can always prove that the right answer is right.

If you can’t prove that a strong word is right, you should choose a neutral one. However, if the sentence contains proof for the stronger word, the stronger word is the right answer. Here’s a sentence that might go with those answer choices from above:

“Gilding the lily” is a 19th-century expression that was first coined to describe the ________ décor adopted by those who were too eager to display their recently acquired wealth; some owners of Beaux Arts homes, for instance, would cover up the beautiful but subtle carvings of flowers around their entranceways with a layer of flashy gold gilt.

There’s a lot of proof here for ostentatious and gaudy. The homeowners were too eager to display their wealth; they covered up beautiful but subtle features of their homes in favor of something more flashy.

This next sentence doesn’t have proof for a strong word, so you should pick the neutral pair:

In the late 19th century, architecture and decoration took a turn for the ________, with many owners of Beaux Arts homes embellishing their entryways with intricate carvings of flowers inscribed with gold gilt.

There’s no proof here that the decoration was too ornate—and if you can’t prove the stronger answer, you can’t pick it. If this is the sentence you’re dealing with, choose ornate and elaborate.

In short, here’s how to avoid charge traps:

  • Pay attention to charge when you learn new GRE words;
  • If you see three similar words in the answers, ask yourself whether they have different charges;
  • If you see two pairs with different charges, only pick what you can prove using the sentence.

If you follow those guidelines, you’ll deepen your vocabulary knowledge and protect yourself against trap answers on GRE Sentence Equivalence! 📝


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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 GRE Sentence Equivalence: Charge Traps 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. 📝


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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 GRE Math Misconceptions appeared first on GRE.

Fuente https://www.manhattanprep.com

GRE Quantitative Comparison Questions, solved exercises step by step; clases particulares GRE en Chile, tutor, tutorials, class, course, The units digit of

entrenamieno

 

GRE Quantitative Comparison Questions

x is a positive integer.

Quantity A
Quantity B
The units digit of 6^x
The units digit of 4^{2x}

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

 

 

 

 

 

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Answer: C

 

Resolución paso a paso:

Este es un ejercicio del item “comparision” del GRE QUANT.
Debemos prestar atención a ¿qué nos piden comparar?
Ojo: a pesar de que en el enunciado hablan de una variable “x”, que se mueve en los valores de los enteros positivos. lo que nos piden comparar, es el valor de la unidad, al desarrollar dos expresiones, una en A y otra en B, en ambas expresiones aparece la variable “x”. Esto es un ejercicio que nos pide “trabajo en la ambigúedad” (no se define un valor fijo para “x”, dado que lo único que se afirma, es que es un entero positivo, esto quiere decir que puede tomar cualquier valor entero positivo).
Además para responder correctamente el alumno debe saber trabajar con potencias y sus propiedades.
No se nos pide comparar “x”, sino la consecuencia de actuar como exponente en una potencia de base entera (6), para ver que resulta en la unidad del número obtenido.

En A, se nos presenta  6^x, debemos ver que ocurre con el valor de la unidad al desarrollar para los distintos valores de exponente entero positivo, haciendo una inspección nos damos cuenta que 6^x, para x=1, es 6 (valor de la unidad 6) , para x=2, es 6*6=..6, es 6 (valor de la unidad), para x=3, es el valor de la unidad de x=2, multiplicado 6, es 6 (valor de la unidad y así sucesivamente.
Conclusión: independiente del valor entero positivo que tome x, el valor de la unidad de 6^x, siempre será 6.

En B, se nos presenta 4^{2x}, debemos ver que ocurre con el valor de la unidad al desarrollar para los distintos valores de exponente entero positivo.
Aplicando una propiedad de potencias, tenemos que    4^{2x}=( 4²)^x= 16^x, para ver el valor de la unidad de 16^x, vasta con analizar cómo se comporta 6^x, este hecho fue analizado en A, es decir estamos en la misma situación de A.

Luego, lo afirmado en A, es igual a lo afirmado en B, ya que en ambos casos, se obtiene un valor único para la unidad, esto es 6.

Respuesta correcta C) 

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