Foundations of GMAT Math

About this deck

By: Ted Jaffe
Created: 2011-09-12
Size: 67 flashcards
Views: 60

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Equations Vs. Expressions

Equations

Must have an equals sign
ex: 2x + 1 = 3

Expressions

When there are variables and/or numbers, but NO equals sign..
ex: 4x + 3

PEMDAS or PE M/D A/S

P = Parenthesis

E = Exponents

M = Multiplication; D = Division

A = Addition; S = Subtraction

Roots

very closely related to exponents
³√64 is the third root 64 AKA the cube root

What multiplied by itself 3 times will equal 64? 4 x 4 x 4 = 64.... ³√64 = 4

²√ or cancels itself out and the number inside the root is left over
√5² = 5 and ³√7³ = 7

Substitution

Insert one equation into another to solve
Steps

1) Isolate one of the variables in one of the equations
2) Substitute the isolated variable in the other to solve
3) Solve for the other variable too

Memorize Cubes ( 1 - 10 )

2³= 8

3³= 27

4³= 64

5³= 125

6³= 216

7³= 343

8³= 512

9³= 729

10³= 1,000

Memorize Cubes ( 11 - 20 )

11^{3 =}1,331 20³ = 8,000

12^{3 =}1,728

13³= 2,197

14³= 2,744

15³= 3,375

16³= 4,096
17³ = 4,913
18³= 5,832
19³ = 6,859

Memorize Fractions (1-10)

1/2 = .5
1/3 = .3333
1/4 = .25
1/5 = .20
1/6 = .1666
1/7 = .1428
1/8 = .125
1/9 = .1111
1/10 = .10

Memorize Fractions (11-20)

1/11 = .0909 1/20 = .05
1/12 = .0833
1/13 = .0769
1/14 = .0714
1/15 = .0666
1/16 = .0625
1/17 = .0588
1/18 = .0555
1/19 = .0526

Quadratic Equations

x²= 4

1. Recognize that the equation may have 2 solutions
2. Know how to find both solutions

FOIL

(x+4) (x+9) --> x²+ 13x + 36

First
Outside
Inside
Last

Why is factoring useful with Quadratics?

x²+ 3x -10 = 0
(x+5)(x-2) = 0
x = -5 or x = 2

How to quickly factor Quadratic equations? Use a diamond.

x²-9x + 18 = 0
18

-3 -6
-9
(x-3) (x-6) = 0

How to Perceive GMAT Math Word Problems: What is an Active Thinker?

The active thinker aggressively seeks out relationships between the various elements of a problem and looks to write equations which can be solved.

Steps to Solve a Word Problem

Identify unknowns and assign variables
Identify relationships and create equations
Identify what the questions is asking for

Integer Rules

Integer + Integer = always an integer
Integer - Integer = always an integer
Integer x Integer = always an integer
Integer / Integer = only if the numerator is divisible by denominator

Divisibility Rules: "2"

An integer is divisible by 2 if the integer is EVEN
i.e. 2, 4, 6, 8, 10, 12...etc

Divisibility Rules: "3"

An integer is divisible by 3 if the SUM of the integers is a MULTIPLE OF 3
i.e. 147 ... 1 + 4 + 7 = 12

12 = 4 x 3

Divisibility Rules: "5"

An integer is divisible by 5 if the integer ENDS IN 0 OR 5
i.e. 75 or 80

Divisibility Rules: "9"

An integer is divisible by 9 if the SUM of the integers in a MULTIPLE OF 9
i.e. 144 .. 1 + 4 + 4 = 9

9 = 9 x 1

Divisibility Rules: "10"

An integer is divisible by 10 if the integer ENDS IN 0
i.e. 8,730

Factors and Divisibility

What numbers is 6 divisible by?

6/1? = 6 yes 6/2? = 3 yes 6/3? = 2 yes
6/4? = 1.5 no 6/5? = 1.2 no
6/6?= 1 yes

]1, 2, 3, and 6 (all factors of 6)

Find Factors using Factor Pairs

Find all Factors of 60...
60
1 60
2 30
3 20
4 15
5 12
6 10 (stop)
10 6

Prime Numbers

Numbers that only have 2 factors

1 and itself

i.e. 2, 3, 5, 7, 11, 13, 17, 19

1 is not prime!
2 is the only even prime number

Prime Factorization

60
4 15
2 2 3 5

2 x 2 x 3 x 5 is the prime factorization of 60

Factor Foundation Rule

If
a is divisible by b (i.e. 12/6)
and
b is divisible by c (i.e. 6/3)
then
a is divisible by c also (i.e. 12/3)

Factor Foundation Rule (reverse order)

If
d has e and f as prime factors (i.e. 90/5 and 90/3 - 5 and 3 are PFs)
then
d is also divisible by e x f (i.e. 90/15)

Finding ALL FACTORS vs. Finding PRIME FACTORS

All factors: use factor pairs

Prime factors: use factor tree

Unknown Numbers and Divisibility ( x / 6 ?)

x
6 ?
2 3
Is x divisible by 3? yes (factor foundation rule)

Is x even? x is divisible by 2 so it must be!

Unknown Numbers and Divisibility ( x / 6 ?)

Is x divisible by 12?

In order for this to true (set ? = 2) since prime factors must all be multiplied to get 12.
Because we don't know if ? = 2 for sure... we cannot say that x must be divisible by 12..

Exponents

7⁵ (7 is the base and 5 is the exponent)

i.e. 7 to the 5th power

Combining Exponential Terms

1. When multiplying exponential terms that share a common base, add the exponents.
i.e. 5²x 5³ = 5⁵

Combining Exponential Terms

2. When dividing exponential terms that share a common base, subtract the exponents.
i.e. 3⁵/ 3³ = 3²There is no rule for adding/subtracting exponents with the same base

More Exponent Rules (very important)

(a²)⁴ = a⁸a⁰ = 1 (anything w/ and exponent of 0 = 1)

a^-2 = 1 / a²

(-3)³ = -27
(-3)⁴ = 81

Roots

√x times √x = x
√2 x √2 = 2

How to manipulate roots:

i.e. √8 x √2 = x

√8 x 2 = √16 = 4

i.e. √27 / √3

√27/3 = √9 = 3

Simplify Exponents

x = √2 x √6 what is x?

x = √12

x = √4x3

x= √4 x √3

x = 2√3 (simplest term)

Simplify Exponents CONTINUED

OR..

x = √12

√12

3 4

2 2

so.. x = √2 x √2 x √3

(√2 x √2 = 2)

=2√3

Simplify Exponents (2)

√360

2 180

2 90

9 10

3 3 2 5

= 6 √10

Simplify Exponents (2) CONTINUED

√360 --> √36 x √10 = 6√10

THIS IS WHAT THE GMAT TESTS.... not just the ability to get the answer.. but who has the fastest and most effective method to find the answer!

Rewriting Bases

5³x 25² = ?
5³x (5²)² = ^?5³x 5⁴ = 5⁷

Solving Algebraic Equations In Exponential Terms

Unknown Bases
x³ = 8
³√x³= ³√8
x = 2

³√x= 8
³√x³= (8)³
x = 512

Unknown Exponents
2^x = 8
2^x = 2^{3
x must be 3}3^x+2 = 27
3^x+2 = 3^{3
x must be 1}

Fractions

3/6 > 3/7 > 3/8
(the pie shrinks as the denominator grows and the numerator remains constant)

Mixed Numbers and Improper Fractions

Mixed Number = both an integer and fractions are in the same number (3 3/4)
Improper Fraction = the numerator is larger than the denominator (5/4)

FDP's (Fractions Decimals and Percents)

1/2 .50 50%

Shifting the Decimal Point

1.23 x 10¹ = 12.3

782.95 / 10¹ = 78.295

43.8723 x 10³ = 43,872.3

57,234 / 10⁴= 5.7234

all of these are backward for negative exponents
i.e. 1.23 x 10^-1 = .123
1.23 / 10^-1= 12.3 (same as the first example)

Decimal Multiplication

0.25 x 0.5 = ?
1) 25 x 5 = 125
move three places.. (.25) (.5)
= .125

2) 0.001 x 0.005 = ?
1 x 5 = 5
move six places
0.000005

Decimal Division

What is 300 / 0.05?

Strategy is to x everything by 100..

300 / .05 = ? --> 300 x 100 / .05 x 100 = 30,000 / 5 = 6,000

Places on a Decimal

457.1235

4 5 7 . 1 2 3 5
Hundreds Tens Ones . Tenths Hundredths Thousandths Ten Thousandths

Switching the Sign in an Inequality

If MULTIPLIED OR DIVIDED BY A NEGATIVE NUMBER...
(try not to do it with a variable unless you know the sign is positive i.e. people, length)

-b/7 > 12
b < 84

-7b > 14
b < 2

Absolute Value

Always solve inside first

Ι 3 - 6 Ι = ?
Ι -3 Ι = 3

Solving Absolute Value Equations

Ι y Ι = 3
y = 3 or -3 (there are two numbers three units away from zero)

Solve:
6 Ι 2x + 4 Ι = 30

1) Isolate the absolute value expression
Ι 2x + 4 Ι = 5

2) Take whats inside the absolute value and set up 2 EQUATIONS

2x + 4 = 5 or -(2x+4) = 6
x = 1/2 or x = 9/2

Circles: Definition

Circle = set of points that are all the same distance from a central point
- every circle has a center

Circles: Radius and Diameter

Radius = the distance between the center and a point on the circle
- all radii in the circle have the same length

Diameter = 2 times Radius (2 x r)

Circles: Circumference

Circumference = measure of the distance around a circle
- the perimeter of the circle

Circumference
-------------------------- = π
Diameter

π d = c

c = 2 π r

Circles: Area

area = π r ²

area = π d

Circles: Arc Length

If a circle is cut... the arc length is the portion of circumference left

Circles: Central Angle

Central Angle / 360
Sector Area / Circle Area
Arc Length / Circumference

Ex: A sector has a radius of 9 and an area of 27π. What is the central angle of the sector?

A = π (9)² -> A = 81π
so 27π / 81π = 1/3
so the sector is 1/3 of the circle
1/3 x 360º = 120º = central angle

Triangles: Basic Rules

x 3

5

The sum of 5 and 3 will always be greater than x ... 2 < x < 8
The difference of 5 and 3 will always be less than x ... 5-3 = 2; x > 2

Triangles: Angles

30º

85º xº

Internal angles must sum to 180º
30 + 85 + x = 180
x = 65

Note:
The longest side is opposite to the longest angle
The smallest side is opposite to the smallest angle

Isoceles Triangle

A triangle that has 2 equal angles and 2 equal sides

Equilateral Triangle

A triangle that has 3 equal angles (all 60º) and 3 equal sides

Triangeles: Perimeter

Perimeter = the sum of all sides (not angles)

Triangles: Area

Area = 1/2 b h

base and height must be perpendicular to each other (90º)

- sometimes you need to fill out the rest of the triangle in order to see the 90º angle to calculate

Right Triangles

Any triangle in which one of the angles is a right angle

2 legs and 1 hypoteneuse

Pythagorean Theorem

a²+ b²= c²Memorize the triplets:

3 - 4 - 5
5 - 12 - 13
8 - 15 - 17
6 - 8 - 10
10 - 24 - 26

Quadrilaterals and Parallelograms (type of Quadrilateral)

Quaadrilateral: any figure with 4 sides

Parallelogram:

4 sided figure
the opposite sides are parallel and equal
the opposite angels are equal

Rectangles (same as squares)

Same properties are parallelograms
- all 4 internal angles are right angles

About this deck

By: Ted Jaffe
Created: 2011-09-12
Size: 67 flashcards
Views: 60

About StudyBlue

“I have been getting MUCH better grades on all my tests for school. Flash cards, notes, and quizzes are great on here. Thanks!”
Kathy