# Math Cheat Sheet

## Basic

### Fractions

A number expressed in the form 𝑎𝑏
Adding and Subtracting with the same denominator:
𝑎𝑏+𝑐𝑏=𝑎+𝑐𝑏
𝑎𝑏−𝑐𝑏=𝑎−𝑐𝑏
Adding and Subtracting with the different denominator:
𝑎𝑏+𝑐𝑑=𝑎𝑑+𝑐𝑏𝑏𝑑
𝑎𝑏−𝑐𝑑=𝑎𝑑−𝑐𝑏𝑏𝑑
Multiplying and Dividing Fractions:
𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑
𝑎𝑏÷𝑐𝑑=𝑎𝑏𝑐𝑑=𝑎𝑑𝑏𝑐   

### ***Decimals\*** 

Is a fraction written in a special form? For example, instead of writing 12 you can write 0.5.  

### ***Mixed Numbers\***

A number is composed of a whole number and a fraction. Example: 223 Converting between improper fractions and mixed numbers: 𝑎𝑐𝑏=𝑎+𝑐𝑏=𝑎𝑏+𝑐𝑏

### ***Factoring Numbers\***

Factor a number means breaking it up into numbers that can be multiplied together to get the original number. Example:12=2×2×3

### ***Integers\*** 

{…,−3,−2,−1,0,1,2,3,…}
Includes: zero, counting numbers, and the negative of the counting numbers

### ***Real Numbers\*** 

All numbers that are on a number line. Integers plus fractions, decimals, and irrationals, etc.) (2‾√,3‾√,π, etc.)  

### ***Order of Operations\*** 

PEMDAS
(parentheses/ exponents/ multiply/ divide/ add/ subtract)  

### ***Absolute Value\***

Refers to the distance of a number from, the distances are positive as the absolute value of a number cannot be negative. |−22|=22

### ***Ratios\***

A ratio is a comparison of two numbers by division. Example: 3:5, or 35 

### ***Percentages\***

Use the following formula to find part, whole, or percent
part =𝑝𝑒𝑟𝑐𝑒𝑛𝑡100×𝑤ℎ𝑜𝑙𝑒

### ***Proportional Ratios\***

A proportion means that two ratios are equal. It can be written in two ways:  
𝑎𝑏=𝑐𝑑 , 𝑎:𝑏=𝑐:𝑑

### ***Percent of Change\***

𝑁𝑒𝑤 𝑉𝑎𝑙𝑢𝑒 – 𝑂𝑙𝑑 𝑉𝑎𝑙𝑢𝑒𝑂𝑙𝑑𝑉𝑎𝑙𝑢𝑒×100%

### ***Expressions and Variables\*** 

A variable is a letter that represents unspecified numbers. One may use a variable in the same manner as all other numbers: **Addition**: 2+𝑎: 2 plus a
**Subtraction**: 𝑦−3 : 𝑦 minus 3
**Division**: 4𝑥 : 4 divided by x
**Multiplication**: 5𝑎 : 5 times a

### ***Distributive Property\*** 

𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐

### ***Equations\*** 

The values of the two mathematical expressions are equal.
𝑎𝑥+𝑏=𝑐

### ***Distance from A to B:\***

(𝑥1−𝑥2)2+(𝑦1−𝑦2)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√

### ***Parallel and Perpendicular lines:\*** 

Parallel lines have equal slopes. Perpendicular lines (i.e., those that make a 90∘ angle where they intersect) have negative reciprocal slopes: 𝑚1 .𝑚2=−1.
Parallel Lines (l ∥ m)

### ***Mid-point of the segment AB:\*** 

M (𝑥1+𝑥22,𝑦1+𝑦22)  

### ***The slope of the line:\*** 

𝑦2−𝑦1𝑥2–𝑥1=𝑟𝑖𝑠𝑒𝑟𝑢𝑛

### ***Point-slope form:\*** 

Given the slope m and a point (𝑥1,𝑦1) on the line, the equation of the line is
(𝑦−𝑦1)=𝑚 (𝑥−𝑥1).  

### ***Slope-intercept form:\***

Given the slope m and the y-intercept b, then the equation of the line is:
𝑦=𝑚𝑥+𝑏.

### ***Factoring:\***

“FOIL”
(𝑥+𝑎)(𝑥+𝑏)
=𝑥2+(𝑏+𝑎)𝑥+𝑎𝑏 “Difference of Squares”
𝑎2−𝑏2=(𝑎+𝑏)(𝑎−𝑏)
𝑎2+2𝑎𝑏+𝑏2=(𝑎+𝑏)(𝑎+𝑏)
𝑎2−2𝑎𝑏+𝑏2=(𝑎−𝑏)(𝑎−𝑏) “Reverse FOIL”
𝑥2+(𝑏+𝑎)𝑥+𝑎𝑏= (𝑥+𝑎)(𝑥+𝑏)

### ***Exponents:\*** 

Refers to the number of times a number is multiplied by itself.
8=2×2×2=23

### ***Scientific Notation:\*** 

It is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.
In scientific notation all numbers are written in this form: 𝑚×10𝑛
**Scientific notation**:
5×100
−2.5×104
5×10−1
2,122456×103

### ***Square:\*** 

The number we get after multiplying an integer (not a fraction) by itself. Example: 2×2=4,22=4

### ***Square Roots:\***

A square root of 𝑥 is a number r whose square is 𝑥:𝑟2=𝑥
𝑟 is a square root of 𝑥

### ***Pythagorean Theorem:\*** 

𝑎2+𝑏2=𝑐2

![img](https://www.effortlessmath.com/wp-content/uploads/2020/04/ttt.png)

### ***Triangles\***

### ***All triangles:***

Area =12 b . h
Angles on the inside of any triangle add up to 180∘.

### ***Equilateral:\*** 

These triangles have three equal sides, and all three angles are 60∘.  

### ***Isosceles:\***

An isosceles triangle has two equal sides. The “base” angles (the ones opposite the two sides) are equal (see the 45∘ triangle above).  

### ***Circles\***

![This image has an empty alt attribute; its file name is circ-1.png](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAPCAQAAAAviQWcAAAAGElEQVR42mP8X89AEWAcNWDUgFEDBokBAH4nFnKVxVNEAAAAAElFTkSuQmCC)

Area =π𝑟2
Circumference =2π𝑟
Full circle =360∘

### ***Rectangles***

![This image has an empty alt attribute; its file name is h-2.png](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAPCAQAAAAviQWcAAAAGElEQVR42mP8X89AEWAcNWDUgFEDBokBAH4nFnKVxVNEAAAAAElFTkSuQmCC)

(Square if *l=w*)
Area=*lw*

![This image has an empty alt attribute; its file name is jk.png](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAPCAQAAAAviQWcAAAAGElEQVR42mP8X89AEWAcNWDUgFEDBokBAH4nFnKVxVNEAAAAAElFTkSuQmCC)

### ***Parallelogram***

(Rhombus if *l=w*)
Area*=lh*
Regular polygons are n-sided figures with all sides equal and all angles equal.
The sum of the inside angles of an n-sided regular polygon is
(𝑛−2).180∘.

### ***Area of a trapezoid:\*** 

𝐴=12ℎ(𝑏1+𝑏2)

### ***Surface Area and Volume of a Rectangular/right prism:\*** 

𝑆𝐴=𝑝ℎ+2𝐵
𝑉=𝐵ℎ

### ***Surface Area and Volume of a Cylinder:***

𝑆𝐴=2π𝑟ℎ+2π𝑟2
𝑉=π𝑟2ℎ

### ***Surface Area and Volume of a Cone*** 

𝑆𝐴=π𝑟𝑠+π𝑟2
𝑉=13 π𝑟2 ℎ

### ***Surface Area and Volume of a Sphere*** 

𝑆𝐴=4π𝑟2
𝑉=43 π𝑟3
(p = perimeter of base B; π 3.14)

### ***Simple interest***:

𝐼=𝑝𝑟𝑡
(*I* = interest, *p* = principal, *r* = rate, *t* = time)

### ***mean***:

Mean: 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎𝑜𝑓 𝑑𝑎𝑡𝑎 𝑒𝑛𝑡𝑖𝑟𝑒𝑠

### ***mode:***

Value in the list that appears most often

### ***range:***

Largest value − smallest value

### ***Median\*** 

The middle value in the list (which must be sorted)
Example: median of
{3,10,9,27,50}=10
Example: median of
{3,9,10,27}=(9+10)2=9.5

### ***Average\***

𝑠𝑢𝑚 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠

### ***Average speed\***

𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒

### ***Probability\***

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
The probability of two different events A and B both happening is:
P(A and B)=p(A) .p(B)
as long as the events are independent (not mutually exclusive).

### ***Powers, Exponents, Roots\***

𝑥𝑎.𝑥𝑏=𝑥𝑎+𝑏
𝑥𝑎𝑥𝑏=𝑥𝑎−𝑏
1𝑥𝑏=𝑥−𝑏
(𝑥𝑎)𝑏=𝑥𝑎.𝑏
(𝑥𝑦)𝑎=𝑥𝑎.𝑦𝑎
𝑥0=1
𝑥𝑦‾‾‾√=𝑥‾‾√.𝑦√
(−1)𝑛=−1, if n is odd.
(−1)𝑛=+1, if n is even.
If 0<𝑥<1, then
0<𝑥3<𝑥2<𝑥<𝑥‾‾√<3𝑥‾‾‾√<1.

### ***Simple Interest\***

The charge for borrowing money or the return for lending it.
Interest = principal × rate × time
OR
𝐼=𝑝𝑟𝑡

### ***Powers/ Exponents\***

### ***Positive Exponents\***

An exponent is simply shorthand for multiplying that number of identical factors. So 43 is the same as (4)(4)(4), three identical factors of 4. And 𝑥3 is just three factors of x, (𝑥)(𝑥)(𝑥).

### ***Negative Exponents\***

A negative exponent means to divide by that number of factors instead of multiplying.
So 4−3 is the same as 143 and
𝑥−3=1𝑥3

### ***Factorials*** 

Factorial- the product of a number and all counting numbers below it.
8 factorial =8!=
8×7×6×5×4×3×2×1=40,320
5 factorial =5!=
5×4×3×2×1=120
2 factorial =2!=2×1=2

### ***Multiplying Two Powers of the SAME Base*** 

When the bases are the same, you find the new power by just adding the exponents
𝑥𝑎.𝑥𝑏=𝑥𝑎+𝑏

### ***Powers of Powers\***

For the power of power: you multiply the exponents.
(𝑥𝑎)𝑏=𝑥(𝑎𝑏)

### ***Dividing Powers\***

𝑥𝑎𝑥𝑏=𝑥𝑎𝑥−𝑏=𝑥𝑎−𝑏

### ***The Zero Exponent\***

Anything to the 0 power is 1.
𝑥0=1





## Advanced