Math Bootcamp

8/13/2019 Math Bootcamp

1/23

GRE/GMAT

MATH REVIEW


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The Princeton Review, Inc.

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

ese 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 difficult questions involve mathfundamentals. To do your best, you also need to be able to perform calculations quickly and accurately. Inthis 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 mostimportant core material. Well discuss:

Number properties

Fractions, decimals, and percentages

Algebra

Exponents and Roots

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


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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:

6

8

7.0012

4

EVENANDODD

An even number is divisible by 2. An odd number is not divisible by 2. Even and odd apply only tointegers!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 figure them out by testing a pair of numbers. Note that there are noeven/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 whetherxis 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


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POSITIVEANDNEGATIVE

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 different, multiplication or division produces a negative result.

Positive Negative = Negative 6 7 = _______

Negative Positive = Negative 2 8 = _______

Indicate whetherxis positive, negative, or if its impossible to determine.

1. x=

( )452

3

535

2

. Positive Negative Cant determine

2. 3x7 is even Positive Negative Cant determine

3. 0 x= 0 Positive Negative Cant determine

PROPERTIESOFZERO

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 undefined. 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


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ABSOLUTEVALUERemember 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? _______ is number represents the absolute value. Teabsolute 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? _____is is the absolute value. Te 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. e symbol

for absolute value is . us, 5 =5 and 5 =5.

Determine the value ofxin 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. is also means that3 is a factor of 18 because 18 3 =6. We can say that 3 and 6 are both factors of 18.

To find 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 ourselveswith positive factors.)

1 and 60

2 and 30

3 and 20

4 and 15

5 and 12

6 and 10

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


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

1. What is the first multiple of 15?

2. Is 21 a multiple of 42?

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

PRIMENUMBERS

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

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

Answer the following questions:

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

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

3. Integeryhas a factornsuch that 1


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

120/ \

6 20/\ /\

2 3 4 5 /\ 2 2

us, the prime factorization of 120 is 2 3 2 2 5.

Break down the following numbers into their prime factorizations.

28 72 154

ORDEROFOPERATIONSWhen youre answering test questions, you must know in what order to perform calculations. e standardacronym for this is PEMDAS, which you can also be remember with the sentence: Please Excuse My DearAunt Sally. More properly it looks like this:

P|E|M D|A S

Pstands forparentheses. Solve expressions in parentheses first.

Estands for exponents. Solve expressions with exponents next.

Mstands for multiplication, and Dstands for division.

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

Astands for addition, and Sstands 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


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FRACTIONS, DECIMALS, AND PERCENTAGESFractions, decimals, and percentages are simply different ways of expressing part-to-whole relationships. Itsimportant 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.

e top part of a fraction is called the numerator, and the bottom is called the denominator, so in2

3, 2 is the

numerator and 3 is the denominator.

Another good way of thinking about fractions is as apart

wholerelationship.

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 fractiongreater than 1, also called an improper fraction. For example,

5

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

would get 21

2. is is called a mixed number.

1. How would you express 42

3as an improper fraction? _______

2. Express 16 as a fraction. _______

A reciprocalis what you get when you flip 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 of5

5? _______

Any number divided by itself is equal to _______.


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

Which is larger,5

8or

6

8?

12

5or

13

5?

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

Which is larger,3

7or

3

8?

7

2or

7

3?

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

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

Which is larger, 4

7or 5

9?

Which is larger,8

3or

13

5?

Reducing FractionsReducing fractions means dividing the top and bottom of the fraction by the same number until no moredivision 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, oryou can cancel first, then multiply.

1.3

4

2

5=

3 2

4 5

=

2.6

4

5

9=

Dividing fractions is also simple. Just flip 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+ =


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Addition and subtraction are a little trickier when you have different denominators, but using the Bowtiemakes 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 off finding common denominators. For other problems, you canuse the Bowtie or common denominators, whichever allows you to solve problems more quickly and accu-rately.

Fraction Tricks

1. Multiplying a number by1

5is the same as dividing it by ______________.

2. Therefore,1

7of 42 = ______________.

3. What is2

7of 42? ______________

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

DECIMALS

Decimals are another way of expressing apart

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 Decimalseres a neat trick for multiplying decimals. Ignore the decimals at first and multiply the numbers as if they

were just plain integers. Next, count how many decimal places were in your original numbers and add themback to the result.

1. 4.21 3.6 =

2. 12.9 25.8 =

3. 4.7 3.1 .44 =


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Dividing DecimalsDividing decimals is a little different from multiplying them. Move the decimal points of both numbers untilthe 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 movingthe 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 = __________

PERCENTAGES

Percentages are one more way of expressing apart

wholerelationship. A percentage is a fraction of 100. Percent

literally meansper centper 100. So 35% is the equivalent of35

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 = __________%


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The Princeton Review, Inc. |


Percent TranslationMany word problems involving percentages can be translated into math equations, which you can thensolve. 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 =difference

original100

If the problem asks for a percent increase, or what percentgreater, 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. In1999, 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?


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Percentage TricksBe careful whenever youre asked to compare, add, or subtract percentages. ere are some sneaky pitfallsyou must avoid.

1. If you increase the price of an item 25% and then reduce the new price by 25%, the

resulting price islower 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 arehonor students, then there are an equal number of male and female honor studentsat the school. True False

Tip Calculatione 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 saveyou 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

7expressed as a decimal is approximately .

2. 4is 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 withand by how much youre rounding. As a rule of thumb, never round by more than 10 percent and try to keepit to 5 percent or less.


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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.

WORKINGWITHEQUATIONSe 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 fundamentalrule.

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

Lets start with some basic equations.

1. 5y4 = 8 + 2y

2. 10 +z= 3z+ 6

3. 6x9 = 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. erefore, to clear a fraction, you need to multiply.

1.

x

39 16+ =

2.8

5

16

z

=

3.2

3

4 8y =


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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 ofxin the following equation? What about the value ofy?

2x+ 7y= 22

What if we have a second equation?

2x4y= 11To solve a linear equation with two variables, you need two equations. Lets look at those last two equationsagain, this time stacked:

2x+ 7y= 22

2x 4y= 11

Solve these pairs of equations forxandy.

1. 2y3x= 2 9y+ 3x= 75

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

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

If 3w+ 3z= 13, and 4w4z= 6, then what is the value of 7wz?


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

Whenever you multiply or divide an inequality by a negative number, you have to flip the direction

of the inequality sign.

1. 4x5 > 43

2. 6x9 < 8x+ 11

3. If xz

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