GMAT Math Review

Real Numbers

they are present on the number line and can be values such as (-2, -0.2, 0, 2/3, √2) . the vales on the left of zero are negative and the values on the right of zero are positive

a number that is inclusive to 1 ≤ n ≤ 4

inclusive to this means that hat the value of n can be any number from 1 to 4, including both 1 and 4. or numbers like 2.5

Absolute Value

denoted as |x| and is the distance between that number and zero on the number line

examples (|5| = 5 or |-7/2| = 7/2

Intergers

a whole number that can be either positive, negative, or zero, but it does not have any fractional or decimal parts. In other words, they’re numbers that belong to the set of whole numbers and their opposites.

Example of Integers

{ -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 }

This set includes:

• Negative integers: -5, -4, -3, -2, -1

• Zero: 0

• Positive integers: 1, 2, 3, 4, 5

consecutive integers

integers that follow each other in order, without any gaps. Each integer is exactly one more than the previous one.

{n, n + 1, n + 2, n + 3, …}

What is a divisor of a number?

A divisor of a number y is any integer x such that y = x * n for some integer n.

If x ≠ 0, what does it mean if x is a divisor of y?

It means that y can be divided by x without leaving a remainder, and y = x * n for some integer n.

Are 4 7 and 8 divisors of 28?

Yes, because 28 = 7 * 4. Both 4 and 7 are divisors of 28.

No, because when you try to divide 28 by 8, the result is 3.5, which is not an integer.

what is a quotient?

When you divide a positive integer y by another positive integer x, you get a quotient. If the division doesn’t result in a whole number, you round down to the nearest whole number. For example, if you divide 28 by 8, you get 3.5. You round it down to 3, which is the quotient.

how do you find the remainder

multiply x (the divisor) by the quotient. Then, subtract that result from y. For example, after dividing 28 by 8:

• The quotient is 3.

• Multiply 8 by 3 to get 24.

• Subtract 24 from 28, and the remainder is 4.

can you divide a smaller integer by a bigger one?

yes when a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer.

Example:

When 5 is divided by 7, the quotient is 0 and the remainder is 5, since 5 = (7)(0) + 5.

what is a prime number?

a positive integer that has exactly two positive divisors: 1 and the number itself. This means it can't be divided by any number other than 1 and itself.

(what are the first 6)

Composite number:

A number greater than 1 that isn’t prime. It can be broken down into smaller prime numbers multiplied together.

explain k and n in exponents

k is the base (the number being multiplied).

n is the exponent (how many times the base is multiplied by itself).

whats the difference between squaring a big and small number?

Squaring a number greater than 1, or raising it to any power greater than 1, gives a larger number.

Squaring a number between 0 and 1 gives a smaller number.

Square roots

a number n is a number x that, when multiplied by itself, gives n. In other words:

x × x = n (or x² = n)

Every positive number has two square roots: one is positive and the other is negative.

• For example, the square root of 9 is both 3 and -3, because:

  • 3 × 3 = 9

  • (-3) × (-3) = 9

Nonnegative square root and absolute value:

• For any number x, the nonnegative square root of x² equals the absolute value of x.

• This means:
  √(x²) = |x|

scientific notation

a number is written as a decimal with only one nonzero digit to the left of the decimal point, multiplied by a power of 10. This makes large or small numbers easier to work with.

Positive exponent

move the decimal point to the right by the number of places equal to the exponent.

ex: 3.2 × 10³ = 3200.

Negative Exponent

move the decimal point to the left by the number of places equal to the absolute value of the exponent.

ex:5.6 × 10⁻² = 0.056.

How do you add or subtract decimals

line up their decimal points. If one decimal has fewer digits to the right of its decimal point than another, insert zeros to the right of its last digit.

how do you multiply decimals

• Ignore the decimal points and multiply the numbers as if they were whole numbers.

• Count the total number of decimal places in both original numbers.

• Place the decimal point in the product so that it has the same total number of decimal places.

Multiply 0.06 × 0.5

• Ignore decimals and multiply 6 × 5 = 30

• Count decimal places:

  • 0.06 has 2 decimal places.

  • 0.5 has 1 decimal place.

  • Total: 2 + 1 = 3 decimal places.

• Place the decimal 3 places from the right: 0.030

  • Final Answer: 0.03

what are the steps to divide decimals

• Make the divisor a whole number

  • Move the decimal to the right in both the dividend (the number being divided) and the divisor (the number you're dividing by), the same number of places.

• Divide as usual (like whole numbers).

• Place the decimal in the quotient

  • The decimal in the answer (quotient) goes directly above the decimal in the dividend.

Positive and Negative Addition Rules

• Two positives always add up to a bigger positive!

• Adding two negatives makes a bigger negative!

zero properties

• Division: 0÷x=

  • Trick: "Zero divided by anything is zero!"

• Undefined Case: x÷0s undefined

  • Trick: "You can’t divide by zero!"

multiplication and dividing properties

• Multiplying two negatives makes a positive

• A mix of signs in multiplication gives a negative!

• Dividing two negatives makes a positive

• Dividing a negative by a positive gives a negative

Multiplying with Same Base (Exponent rule)

KKeep base the same, then add exponents, then solve

Dividing with the Same Base(Exponent rule)

Keep base the same, then subtract exponents, then solve.

Power of a Power (Multiply Exponents rule)

Power of a Product (Distribute Exponents)

Distribute exponent to each factor, then solve.

(2×3)^4= 2^4×3^4

=16×81

=1296

Power of a Quotient (Divide Bases Separately)

Distribute the exponent to both numerator and denominator, then solve.

(4/2)³ = 4³ / 2³

= 64/8

= 8

Algebra

uses basic arithmetic (addition, subtraction, multiplication, division) but involves the idea of unknown values.

Algebraic expression:

his is a combination of variables, constants, and arithmetical operations (like addition, subtraction, multiplication, division).

For example, 3x+2 is one

translate x + y

x added to y

x increased by y

x more than y

x plus y

the sum of x and y

the total of x and y

translate x − y

x decreased by y

difference of x and y

y fewer than x

y less than x

x minus y

x reduced by y

y subtracted from x

translate xy

x multiplied by y

the product of x and y

x times y

translate x/y

x divided by y

x over y

the quotient of x and y

the ratio of x to y

translate x^y

x to the power of y

x to the yth power

Constant term vs coefficient

• Constant Term: A term that has no variable. For example, 555 is a constant term because it's just a number with no variable attached.

• Coefficient: This is the constant part of a term that has a variable. For example, in 3, 3 is the coefficient of the term, because it multiplies the variable x.

polynomial

is an algebraic expression that involves sums of terms, and each term has a variable raised to some power, with the variable being multiplied by a coefficient.

what are the degrees of polynomials?

• First-degree polynomial (linear polynomial): If the highest power of the variable is 1, it’s called a linear polynomial

• Second-degree polynomial (quadratic polynomial): If the highest power of the variable is 2, it’s called a quadratic polynomial.

• Higher-degree polynomials: If the highest power is 3, it’s called a cubic polynomial, and so on.

what are the ways to simplify algebraic expressions

combining like terms or factoring

Combining Like Terms

• Like terms are terms that have the same variable raised to the same power.

• You add or subtract the coefficients while keeping the variable part unchanged.

3x+5x = (3+5)x =8x

or

4y²−2y²= (4−2)y² =2y²

Factoring

• Factoring involves rewriting an expression as a product of its factors.

• Look for common factors in all terms and factor them out.

6x+9= 3(2x+3)

or

x² +5x+6= (x+2)(x+3)

Multiplying Algebraic Expressions

distribute each term from the first expression to each term in the second expression.To evaluate an expression, replace variables with given values and simplify

algebraic equation

s a mathematical statement where two algebraic expressions are set equal to each other. The goal is to find the values of the variables that make the equation true—these values are called solutions or roots.

Two equations are equivalent….

if they have the same solution(s).

linear equation

an equation that shows a straight-line relationship between variables. In simpler terms, it’s an equation where the highest exponent of the variables is 1.

solve a linear equation

you want to isolate the unknown (like x) on one side of the equation. You do this by performing the same operation on both sides of the equation to keep it balanced.

• Adding/Subtracting the same number on both sides keeps the equality balanced.

• Multiplying/Dividing by the same nonzero number on both sides also keeps it balanced.

When you have two linear equations with the same two unknowns (like x and y), there are three possible outcomes:

1. Infinitely Many Solutions (Equations are equivalent):

If the two equations are equivalent (basically, they represent the same line), then they have infinitely many solutions. This happens when the two equations describe the same relationship between the variables.

2. No Solution (Contradiction)

If the two equations aren’t equivalent, they can never meet at any point. This is called a contradiction.

3. One Unique Solution

If neither a trivial equation (like 0 = 0) nor a contradiction happens, the two equations are not equivalent but do intersect at exactly one point. This means there is one unique solution for x and y.

Substitution Method for Solving Two Linear Equations

• Solve one equation for one variable.

• Substitute this expression into the other equation.

• Solve the new equation for the remaining variable.

• Substitute the value back into either original equation to find the other variable.

ex:

2x+3y=12x and x -y =3

1) Solve for x in the second equation: x= y +3

2) Substitute into the first equation: 2(y+3)+3y=12

3) solve for y: 2y+6+3y=12 —> 5y = 6 → y=6/5 =1.2

4) now substitute into second equation x = 1.2+3 = 4.2

solutions: x=4.2 and y=1.2

Elimination Method for Solving Two Linear Equations

• Multiply each equation to make the coefficients of one variable the same (ignoring the sign).

• Add or subtract the equations to eliminate one variable.

• Solve for the remaining variable.

• Substitute the value of the variable back into one of the original equations to find the other variable.

ex: 6x+5y=29 and 4x-3y= -6

1) multiply equation 1 by 3 and 2 by 5 to get:

18x+15y=87 and 20x-15y=-30

2) add the equations to get: 38x=57

3)solve x = 57/38 = 3/2

solve y:

how to solve factoring equations

• Rearrange so one side is 0.

• Factor the nonzero side.

• Set each factor to 0 and solve.
  (Since if xy=0xy = 0xy=0, then x=0x = 0x=0 or y=0y = 0y=0)

quadratic equation standard form

ax² + bx + c = 0,

where a, b, and c are real numbers and a ≠ 0.

How many roots can a quadratic equation have

They can have two, one, or no real roots:

• Two roots: When the equation factors into two different numbers

• One root: When it factors into the same number twice,

• No real root: If the equation results in an impossible statement like x²+4=0 since

• X² much always be greater than 0

Factor a² – b²

(a – b)(a + b).

inequality statements

≠ is not equal to

> is greater than

≥ is greater than or equal to

< is less than

≤ is less than or equal to

A function is a rule

that takes an input (a number) and gives exactly one output.

• It’s written using a letter like f(x) or g(x)

• It’s notation shows that a number is being plugged into the function.

ex: f(x)=x²+3

• f(2)= 2²+3= 7

• This means when x=2 function gives 7.

A key rule of functions

:Each input has at most one output. But different inputs can have the same output.

Example:

If f(x)=x² then:

• f(2)=2² = 4

• f(−2)=(−2)² =4

Even though the inputs 2 and -2 are different, they give the same output (4)

straight line in the coordinate plane equation

y=mx+b

vertical line in the coordinate plane equation

x=a

Slope formula

• m=change in y /change in x

• = y2​−y1​​ / x2​−x1​

A ratio compares

two numbers by division. It can be written in three ways:

• x:y (with a colon)

• x/y​ (as a fraction)

• "x to y" (in words)

Order matters in a ratio!

A proportion

is when two ratios are equal.

2/3 = 8/12

Finding a Percent of a Number

multiply the number by the percent written as a fraction or decimal.

ex 20% of 90 =

90 (20/100) = 90(1/5) = 18

Percent (%) = "per hundred" (out of 100).

• 37% blue houses → 37 out of every 100 houses are blue.

• 150% blue vs. red → 150 blue for every 100 red → Ratio: 3:2.

• 0.5% pink vs. blue → 0.5 pink per 100 blue → Ratio: 1:200.

• 12.5% orange vs. blue → 12.5 orange per 100 blue → Ratio: 1:8.

4o

To find the percent increase

new - old / old

To find price decrease

old-new/ old

When a price is discounted by n% (percent), the new price is

(100 − n)% of the original price.

A customer paid $24 for a dress after a 25% discount. What was the original price?

100-0.25 = 0.75

0.75p = 24 —> p = 24/0.75 = 32

two discounts

are applied successively, they do not simply add together. Instead, each discount is applied to the already reduced price, which leads to a slightly smaller overall discount than just adding them directly.

• First Discount: A 20% discount means that the price is reduced to 80% of the original price. Mathematically, this is represented as:

  New Price=0.8p where p is the original price.

• Second Discount: A 30% discount is now applied to the new price (not the original price). This means the new price is reduced to 70% of its current value:

  Final Price=0.7×0.8p=0.56p

• Overall Discount: The final price is 56% of the original price, meaning the total discount is:

  100%−56% = 44%

Gross Profit

revenue - expenses or

selling price - cost

formula for calculating the selling price when given the cost price and the desired gross profit percentage is:

s= C+(P×C)

where:

• s = Selling price

• C = Cost price

• P = Profit percentage (expressed as a decimal)

Simple annual interest on a loan or investment is based only on the original loan or investment amount (the principal).

(principal) × (interest rate) × (time).

Compound interest is based on the principal plus any interest already earned.

Compound interest over n periods

= (principal) × (1 + interest per period)^n – principal.

compound intrest formula

A=P( 1+ r/n )^nt

how do you solve some word problems with percents and fractions

use a table idiot

Distance =

rate x time

How do you find an object’s average travel speed?

divide the total travel distance by the total travel time.

ex: On a 600-kilometer trip, a car went half the distance at an average speed of 60 kilometers per hour (kph), and the other half at an average speed of 100 kph. The car didn’t stop between the two halves of the trip. What was the car’s average speed over the whole trip?

1) find the total travel time: 300/60 = 5 hours then 300/100 = 3 hours —> 3+5 = 8

2) find the average speed : 600/8 = 75km/hr

What is a work problem and its formula

usually says how fast certain individuals work alone, then asks you to find how fast they work together, or vice versa.

Formula = 1/r + 1/s = 1/h

where:

r= is how long an amount of work takes a certain individual

s= how long that much work takes a different individual,

h= how long that much work takes both individuals working at the same time.

set

a collection of numbers or other things. The things in the set are its elements. A list of a set’s elements in a pair of braces stands for the set. The list’s order doesn’t matter.

If all the elements in a set S are also in a set T

then S is a subset of T.

Union

two sets A and B is the set of all elements that are each in A or in B or both.

intersection

of two sets A and B is the set of all elements that are each in both A and B.

Two sets sharing no elements are

disjoint or mutually exclusive.

The formula for the number of elements in the union of two sets S and T is:

∣S∪T∣= ∣S∣ + ∣T∣ − ∣S∩T∣

permutation

any sequential ordering of a set’s elements, and is a way to choose elements one by one in a certain order.

combination

is a way of selecting items from a group without considering the order.

Discrete Probability

deals with experiments that have finitely many possible outcomes.

• Sets and counting methods help determine probabilities.

• An event is a set of possible outcomes from an experiment.

• Example: In rolling a die, the event "rolling an even number" is the set {2, 4, 6}.

Probability of an Event

• P(E) represents the probability of event E, where:

  • 0 ≤ P(E) ≤ 1

• If E is impossible (empty set): P(E)=0

• If E is certain (all outcomes): P(E)=1

• If E is possible but uncertain: 0 < P(E) <1

• If F⊆ E then P(F) ≤ P(E)

mutually exclusive

Two events, E and F, cannot happen at the same time. This means they have no outcomes in common.

Independent Events

Two events, E and F, are this if the occurrence of one does not affect the probability of the other.

Dependent Events

Two events, E and F, are this f the occurrence of one affects the probability of the other.

How to Round Numbers for Estimation

• Round down: Delete digits to the right of the place you want to round to.

• Round up: Add the nearest multiple of 10 and then round down.

• To the nearest 10n: Check the next digit. If it's 5 or higher, round up. If less than 5, round down.

How do you round 7651.4 to the nearest hundred?

• Look at the tens place (5 in 7651.4).

• Since it's 5 or higher, round up.

• Add 100 to 7651.4 → 7751.4.

• Drop digits after the hundreds place → 7700 is the nearest hundred.

What is range estimation and when should it be used?

• A range estimation gives a possible range of values for an expression instead of a single estimate.

• Example: If decimals in an equation are between 2 and 3, the range of possible values is between the lowest and highest values in that range.