Tag Archive for 'GRE math'

GRE Multiple Choice Math Problem – Speed Trap

Do you know how fast you were going on that problem?

Do you know how fast you were going on that problem?

On every GRE Math section, the test makers try to come up with a few extremely difficult problems that will leave even the cleverest students scratching their heads. The really evil part, though, is that even these problems can be solved in under a minute without a calculator – if you know what to do. This means that once you “figure out the trick,” these difficult problems become easy. So, while those test makers are busy cackling with sadistic glee, let’s see if we can’t beat them at their own game.

Consider the following problem:

If x and y are integers and y < 20, for exactly how many ordered pairs (xy) will x^2 = y?

A) 4

B) 5

C) 7

D) 8

E) 9

This one actually doesn’t seem so bad, does it?

1^2 = 1

2^2 = 4

3^2 = 9

4^2 = 16

5^2 = 25 > 20

So we’ve got (1,1), (2,4), (3,9), and (4,16). Answer choice A, right? Not so fast! You forgot that the square of a negative number is also positive, so for every y, there are two x values: one positive and one negative. So really our list should look like this:

(1,1) and (-1,1)

(2,4) and (-2,4)

(3,9) and (-3,9)

(4,16) and (-4,16)

So the answer is D, right? Wrong again! There’s one last square you forgot:

0^2 = 0

Thus, there are in fact 9 pairs: the eight already mentioned, plus (0,0). Thus, the correct answer is actually choice E.

On the GRE, sometimes slow and steady does win the race.

On the GRE, sometimes slow and steady does win the race.

Was there actually anything hard about this question? Not really. However, if you were going fast and running out of time, you might have easily made one of the careless errors above. Note that 4 and 8 are traps set for students who see this problem, think it’s easy, and then blow through it too fast without thinking carefully (if you forgot the negatives but remembered 0, there’s also choice B, 5). If you get toward the end of a math section and see a problem that looks really easy, be careful – there’s probably more to it than meets the eye. Sometimes it’s just as bad to spend too little time on a problem as it is to spend too much, so make sure you don’t go too fast through any “easy” problems at the end of a math section.

Check back here each week for more problems and the tricks you need to solve them! Also, remember that you can find out all the tricks from experts like me with a Test Masters course or private tutoring. Until  then, keep up the good work and happy studying!

GRE Multiple Choice Math Problem – Prime Factorization

Math problems that appear hard but are secretly easy? Most excellent!

Math problems that appear hard but are secretly easy? Most excellent!

On every GRE Math section, the test makers try to come up with a few extremely difficult problems that will leave even the cleverest students scratching their heads. The really evil part, though, is that even these problems can be solved in under a minute without a calculator – if you know what to do. This means that once you “figure out the trick,” these difficult problems become easy. So, while those test makers are busy cackling with sadistic glee, let’s see if we can’t beat them at their own game.

Consider the following brain teaser:

What is the least common multiple of 18 and 14 that is also a perfect square?

A simple enough question. But how do you figure it out? Could we just multiply 18 by 14?

18*14 = 252

That’s definitely a common multiple of 18 and 14, but is it a perfect square?

252^(1/2) = 15.87450787…

Should’ve known it wouldn’t be that easy. This IS a hard problem after all. Maybe we should multiply 18^2 by 14^2:

(18^2)*(14^2) = 63504

Well, that works, but is it the LEAST common multiple? Don’t look at me! How should I know? Hmm…something tells me it probably isn’t. Maybe we could just go through the multiples of 18 trial and error style?

1*18 = 18. Not divisible by 14, not a perfect square.

2*18 = 36. It is a perfect square, but not divisible by 14.

3*18 = 54. Not divisible by 14, not a perfect square.

4*18 =

Time is running out, my pretty!

Time is running out, my pretty!

This is clearly going to take too long. We’re running out of time on this section – whatever shall we do? Never fear – prime factorization is here! Remember, all GRE math problems can be solved in under a minute without a calculator. If you know what to do, hard problems like this one become fast and easy. In this case, we should begin by finding the prime factors of both 18 and 14:

18 = 9*2 = 3*3*2

14 = 7*2

Remember, to find the prime factors of a number means to write out the numbers as products of only prime numbers like 2, 3, 5, 7, 11, 13, etc. In this case, the prime factors of 18 are 3, 3, and 2, and the prime factors of 14 are just 7 and 2. Now, if we just wanted to find the least common multiple of 18 and 14, we would multiply all the prime factors of 18 by all the prime factors of 14, but leave out any duplicates that appear in both lists. In this case you would multiply:

3*3*2*7 = 126

This works, because

(3*3*2)*7 = (18)*7 = 126

3*3*(2*7) = 3*3*(14) = 126

Note that you need only one 2 in 3*3*2*7, since both the prime factors of 18 (3*3*2) and 14 (7*2) include a 2. This is all well and good, except that 126 is not a perfect square:

126^(1/2) = 11.22497216…

So, what are we to do? Note that a perfect square multiplied by a perfect square is also a perfect square:

4*25 = 100

This means that we can pair up all of the unique prime factors of 126 so that we have a bunch of squares multiplied by each other. Thus,

3*3*2*7 = 126

becomes

(3*3)*(2*2)*(7*7) = 1764

We have to add an extra 2 and an extra 7 in order to make sure those numbers are part of perfect square pairs (we don’t need to add any extra 3s because there are already two of them). Sure enough, 1764 is a perfect square:

1764^(1/2) = 42

and 1764 is a multiple of both 18 and 14:

1764/18 = 98

1764/14 = 126

But, how can we be 100% sure that 1764 is the least common multiple? Well, think about it. Every perfect square that isn’t the square of a prime number will always be prime factorized into pairs like the ones above ((3*3)*(2*2)*(7*7) = 1764). Consider the following examples:

16 = 4*4 = (2*2)*(2*2)

36 = 6*6 = (3*2)*(3*2) = (3*3)*(2*2)

144 = 12*12 = (3*2*2)*(3*2*2) = (3*3)*(2*2)*(2*2)

You may have figured out this problem, but I'll get you next time! Next time!

You may have figured out this problem, but I’ll get you next time! Next time!

 

Et cetera. It’s inevitable. If the prime factors of a perfect square aren’t two prime numbers, then they will consist of multiple pairs of other prime numbers. Now, is there any way we could remove any pairs or make any of these factors smaller without breaking the conditions of finding the least common multiple of 18 and 14 that is also a perfect square?

(3*3)*(2*2)*(7*7) = 1764

I don’t think so. Remove or change one number and the product will either no longer be a perfect square or no longer be a multiple of both 18 and 14. Thus, 1764 must be the least common multiple of both 18 and 14 that is also a perfect square.

To sum up, the “trick” to solving a problem like this is to:

-find the prime factors of the two numbers in question

-pair them up

-multiply

If you know what to do, it takes about 30 seconds to solve this problem. So you see, with practice, even the hardest problems on the GRE become easy. Check back here each week for more extra hard problems and the tricks you need to solve them! Also, remember that you can find out all the tricks from experts like me with a Test Masters course or private tutoring. Until  then, keep up the good work and happy studying!

Quantitiative Comparison Angles of a Triangle

The Quantitative Comparison question type on the GRE can be very challenging. Essentially, you are given some information (usually in the form of a sentence, equation, or picture) and then two quantities. You are then asked to determine:

A if the quantity in Column A is greater;

B if the quantity in Column B is greater;

C if the two quantities are equal;

D is the relationship cannot be determined from the information given.

Like most questions on the GRE Quantitative section, the concepts involved in solving this question type are not of themselves very advanced, you just have to be careful in their application. Let’s go through an example problem:

First, look at the triangle formed by the points T, U, and V. We are given two angles of this triangle. Remember that the sum of the angles for a triangle will always equal 180°. This means we can subtract the measures of the two marked angles from 180° to find the third angle.

180° – 120° – 25° = 35°

This means that the angle at point U has a measure of 35°.

Now, the key to this problem is that the triangle formed by points R, S, and V has the same angle measurements as the triangle formed by points T, U, and V. The angle at point S is equal to the angle at point U. These two triangles are what are known as similar triangles. Similar triangles are triangles that have equal angles, and whose sides are proportional to each other. In other words, similar triangles will have the same shape, though not necessarily be the same size.

It may seem obvious from the picture that these triangles are similar, but how can we be certain? Generally, you will not want to only rely on the shape of the figure in the picture. Figures on the exam may not be drawn to scale.

We can be certain because of a rule of triangles known as the Side-Angle-Side rule. Here is the rule:

  • First, the two triangles must have an equal angle.
  • Next, the sides that form the angle on one triangle are equal to the sides that form the angle on the other triangle. Either this, or there is a common ratio between the sides of the two triangles.
  • If these two conditions are true then the triangles are similar.

In this case, the two triangles share the 120° angle at point V. Also, the sides forming that angle are in proportion to each other. Notice that line segment RT is equal to segment TV. This means that line segment RV is exactly twice the length of line segment TV. Since line segments VU and US are equal to each other, line segment SV is exactly twice the length of line segment VU. These two triangles pass the Side-Angle-Side rule and must be similar.

Since the triangles are similar, the angle at point S will be equal to the angle at point U. This means that the value of x is 35°. This is greater than 30, which means that the answer is A.

Quantitative Comparison Example Problem

The Quantitative Comparison question type on the GRE can be very challenging. Essentially, you are given some information (usually in the form of a sentence, equation, or picture) and then two quantities. You are then asked to determine:

A if the quantity in Column A is greater;

B if the quantity in Column B is greater;

C if the two quantities are equal;

D if the relationship cannot be determined from the information given.

Like most questions on the GRE Quantitative section, the concepts involved in solving this question type are not of themselves very advanced, you just have to be careful in their application. In fact, sometimes there is little or no need at all to do any “real” math; let’s take a look at the following GRE Quantitative Comparison Example Problem:

With geometry problems, it always helps to start with a formula. In this case, the problem is involving the volume of a cylinder. The formula for this is pi times the radius squared times the height.

Volume of a cylinder = πr2h

Therefore, to find the volume of a cylinder you must find both the radius of the cylinder and the height.

In this problem, two cylinders are given. Both cylinders have a different radius, and so it might be assumed that they must have a different volume. However, no information about the height is given. It is possible that the cylinder with the smaller radius has a larger height and might be the larger cylinder. Since there is no way of knowing how these cylinders compare to each other, the answer is D.

Test Masters offers the most comprehensive and successful GRE course available; every Test Masters GRE course, whether it is online or in-class, comes with a 10 point Score Increase Guarantee.

Quantitiative Comparison Word Problem

The Quantitative Comparison question type on the GRE can be very challenging. Essentially, you are given some information (usually in the form of a sentence, equation, or picture) and then two quantities. You are then asked to determine:

A if the quantity in Column A is greater;

B if the quantity in Column B is greater;

C if the two quantities are equal;

D is the relationship cannot be determined from the information given.

Like most questions on the GRE Quantitative section, the concepts involved in solving this question type are not of themselves very advanced, you just have to be careful in their application. Let’s go through an example problem:

Word problems require you to translate words into math. You will want to learn key phrases that often come up and their math equivalents. With this problem, we will rewrite the phrase “Steve purchased 3 more than half as many pencils as Joanne” into math. To start, Let the variable S mean the number of pencils Steve purchased.

“Steve purchased 3 more than half as many pencils as Joanne”

S = “3 more than half as many pencils as Joanne”

The phrase “more than” means addition.  The value of 3 is added to “half as many pencils as Joanne.”

S = “3 more than half as many pencils as Joanne”

S  = 3 + “half as many pencils as Joanne”

“Half as many” means half, or divided by two. Remember that the number of pencils purchased by Joanne is represented by p

S  = 3 + “half as many pencils as Joanne”

S  = 3 + “half of p

S  = 3 + (p/2)

Now, simplify by combining 3 and (p/2). Remember that to add fractions, you need a common denominator.

The number of pencils Steve purchased, or S, is the same as .  Therefore, the two values in the columns are equal and the answer is C.

Test Masters offers the most comprehensive and successful GRE course available; every Test Masters GRE course, whether it is online or in-class, comes with a 10 point Score Increase Guarantee.

GRE Graph Analysis Word Problem

For which of the following years was the ratio of the median sale price of a new home minus the median sale price of an existing home to per capita income least?

(A) 1960

(B) 1965

(C) 1970

(D) 1975

(E) 1980

The question asks for which year “the ratio of the median sale price of a new home minus the median sale price of an existing home to per capita income was least.” In other words, what is the smallest difference between the ratio of median sale price of a new home to per capita income and the ratio of median sale price of an existing home to per capita income.

TIP: Don’t allow yourself to be confused by complex phrasing, this question is all about being able to read the chart.

In the context of this question, the “ratio of the median sale price” simply refers to the lines of the graph; each line is a representation of the ratio of the median sale price (to per capita income), one for existing homes and the other for new homes. If all those words are confusing you, then just get rid of them:

For which of the following years was the ratio of the median sale price of a new home minus the median sale price of an existing home to per capita income least? Or,

For which of the following years was the price of a new home minus the price of an existing home least?

See how much more simple that question looks! Now all you have to do is determine the different ranges of the different options, by subtracting the … wait! No you don’t. Just look at the lines, there is no need to actually involve any math here. Since the difference between the two lines is represented by the space or gap between them, we can simply look and see for which year the gap was smallest. The smallest gap occurs in the year 1970. Therefore, the correct answer is (C).

Find more example problems here!

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GRE Math – Fun with Averages!

grumpy catStudying for the GRE can be tough. In the mean time, let’s make sure your math score is above average by reviewing averages! Consider the following problem:

The average (arithmetic mean) of six numbers is 14. After one of the numbers is removed, the average (arithmetic mean) of the remaining numbers is 16. What number has been removed?

To solve this problem, all you need to remember is the definition of an average:

average = (sum of terms)/(number of terms)

Multiplying both sides by the number of terms, we get:

average(number of terms) = sum of terms

First, let’s figure out the sum of the terms when the average was 14:

14(6) = 84

Next, let’s do the same for the situation in which the average is 16:

16(5) = 80

The difference between the two sums must be the number that was taken out:

84 – 80 = 4

Thus, the answer is 4. That’s all there is to it! Now, try the following problem and post the answer in the comments below:

The average (arithmetic mean) of four numbers is 23. After one of the numbers is removed, the average (arithmetic mean) of the remaining numbers is 15. What number has been removed?

Good luck, and happy studying!

GRE Example Problem – Order of Operations

GRE Math isn't so scary; just try this GRE example problem.

GRE Math isn’t so scary; just try this GRE example problem.

Learn more about what you need to know to do well on GRE Math by taking some time to complete this GRE Math example problem.

If L = (a – b) – c and R = a – (b – c), then L – R = ?

This example problem is an exercise in basic mathematical principles, particularly the Order of Operations and your understanding of the Commutative, Associative, and Distributive Laws of mathematics. Let’s do a brief review:

The Commutative Law essentially states that, when you add or multiply, you can swap the order of numbers and get the same answer. So, for example:

Addition:             X + Y = Y + X

Multiplication:  A x B = B x A

The Associative Law states pretty much the same as the Commutative Law with the additional declaration that when you are multiplying and adding groups of numbers, the grouping of those numbers is irrelevant. So, for example:

Addition:             (X + Y) + Z = (Z + Y) + X

Multiplication:  (A x B) x C = (C x B) x A

The Distributive Law says you get the same answer when you multiply a number by a group of numbers added together or multiply each number separately and then add them together. So, for example:

A x (B + C) = AB + AC

This might be easier to understand with actual numbers:

3 x (4 +5) = 3(4) + 3(5)

3 x 9 = 12 + 15

27 = 27

The Order of Operations determines the order in which certain mathematical operations act. The actual order of operations is Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. A particularly useful mnemonic device to remembering this (rather than memorizing the acronym PEMDAS) is “Please Excuse My Dear Aunt Sally.”

Okay, now duly armed, let’s return to the question above:

L – R = [(a – b) – c] – [a – (b – c)]

Notice that each equation has been bracketed off from the other. This is not because you cannot add or subtract these equations; it is only to signify and help you recognize that you are, in fact, beginning with and looking at the two different variables, L and R. Mainly, in this problem, brackets will help you keep track of which numbers are positive and negative.

In order to solve this problem, the first thing you should do is distribute the negative in front of the equation R represents, a – (b – c). The reason for this is that this equation includes two subtractions; so, when you subtract R from L, you will inevitably subtract a negative. Subtracting a negative turns that negative into a positive number. Observe:

L – R = [(a – b) – c] – [a – (b – c)]

L – R = [(a – b) – c] – [a – b + c]

L – R = [(a – b) – c] – a + b – c

After having successfully distributed the negative, the Commutative and Associative Laws, and the Order of Operations, tells us that we are free to solve this problem with no more hang ups:

L – R = a – b – c – a + b – c

You can reorganize for coherency:

L – R = a – a + b – b – c – c

L – R = 0 + 0 – c – c

L – R = -c – c

L – R = -2c

Thus, the answer is -2c.

Find more GRE example problems here. Have a question about GRE Math or graduate school admissions? Ask the experts at Test Masters!