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- Foundations of GMAT Math

Ted J.

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

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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 - √5
^{2}= 5 and^{3}√7^{3}= 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

(-3)^{2 }vs - 3^{2}

(-3)^{2 }= 9

-3^{2 }= - 9 ..... - (3^{2})

Memorize Cubes ( 1 - 10 )

2^{3 }= 8

3^{3 }= 27

4^{3 }= 64

5^{3 }= 125

6^{3 }= 216

7^{3 }= 343

8^{3 }= 512

9^{3 }= 729

10^{3 }= 1,000

Memorize Cubes ( 11 - 20 )

11^{3 = }1,331 20^{3} = 8,000

12^{3 = }1,728

13^{3 }= 2,197

14^{3 }= 2,744

15^{3 }= 3,375

16^{3 }= 4,096

17^{3} = 4,913

18^{3 }= 5,832

19^{3} = 6,859

17

18

19

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

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

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

x^{2 }= 4

1. Recognize that the equation may have 2 solutions

2. Know how to find both solutions

1. Recognize that the equation may have 2 solutions

2. Know how to find both solutions

FOIL

(x+4) (x+9) --> x^{2}+ 13x + 36

First

Outside

Inside

Last

First

Outside

Inside

Last

Why is factoring useful with Quadratics?

x^{2}+ 3x -10 = 0

(x+5)(x-2) = 0

x = -5 or x = 2

(x+5)(x-2) = 0

x = -5 or x = 2

How to quickly factor Quadratic equations? Use a diamond.

x^{2}-9x + 18 = 0

18

18

-3 -6

-9

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

-9

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

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

Steps to Solve a Word Problem

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

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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

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

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**

4 15

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)

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)

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

Prime factors: use factor tree

Unknown Numbers and Divisibility ( x / 6 ?)

x

6 ?

2 3

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

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

7^{5} (7 is the base and 5 is the exponent)

i.e. 7 to the 5th power

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. 5^{2 }x 5^{3} = 5^{5}

i.e. 5

Combining Exponential Terms

2. When dividing exponential terms that share a common base, subtract the exponents.

i.e. 3^{5 }/ 3^{3} = 3^{2}There is no rule for adding/subtracting exponents with the same base

i.e. 3

More Exponent Rules (very important)

(a^{2})^{4} = a^{8}a^{0} = 1 (anything w/ and exponent of 0 = 1)

a^{-2} = 1 / a^{2}

(-3)^{3} = -27

(-3)^{4} = 81

a

(-3)

(-3)

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

OR

√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

5^{3 }x 25^{2} = ?

5^{3 }x (5^{2})^{2} = ^{?}5^{3 }x 5^{4} = 5^{7}

5

Solving Algebraic Equations In Exponential Terms

x

x = 2

x = 512

2

2

3

Fractions

3/6 > 3/7 > 3/8

(the pie shrinks as the denominator grows and the numerator remains constant)

(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)

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 10^{1} = 12.3

782.95 / 10^{1} = 78.295

43.8723 x 10^{3} = 43,872.3

57,234 / 10^{4}_{ }= 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)

782.95 / 10

43.8723 x 10

57,234 / 10

all of these are backward for negative exponents

i.e. 1.23 x 10

1.23 / 10

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

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

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

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

(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

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

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

- 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)

- 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

- the perimeter of the circle

Circumference

-------------------------- = π

Diameter

π d = c

c = 2 π r

Circles: Area

area = π r ^{2}

area = π d

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

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)

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

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

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

2 legs and 1 hypoteneuse

Pythagorean Theorem

a^{2 }+ b^{2 }= c^{2}Memorize 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:

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

- all 4 internal angles are right angles

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