# Arithmetic

## Remainders

The Remainder is the number leftover after you divide two numbers.

DividendDivisor=Quotient+Remainder

Example: 5/2=4, with a remainder of 1

Note that the remainder can never be larger than the divisor.

## Fractions/Percents

(ab)/(cd)=(a*d)/(b*c)

(a/b)+(c/d)=(a*d)/(b*d)+(c*b)/(b*d)=(a*d+c*b)/(b*d)

(a/b)*(c/d)=(a*c)/(b*d)

x%=x/100

Percent Change= Final Value – Initial ValueInitial Value*100

Common Fractions

 1/2=0.5 1/3=0.3 1/4=0.25 1/5=0.2 1/6=0.16 1/8=0.125 1/20=0.05 1/25=0.04

## Roots

√x is called the Square Root. The answer is the number that is x when multiplied by itself.

3√x is called the Cubic Root. The answer is the number that is x when multiplied by itself twice.

Perfect Squares:

 Base Square Cube 1 1 1 2 4 8 3 9 27 4 16 64 5 25 125 6 36 216 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225

## Order of Operations

Use the order of operations to determine which mathematical operations to carry out first:

PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction)

Example: (4-2)*3=(2)*3 BUT 4-2*3=4-6=-2

## Word Problems

Distance= Rate*Time

Pay=Wage Rate*Time

Addition: adds/ together/ more than/ combine/ total of/ sum/ and/ plus/ increased by

Subtraction: decreased by/ minus/ less than/ fewer than/ difference of/ difference between/ less than/ subtract

Multiplication: times/ multiplied by/ product of/ increased by a factor of

Division: ratio of/ per/ quotient of/ out of/ divided by/ decreased by a factor of

Equals Sign: is/ are/ was/ were/ will be/ yields/ gives/ results/ total is/ comes to/ as much as/ the same as

0!=1

1!=1

2!=2*1=2

3!=3*2*1=6

4!=4*3*2*1=24

5!=5*4*3*2*1=120

x!=x*(x-1)*…*1

## Exponents

x^2=x*x

x^3=x*x*x

(x^y)*(x^z)=x^(y+z)

(x^y)^z=x^(y*z)

x^y/x^z=x^(y-z)

## Factoring

Factor–a whole number that when multiplied by another whole number, yields a given number

Prime factor–a factor that cannot be divided evenly by any number except for one and itself

Greatest Common Factor–the largest factor that two numbers share in common; find by factoring both numbers, finding the prime factors they have in common, and then multiplying those factors

Least Common Multiple–the smallest number of which two numbers are both a factor; find the greatest common factor, find the leftover prime factors from both numbers, and multiply everything together

When factoring, represent a number as the product of all of its prime factors, with each factor taken to the power of the number of times it appears in the factorization.

Example: 50=52*2, 48=24*3

Factoring Tricks:

1–all numbers

2–ends with 2, 4, 6, 8, 0

3–sum of the digits is divisible by 3

4–last two digits form a number divisible by 4

5–ends with 5, 0

6–factor by 2 and 3

7–double last digit and subtract from the rest of the number; if the difference is divisible by 7, so is the number

8–last 3 digits form a number divisible by 8

9–sum of the digits is divisible by 9

10–ends with 0

## Interest

### Simple Interest

Return stays the same each period

Amount in Account=Amount Deposited+Amount Deposited*Rate*Number of Periods

### Compound Interest

Return grows each period

Amount in Account=Amount Deposited*(1+Rate)^Number of Periods

## Work

Used when two people or entities are working together on a project.

1/(Combined Rate)=1/Rate1+1/Rate2

# Algebra

## Systems of Equations

Algebraic expressions with the form (a,b,c,d,e, and f are constants):

a*x+b*y=c

d*x+e*y=f

Method 1:

Isolate one of the variables in one equation and then substitute for that variable in the other equation.

3x+4y=17

5x+3y=21

y=(21-5x)/3

3x+(4/3)*(21-5x)=17 →9x+84-20x=51→-11x=-33→x=3

y=(21-5*x)/3=2

Method 2:

Multiply the equations by constants, so that they have the same coefficient on one of the variables. Then subtract one of the equations from the other

3x+4y=17

5x+3y=21

15x+20y=85

-(15x+9y=63)

11y=22y=2

3x+4*2=17x=3

### Infinite Solutions

Sometimes, one equation is a multiple of the other. In this case, the system cannot be solved and there are infinite solutions.

Example: 3x-y=4 and 6x-2y=8

Oftentimes, this won’t be obvious at first.

### FOIL

First, Outside, Inside, Last–the method of multiplying two binomial expressions; multiply the first expression in each binomial, then the two outside ones, then the two inside ones, and then the last two in each binomial. Finally, add everything together.

Example: (x-3)(x+2)=x^2+2x-3x-6=x^2-x-6

Step 1: Get the equation in the form ax^2+bx+c=0

Step 2:  Try to factor the equation into the form (x+a)(x+b)=0

Step 3: Solve: x=-a,-b

If the quadratic has the form, x^2a^2=0, then (x+a)(x-a)=0; in other words, remember that, because squares are always positive, if you have x^2=NUMBER, then x is the positive and negative square root of that number.

If the quadratic has the form, x^2+2ax+a^2=0, then (x+a)^2=0

If the quadratic has the form, x^2-2ax+a^2=0, then (x-a)^2=0

General formula to solve a quadratic equation:

## Inequalities

Solve inequalities like equalities, except when multiplying or dividing by a negative number. In that case, flip the inequality sign: >< and .

In 3-part inequalities, sometimes called boundary expressions (ex: x<y<z), perform each operation on all 3 parts.

## Absolute Values

|x|=x and |-x|=x

Example: |3|=3 and |-3|=3

In an algebraic equation, isolate the absolute value term and then solve two equations: one with the original equation; one with one side of the equation multiplied by (-1).

Example:

|x-3|+5=7 -> |x-3|=2 & |x-3|=-2

x=5,1

# Geometry

## Circles

Area of a sector of a circle = (angle/360)*pi*radius^2

Length of an arc along a circle = (angle/360)*2*pi*radius

## Triangles

Rules of Triangles:

All angles in a triangle add up to 180 degrees

No one side can be greater than the sum of the other two sides

Area=12*Base*Height

Perimeter=Side1+Side2+Side3

In right triangles (one angle is 90 degrees), the Pythagorean Theorem relates the sides:

Special Right Triangles (if you know the angles and one of the sides, you can quickly calculate the other side; if you know two of the sides and that it is a right triangle, you can quickly calculate the angles):

Common Right Triangles (many triangles on the GMAT have sides with these ratios:

3-4-5, 5-12-13, 8-15-17, 7-24-25

Area=Side^2

Perimeter=4*Side

### Rectangle

Area=Length*Width

Perimeter=2*Length+2*Width

### Parallelogram

Area=Base*Height

Perimeter=2*Length+2*Width

### Trapezoid

Area=12*(Parallel Side1 + Parallel Side2)*Height

## 3-Dimensional

### Box/Rectangular Prism

Surface Area of a Rectangle=2*Length*Width+2*Length*Height+2*Width*Height

Surface Area of a Cube=6*Side2

Volume=Length*Width*Height

### Other Shapes and 3-D Figures in General:

Surface Area=sum of area of all sides

Volume=area of base times height of shape

## Intersecting Lines and Angles

When two lines intersect, opposite angles created by the intersection are equal.

When a line runs through two parallel lines, it creates equal angles with each line:

Two angles that create a straight line together sum to 180 degrees:

## Lines and Angles

### XY Plane

Equation of a line: y=slope*x+y_intercept

Slope=(Change in Y)/(Change in X)

The Y Intercept is the point on the line where X=0

The mid-point between two points is the average of the x-coordinates of those points and the average of the y-coordinates of those points.

### Parallel and Perpendicular Lines

Parallel lines have the same slope

The slopes of two perpendicular lines are negative inverses (ab and -ba)

# Number Properties

## Odds and Evens

Odd+-Odd=Even

Odd+-Even=Odd

Even+-Even=Even

Odd*Odd=Odd

Even*(Even/Odd)=Even

+*+=+

+/+=+

+*-=-

+/-=-

-*-=+

-/-=+

## Squaring and Cubing Numbers

### Squares

Positive numbers greater than 1 get bigger

Positive numbers between 0 and 1 get smaller

Negative numbers between 0 and -1 get larger, but smaller in magnitude, and become positive

Negative numbers less than -1 get larger and greater in magnitude, and become positive

### Cubes

Positive numbers greater than 1 get bigger

Positive numbers between 0 and 1 get smaller

Negative numbers between 0 and -1 get larger, but smaller in magnitude, and stay negative

Negative numbers less than -1 get smaller, but greater in magnitude, and stay negative

## Sets of Consecutive Integers

### Counting Number of Integers in a Set

1. Subtract the bottom number from the top number
2. Add 1 to include the end number

This also works for sets with numbers that are more than 1 apart, as long as the difference between each number is consistent. In cases like this, first divide the top and bottom numbers by the interval between each number.

### Factors of Consecutive Integers

In a set of x consecutive integers, there must be a number with every number 1 through x as a factor. Thus, the product of a set of consecutive integers, must have the numbers 1 through x as factors.

### Average and Sum of Consecutive Integers

Calculate the average by averaging the first and last integers in the set.

Calculate the sum by multiplying this average by the total number of integers in the set.

In a set of consecutive integers, the median and average will be the same.

# Probability, Statistics, and Combinatorics

## Basics of Probability

Probability(Event)=Number of Favorable OutcomesTotal Number of Possible Outcomes

Probability(Event)<=1

Probability(Event)+Probability(No Event)=1

If A and B are two independent events (meaning the occurrence of one does not affect the probability of the other occurring): Probability(A & B)=Probability(A)*Probability(B)

If A and B are two mutually exclusive events (meaning both cannot occur): Probability(A or B)=Probability(A)+Probability(B)

More generally: Probability(A or B)=Probability(A)+Probability(B)-Probability(A & B)

## Combinatorics

### Permutation

Calculates number of smaller sets of k items from a set of n items. In Permutation, the order that you arrange the k items in matters. For example, in a set of numbers {1,2,3,4}, the 2 item Permutations would be: {1,2}, {2,1}, {1,3}, {3,1}, {1,4}, {4,1}, {2,3}, {3,2}, {2,4}, {4,2}, {3,4}, and {4,3}.

Permutation formula: Permutations of k items out of n=n!/(n-k)!

### Combination

Calculates number of smaller sets of k items from a set of n items. In Combination, the order that you arrange the k items in does not matter. For example, in a set of 4 numbers {1,2,3,4}, the 2 item Combinations would be: {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}.

Combination formula: Combinations of k items out of n=n!/((n-k)!k!)

## Statistics

### Mean

Also called the Average.

Calculated by summing all numbers in the set and dividing by the total number of numbers in the set.

### Median

The middle number in a set of numbers that are in order. In sets with an even number of elements, the median is the average of the two middle numbers.

### Range

The difference between the largest and smallest numbers in a set. Represents how spread out the numbers in the set are.

### Mode

The most common element in the set. There can be multiple modes.

### Standard Deviation and Variance

Standard Deviation represents the dispersion of the data. Standard Deviation is the square root of Variance. Variance is difficult to interpret on its own, but is used to calculate Standard Deviation.

Variance is calculated as the average of the squares of the difference between each element of the set and the mean.

`*Note: I do not condone actual cheating in any way, shape or form. This sheet is intended as a relatively concise summary of GMAT math. Not an actual resource for cheating.`