# Foundations of GMAT Math

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- Foundations of GMAT Math

**Created:**2011-09-12

**Last Modified:**2011-09-29

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

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

- 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

- 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

^{2 }vs - 3

^{2}

^{2 }= 9

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

^{2})

^{3 }= 8

^{3 }= 27

^{3 }= 64

^{3 }= 125

^{3 }= 216

^{3 }= 343

^{3 }= 512

^{3 }= 729

^{3 }= 1,000

^{3 = }1,331 20

^{3}= 8,000

^{3 = }1,728

^{3 }= 2,197

^{3 }= 2,744

^{3 }= 3,375

^{3 }= 4,096

17

^{3}= 4,913

18

^{3 }= 5,832

19

^{3}= 6,859

1/3 = .3333

1/4 = .25

1/5 = .20

1/6 = .1666

1/7 = .1428

1/8 = .125

1/9 = .1111

1/10 = .10

1/12 = .0833

1/13 = .0769

1/14 = .0714

1/15 = .0666

1/16 = .0625

1/17 = .0588

1/18 = .0555

1/19 = .0526

^{2 }= 4

1. Recognize that the equation may have 2 solutions

2. Know how to find both solutions

^{2}+ 13x + 36

First

Outside

Inside

Last

^{2}+ 3x -10 = 0

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

x = -5 or x = 2

^{2}-9x + 18 = 0

18

-9

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

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

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

- 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

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

- 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

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

- 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

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

- 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

60

1 60

2 30

3 20

4 15

5 12

6 10 (stop)

10 6

- 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

4 15

**2 2 3 5**

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)

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)

Prime factors: use factor tree

6 ?

2 3

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

- 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..

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

i.e. 7 to the 5th power

i.e. 5

^{2 }x 5

^{3}= 5

^{5}

i.e. 3

^{5 }/ 3

^{3}= 3

^{2}There is no rule for adding/subtracting exponents with the same base

^{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

- √x times √x = x
- √2 x √2 = 2

- i.e. √8 x √2 = x
- √8 x 2 = √16 = 4
- i.e. √27 / √3
- √27/3 = √9 = 3

^{3 }x 25

^{2}= ?

5

^{3 }x (5

^{2})

^{2}=

^{?}5

^{3 }x 5

^{4}= 5

^{7}

**Unknown Bases**

x

^{3}= 8

**x**

^{3}√^{3 }=

**8**

^{3}√x = 2

^{3}√x

^{ }= 8

^{3}√x

^{3 }= (8)

^{3}x = 512

**Unknown Exponents**

2

^{x}= 8

2

^{x}= 2

^{3x must be 3}3

^{x+2}= 27

3

^{x+2}= 3

^{3x must be 1}

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

Improper Fraction = the numerator is larger than the denominator (5/4)

^{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)

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

Strategy is to x everything by 100..

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

4 5 7 . 1 2 3 5

Hundreds Tens Ones . Tenths Hundredths Thousandths Ten Thousandths

(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

Ι 3 - 6 Ι = ?

Ι -3 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

- every circle has a center

- all radii in the circle have the same length

Diameter = 2 times Radius (2 x r)

- the perimeter of the circle

Circumference

-------------------------- = π

Diameter

π d = c

c = 2 π r

^{2}

area = π d

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

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

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

- A triangle that has 2 equal angles and 2 equal sides

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

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

2 legs and 1 hypoteneuse

^{2 }+ b

^{2 }= c

^{2}Memorize the triplets:

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

Parallelogram:

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

- all 4 internal angles are right angles

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