09 Math Bootcamp, GRE prep!

800-2Review (800-273-8439) PrincetonReview.comTest names are the trademarks of their respective owners, who are not affiliated with The Princeton Review. The Princeton Review is not affiliated with Princeton University.

The Princeton Review, Inc. | 1

WELCOMEIf its been years since youve taken a math class, then you might feel that your computation skills and math knowledge are a little rusty. Its probably been some time since youve regularly done multiplication and long division without a calculator handy. You may not remember the precise denition of a prime number or how to solve an inequality.

These sorts of skills and knowledge form the basis of whats tested in GRE and GMAT math sections. Many questions test your knowledge of math concepts, and even the most dicult questions involve math fundamentals. To do your best, you also need to be able to perform calculations quickly and accurately. In this seminar, we will help bring you back up to speed by reviewing math basics.

We cant cover everything you need to know about math in just a few hours, so well focus on the most important core material. Well discuss:

Number properties

Fractions, decimals, and percentages

Algebra

Exponents and Roots

Rest assured that the prework and lessons in your course oer comprehensive coverage of all the math tested on your exam. Youll build on what you learn in this session, and you will be prepared for anything you might encounter on the GRE or GMAT.

2 | The Princeton Review, Inc.

GMAT/GRE Math Review

NUMBER PROPERTIES

INTEGERS

Integers are whole numbers including negative numbers and 0. No fractions; no decimals.

Examples of Integers Examples of Non-integers

14, 5, 0, 39, 1438 23.6, 73

, 3

4, 59.475

Circle the integers below:

68

7.00 12

4

EVEN AND ODD

An even number is divisible by 2. An odd number is not divisible by 2. Even and odd apply only to integers! Th eres no such thing as an even or odd fraction.

Rules for Operations with Even and Odd NumbersIf you forget these rules, you can always gure them out by testing a pair of numbers. Note that there are no even/odd rules for division. Division often produces fractions, which are neither even nor odd.

Rule Example

Even Even = Even 4 6 = _______

Even Odd = Even 8 7 = _______

Odd Odd = Odd 3 5 = _______

Even Even = Even 12 + 6 = _______

Odd Even = Odd 17 6 = _______

Odd Odd = Even 7 + 5 = _______

Indicate whether x is even, odd, or if its impossible to determine:

1. 4x + 7 is odd Even Odd Cant determine

2. x = 32,455 2,021 Even Odd Cant determine

3. x = even number even number Even Odd Cant determine



POSITIVE AND NEGATIVE

Negative numbers are less than 0. Positive numbers are greater than 0.

Rules for Multiplication and DivisionIf the signs are the same, multiplication or division produces a positive result.

Positive Positive = Positive 3 7 = _______

Negative Negative = Positive 30 10 = _______

If the signs are di erent, multiplication or division produces a negative result.

Positive Negative = Negative 6 7 = _______

Negative Positive = Negative 2 8 = _______

Indicate whether x is positive, negative, or if its impossible to determine.

1. x =

( )45

2

3

5352

. Positive Negative Cant determine

2. 3x 7 is even Positive Negative Cant determine

3. 0 x = 0 Positive Negative Cant determine

PROPERTIES OF ZERO

Zero has some special properties that are important to remember:

0 is an integer.

0 is an even number.

0 is neither positive nor negative.

0 times anything is equal to 0.

0 divided by anything is equal to 0.

Anything divided by 0 is unde ned. Division by 0 is impossible.

Determine whether the following statements are true or false.

1. All odd numbers are positive or negative. True False

2. All even numbers are positive or negative. True False

3. 1 0 = 0 True False



ABSOLUTE VALUERemember the number line?

6 5 4 3 2 1 0 1 2 3 4 5 6

How many units away from 0 is the number 5? _______ Th is number represents the absolute value. Th e absolute value of any positive number is simply the number itself.

How about negative numbers? Whats the absolute value of 5? How many units away from 0 is 5? _____Th is is the absolute value. Th e absolute value of any negative number is the positive equivalent of the number.

Absolute value is a term that represents how far a number is from 0 on the number line. Th e symbol for absolute value is . Th us, 5 = 5 and 5 = 5.

Determine the value of x in the following:

1. x = 3

2. x = 244

3. x = 6

FACTORS

Factors are numbers that divide evenly into a given number.

For example, 6 is a factor of 18 because 6 divides evenly into 18 three times. 18 6 = 3. Th is also means that 3 is a factor of 18 because 18 3 = 6. We can say that 3 and 6 are both factors of 18.

To nd all the factors of a number, start with 1 and the number itself. Move up in pairs until the num-bers converge.

Here are all the factors of 60. (In ordinary math and on standardized tests, we only concern ourselves with positive factors.)

1 and 60

2 and 30

3 and 20

4 and 15

5 and 12

6 and 10

Th ats it. Notice that the pairs have gotten closer together. Th ere are no factors between 6 and 10, so you know youve found them all.



List all the factors of the following numbers:

36 45 72

MULTIPLES

A multiple is the result of multiplying two positive integers. To put it another way, a multiple is a number that is divisible by a factor.

Multiples are the ip side of factors. Because 6 is a factor of 18, 18 is a multiple of 6. While you can list all the factors of a number, you cant list all the multiples, because they go on forever. Th e multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, ... never ending.

1. What is the rst multiple of 15?

2. Is 21 a multiple of 42?

3. What is the lowest common multiple of 6 and 8?

PRIME NUMBERS

A prime number is a number with only two factorsone and itself. Th e number 1 is not prime.

Th e rst ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Prime numbers only refer to positive integerstheres no such thing as a negative prime number or a prime fraction. Two is the rst prime num-ber, and its the only even prime number. All other prime numbers are odd.

Answer the following questions:

1. x is the product of an even single-digit prime number and an odd single-digit prime number. What are the possible values of x? _______

2. How many prime numbers are there between 30 and 50? _______

3. Integer y has a factor n such that 1 < n < y. Is y prime? _______



PRIME FACTORIZATIONBreaking a number down into its prime factors can be handy in a number of situations. To do so, create a factor tree and keep breaking the numbers into factors until you have prime numbers.

120/ \

6 20/\ /\

2 3 4 5 /\ 2 2

Th us, the prime factorization of 120 is 2 3 2 2 5.Break down the following numbers into their prime factorizations.

28 72 154

ORDER OF OPERATIONSWhen youre answering test questions, you must know in what order to perform calculations. Th e standard acronym for this is PEMDAS, which you can also be remember with the sentence: Please Excuse My Dear Aunt Sally. More properly it looks like this:

P|E|M D|A S

P stands for parentheses. Solve expressions in parentheses rst.

E stands for exponents. Solve expressions with exponents next.

M stands for multiplication, and D stands for division.

Do all the multiplication and division together in the same step, going from left to right.

A stands for addition, and S stands for subtraction.

Do all the addition and subtraction together in the same step, going from left to right.

Perform the following operations:

1. 6 150 3 2 3 5 42

+( ) + = 2. 7 3 15 8 4 8 44 42 ( ) + = 3. ( )( ) + =4 10 25 3 22



FRACTIONS, DECIMALS, AND PERCENTAGESFractions, decimals, and percentages are simply dierent ways of expressing part-to-whole relationships. Its important to be comfortable working with all three.

FRACTIONSFractions are expressed with a fraction bar, which is also a division bar. When we write the fraction

2

3, were

really saying 2 3.

The top part of a fraction is called the numerator, and the bottom is called the denominator, so in 2

3, 2 is the

numerator and 3 is the denominator.

Another good way of thinking about fractions is as a part

whole relationship.

If there are 5 men in a room and 7 women, what fraction of the room is male? _______

When the numerator is greater than the denominatorand theyre both positiveyou have a fraction

greater than 1, also called an improper fraction. For example, 5

2 is greater than 1. If you did the division, you

would get 21

2. This is called a mixed number.

1. How would you express 42

3 as an improper fraction? _______

2. Express 16 as a fraction. _______

A reciprocal is what you get when you ip a fraction upside down.

1. What is the reciprocal of 16? _______

2. What is the reciprocal of

1

1

4? _______

3. What is the value of 5

5? _______

Any number divided by itself is equal to _______.



Comparing FractionsAnswering test questions may require you to compare fractions and determine which quantity is greater.

Which is larger, 5

8 or

6

8?

12

5 or

13

5?

When the numerator of a fraction changes, and the denominator remains the same, the value of the fraction will go in the ______________ direction as the numerator.

Which is larger, 3

7 or

3

8?

7

2 or

7

3?

When the denominator of a fraction changes, and the numerator remains the same, the value of the fraction will go in the ______________ direction as the denominator.

The Bowtie makes it easy to compare fractions. Multiply diagonally up (opposing denominators and numer-ators). The side with the larger product has a greater value.

Which is larger, 4

7 or

5

9?

Which is larger, 8

3 or

13

5?

Reducing FractionsReducing fractions means dividing the top and bottom of the fraction by the same number until no more division is possible. Reduce the following fractions:

18

126

12

60

23

56

Multiplication and DivisionMultiplying fractions is a snap. Just multiply across the top and bottom. You can multiply and reduce, or you can cancel rst, then multiply.

1. 3

4 2

5 = 3 2

4 5

=

2. 6

4 5

9 =

Dividing fractions is also simple. Just ip the second fraction upside down and multiply.

1. 3

5 5

8 =

2. 9

5

3

10 =

Addition and SubtractionWhen the denominators are the same, simply add or subtract the numerators.

3

11

5

11+ =



Addition and subtraction are a little trickier when you have dierent denominators, but using the Bowtie makes the calculation easy.

1. 2

9

3

4+ =

2. 5

4

1

3 =

3. 7

9

5

2

5

6

2

3+ + =

On problems like the last one, youre best o nding common denominators. For other problems, you can use the Bowtie or common denominators, whichever allows you to solve problems more quickly and accu-rately.

Fraction Tricks

1. Multiplying a number by 1

5 is the same as dividing it by ______________.

2. Therefore, 1

7 of 42 = ______________.

3. What is 2

7 of 42? ______________

One way to take a fraction of a number is to divide by the ______________ and then multiply by the ______________.

DECIMALSDecimals are another way of expressing a

part

whole relationship. Any fraction can be converted into a decimal

and vice versa.

Adding and Subtracting DecimalsTo add and subtract decimals, line up the decimal points, and add or subtract as you normally would.

1. 46.15 + 127.74 =

2. 8.654 3.27 =

Multiplying DecimalsTheres a neat trick for multiplying decimals. Ignore the decimals at rst and multiply the numbers as if they were just plain integers. Next, count how many decimal places were in your original numbers and add them back to the result.

1. 4.21 3.6 =

2. 12.9 25.8 =

3. 4.7 3.1 .44 =



Dividing DecimalsDividing decimals is a little dierent from multiplying them. Move the decimal points of both numbers until the divisorthe number youre dividing byis an integer.

1. 148.75 42.5 )425 14875. .

2. 12.8 6.25 )625 128. .

3. 21.6 8 )8 216.

Place ValueIts useful to understand place value when dealing with decimals. What does each digit represent in the number 4368.3279?

4 3 6 8 3 2 7 9

_____ _____ _____ _____ _____ _____ _____ _____

.325 expressed as a fraction is __________.

Which is larger, the hundreds digit of 10,674, or the hundredths digit of 1.438?

Decimal TricksOne of the neat things you can do with decimals is multiply and divide numbers by 10 simply by moving the decimal point. Multiplying by 10 moves the decimal one place to the right, and dividing by 10 moves the decimal one place to the left.

1. 13.458 100 = __________

2. 142 10 = __________

PERCENTAGESPercentages are one more way of expressing a

part

whole relationship. A percentage is a fraction of 100. Percent

literally means per centper 100. So 35% is the equivalent of 35

100. Its also equivalent to the decimal .35.

Percentages can thus be converted easily into decimals and fractions, and vice versa.

To convert a percentage into a decimal, move the decimal point two spaces to the __________.

To convert a decimal into a percentage, move the decimal point two spaces to the __________.

1. 121

2% = __________

2. 1.36 = __________%



Percent TranslationMany word problems involving percentages can be translated into math equations, which you can then solve. Heres a quick English-Math dictionary.

English Math

percent 100

of

what x, y (any variable)

is =

1. What is 15% of 60?

2. 20 is what percent of 125?

3. 30% of 50% of 60 is equal to __________.

Percent Change

To calculate percent change, use the formula:

Percent change = differenceoriginal

100

If the problem asks for a percent increase, or what percent greater, then the original number is the _______one. (You can only increase if you started smaller than you ended up.)

If the problem asks for a percent decrease, or what percent less, then the original number is the __________ one. (You can only decrease if you started larger than you ended up.)

1. In 1995, only 2,400 homes in Belmont County had high-speed internet access. In 1999, 3,000 homes had high-speed internet access. What was the percent increase in Belmont County of homes with high-speed internet access?

2. Jasons music collection contains 600 CDs. Sandras music collection contains 500 CDs. Sandras music collection is what percent smaller than Jasons?

3. Which is larger, the percent increase from 4 to 5, or the percent decrease from 5 to 4?



Percentage TricksBe careful whenever youre asked to compare, add, or subtract percentages. There are some sneaky pitfalls you must avoid.

1. If you increase the price of an item 25% and then reduce the new price by 25%, the resulting price is lower than/higher than/the same as (circle one) the original price.

2. If you discount an item 10% and then reduce the price by another 10%, the total dis-count is 20%. True False

3. If 32% of the boys in a school are honor students and 32% of the girls in a school are honor students, then there are an equal number of male and female honor students at the school. True False

Tip CalculationThe technique you use to calculate a 15% or 20% tip can be used in other situations with percentages.

Restaurant check = $52.40

Desired tip = 15%

10% of 52.40 = __________

5% of 52.40 = __________

15% of 52.40 = __________

Now try these:

1. 30% of 75 =

2. 12% of 1,200 =

BALLPARKINGBallparking, or estimating, is a very important skill for standardized tests. Not only does this technique save you from considering answer choices that, with a little thought, are clearly too large or too small to be cor-rect, but it also allows you to use easier numbers in your calculations.

1. 3

7 expressed as a decimal is approximately .

2. 4 is approximately equal to .

3. .247 40 is approximately equal to .

How to Round NumbersIn order to know whether its safe to round a number, consider the size of the number youre starting with and by how much youre rounding. As a rule of thumb, never round by more than 10 percent and try to keep it to 5 percent or less.



Are the following roundings are safe to use?

1. 225 to 300?

2. 246,000 to 250,000?

3. 53 to 50?

4. 13 to 10?

5. .343 to .35?

ALGEBRAIf its been a long time since you had algebra in high school, you might be a little rusty, but with a little prac-tice youll be back in the swing of things.

WORKING WITH EQUATIONSTh e most basic part of algebra is manipulating equations. Often the goal is to solve for a variable, but some-times you simply need to shift terms around. Manipulating equations starts with the following fundamental rule.

Whatever you do to one side of the equation, you must do to the other side.

Lets start with some basic equations.

1. 5y 4 = 8 + 2y

2. 10 + z = 3z + 6

3. 6x 9 = 5

Clearing FractionsWhen an equation contains a fraction, you usually need to clear the fraction in order to solve the equation. Remember that a fraction bar signals division. Th erefore, to clear a fraction, you need to multiply.

1. x

39 16+ =

2. 8

516

z=

3. 2

34 8y =



Cross MultiplicationWhen you have two fractions set equal to each other, you can cross-multiply.

1. 3

2

2

5y=

2. 5

1

4

3

x

x =

Simultaneous EquationsCan you solve for the value of x in the following equation? What about the value of y?

2x + 7y = 22

What if we have a second equation?

2x 4y = 11

To solve a linear equation with two variables, you need two equations. Lets look at those last two equations again, this time stacked:

2x + 7y = 22

2x 4y = 11Solve these pairs of equations for x and y.

1. 2y 3x = 2 9y + 3x = 75

2. 3x + 2y = 8 12x + 5y = 47

Pay attention to what youre being asked to nd. Sometimes you dont need to solve for the individual vari-ables.

If 3w + 3z = 13, and 4w 4z = 6, then what is the value of 7w z ?



InequalitiesFor the most part, you can manipulate inequalities as you do equations. Th ere is one extra rule to remem-ber.

Whenever you multiply or divide an inequality by a negative number, you have to ip the direction of the inequality sign.

1. 4x 5 > 43

2. 6x 9 < 8x + 11

3. If xz

< 3 , is xz < 3 ?

EXPONENTSExponents are shorthand for multiplication. Instead of writing 3 3 3 3, we write 34. Th is means that you can always expand an exponent into a multiplication problem if you need to.

x x2 3i =

When you multiply with the same base, you __________ the exponents.

y

y

4

2=

When you divide with the same base, you __________ the exponents.

z23( ) =

When you raise an exponent to another power, you __________ the exponents.

Together, these rules give you the acronym MADSPM. Try these problems.

1. w w4 =

2. z

z

7

6=

3. y 44( ) =

4. x x

x

5 7

4

3

i

=



Special Exponent RulesAny number raised to the rst power is __________.

1 raised to any power is __________.

0 raised to any power is __________.*

Any number raised to the 0 power is __________.*

*00 is undened, but this isnt tested on standardized tests.

What is the value of ( )3 2 ? __________ What is the value of ( )3 3 ? __________

Any negative number raised to an even power is __________. Any negative number raised to an odd power is __________.

Which is larger, ( )5 32 or ( )6 31 ? If x2 16= , what is the value of x? __________

Be careful. Variables raised to even powers have two solutions.

If y2 25 , what is the value of y? __________

Watch out for inequalities that contain variables. Make sure you consider the negative solution as well as the positive one.

Which is larger, 1

2

2

or

1

2?

When you apply exponents to proper fractions between 0 and 1, they become __________.



Negative and Fractional ExponentsNegative and fractional exponents look much scarier than they really are.

1. x 2 = __________

2. 2 3 = __________

3. y

y

2

6 = __________

Negative exponents have nothing to do with negative numbers. When you see a negative exponent, ip the number upside down to get the reciprocal.

1. x1

2 = __________

2. 41

2 = __________

3. 81

3 = __________

4. 82

3 = __________

If you see a fractional exponent, remember that the bottom of the fraction is the root, and the top is the exponent. One last thing to remember is that negative and fractional exponents obey all the MASDPM rules that we looked at a little while ago.

y y3

2 2 = _______

Factoring and Canceling ExponentsSince exponents are all about multiplication, you can break an exponent problem into factors that produce your original number.

1. 12

3

9

6= __________

2. 10 5

2

5 3 = __________

3. 15

25

4

2= __________



SQUARE ROOTS

Th e sign denotes the positive square root only. Th e sign is also sometimes called the radicalsign.

25 = __________

x2 9= gives the same information as x = 9 . True False

Combining Square Roots

Th e only time you can add or subtract square roots is when you have the same number under the square root sign.

2 5 4 5+ = _______Multiplying and dividing square roots, however, is easy.

1. 12 3 = 12 3 =

2. 75

3= 75

3=

3. 3 15

5

= 3 15

5

=

Factoring Square RootsBecause you can combine square roots with multiplication, you can also split them apart by factoring. Th is is important for simplifying square roots, because you have to pull out any perfect square factors from the square root sign. Keep an eye out for the following perfect squares:

Th e Perfect Squares up to 122

1 12 = 2 42 = 3 9

2 = 4 162 =

5 252 = 6 36

2 = 7 492 = 8 64

2 =

9 812 = 10 100

2 = 11 1212 = 12 1442 =

Lets simplify a few square roots by factoring out the perfect squares.

1. 75 =

2. 48 =

3. 2 45 =



Estimating Square RootsYou can estimate the value of square roots by looking for the nearest perfect squares to the number under the

radical sign. Lets work with 44 .

44 lies between which two perfect squares? __________ and __________

Therefore, 44 lies between which two numbers? __________

44 Estimate the value of the following square roots:

1. 72

2. 3 18

3. 6 24

Rationalizing Square Roots

Sometimes you nd a fraction with a radical in the denominator, such as 10

2. Th eres nothing wrong with

this if youre going to do something with the fraction, for example, multiply it, add to it, or compare it to

something else. However, the fraction is not considered fully reduced. In order to reduce it, you need a little

trick called rationalizing the denominator.

Rationalize a root in the denominator of a fraction by multiplying the numerator and denominator by the root. Because youre multiplying the fraction by 1, you wont change the value.

x

y

x

y

y

y

x y

y= =

10

2 =

10

2

2

2 =

Try one yourself:

5

3=



Comparing Square RootsTo compare two numbers that contain square roots, square both numbers to clear out the radicals.

Which is larger, 3 6 or 5 2 ?

3 62

=

5 22

=

Which is larger, 8

2 or 6?

THE DISTRIBUTIVE LAWWhen solving problems, its often useful to rearrange expressions. Th e distributive law allows you to factor common terms from expressions.

Th e distributive law states:

a b c ab ac+( ) = +For example, 3( 2 + 5) = (3 2) + (3 5) = 21.

1. 3 4 5+( ) = 2. 3 9xz zw =

3. 5 10

2

xy x

y

++

=

4. x x2 3+ =

5. 4 2

2

5 5

3

y y

y

=



KEEP WORKINGWe hope youve found this session useful. Here are some ideas for how to continue to build your math knowledge.

MAKE FLASHCARDSWhen you encounter a term or formula, create a ashcard. Put the term or formula on one side and the denition or explanation on the back. Carry a few cards at a time, and quiz yourself whenever you have a few free minutes.

USE YOUR MATH SKILLSPractice the techniques youve learned here in your everyday life. If youre shopping and an item is advertised as 20 percent o, quickly calculate the discount. When youre at the grocery store, try Ballparking your total bill. At work, add up the budget numbers without a calculator, and then use the calculator to check your results when youre done. The more you practice math skills, the more procient youll become.

ANALYZE TEST QUESTIONSWhen youre doing homework, analyze problems before you solve them. Ask yourself:

What concepts is this problem testing?

What rules do I have to apply to solve the problem?

What kinds of calculations do I have to perform?The better you get at spotting what a question is testing, the better and faster you will become at solving math problems.

Documents

09 Math Bootcamp, GRE prep!