Math Formulas

Simplifying Fractions Example: 12/28

Simplify 12/28 1. LCM = 4 ( 4 goes into both 12 and 28 equally) 12 divided by 4 = 3 and 18 divided by 4 = 7 2. Answer equals 3/7 3. It is already in smallest form

How to find tip

percent you want to tip x selling price

Parenthesis Tip

when a number is directly outside of a parenthesis it means you multiply all values within the parenthesis by it -( x + b) means you multiply by -1 + ( x + b) means you multiply by 1

Rules of negatives and positives

(negative) + (negative) = Positive Negative + Positive = depends on which is further from zero Positive + Positive = Positive Negative x Negative = Positive Positive x Positive= Positive Negative x Positive = Negative Negative divided by Negative= Positive Negative divided by Positive= Negative Positive divided by Positive = Positive

How to find tip example: You spent 28$ at Chili's and want to tip 10%, how much will you tip? What is the final price?

1. 0.10 x 28= 2.80$ You will tip 2.80$ 28 + 2.80 = 30.80$ is the final price

Find the tax amount example: If you pay 480 for new point shoes and the sales tax is 10% how much in tax are you playing? What is the final price?

1. 0.10 x 480 = 48 So you will be paying 48$ in tax 2. final price = taxable amount + tax 480 + 48 = 528$ So the final price is 528$

Finding the Part example: what is 30% of 60

1. 0.30 x 60 = 18 Part = 18

Percent increase and decrease example: The price of a bouquet increased from 15$ to 20$ what is the increase?

1. 20 - 15 = 5 2. 5 divided by 20 = 0.25 3. 0.25 x 100 = 25 So the price increased 25%

To find discount example: A shirt cost 25$ and you have a 10% discount coupon, how much is taken off? And what is the selling price?

1. 25 x 0.10= 2.50 2.50$ is taken off. Selling price= original price - discount 25 - 2.50= 22.50$

Finding a Percentage example: 30 is what percent of 120?

1. 30 divided by 120 = 0.25 2. 0.25 x 100 = 25 So 30 is 25% of 120

Finding the Base Example: 70 is 25% of what number?

1. 70 divided by 0.25= 280 Base = 280

Adding mixed numbers example: 3 2/3 + 1 2/5

1. Add whole numbers 3+1= 4, set aside 2. Add fractions 2/3 + 2/5 = 16/15 (see flashcard) 3. Convert to mixed number 16/15 = 1 1/15 4. Add mixed number and the whole number 1 1/15 + 4 = 5 1/15

Adding Mixed Numbers

1. Add whole numbers then set aside 2. Add both fractions separate from their whole numbers, this will result in an improper fraction 3. Convert improper fraction into mixed number 4. Add mixed number to the sum of the whole number

simplifying ratios example: 12:18

1. Both 12:18 are both divisible by 6, six goes into 12 two times, 6 goes into 18 three times. Answer: 2:3

Fraction multiplied by exponent example: (5x^5/2x^4)^2

1. Cancel out common factor ( when two variables are the same, but ones exponent is lesser, than it can be subtracted from the greater exponent to be cancelled out.) x^4 is a factor of x ^5 so we can subtract to cancel out x^4 and get x (since x = 1 and 4-5 = 1) 2. Raise all numbers and variables to the power of the exponent (5^2)= 25 (x^2)= x^2 (2^2)=4 Answer : 25x^2/4

Fraction Multiplied by an exponent

1. Cancel out common factors 2. Simply bring all numbers and variables in the fraction up to the exponent 3. if there is an exponent within the fraction, simply multiply them.

Dividing Exponents

1. Cancel out common factors (if an divisor integer can divide an dividend and vice verse it cancels itself out and the other integer becomes the product) 2. Subtract exponents if the dividend exponent is greater X ^ a - b ( product becomes the dividend) If the divisor exponent is greater 1 / X b - a (product becomes the divisor) (a = dividend b = divisor) 3. Continue until in simplest form

Subtracting Mixed Numbers example 2 1/5 - 1 2/3

1. Convert both into improper fractions 2 1/5 = 11/5 and 1 2/3 = 5/3 so the equation becomes 11/5 - 5/3 2. Make denominators "like" 33/15 - 25/15 3. subtract across numerators 25-33= 8 Answer = 8/15

Subtracting Mixed Numbers

1. Convert both mixed numbers into improper fractions 2. Make both denominators "like" ( see adding and subtracting fractions) 3. Subtract across numerators, denominator stays the smae. 4. simplify if needed.

Multiplying Mixed Numbers

1. Convert both mixed numbers into improper fractions 2. Multiply across numerators 3. multiply across denominators 4. simplify fraction 5. Convert to mixed fraction

Multiplying Mixed number example: 3 1/3 x 2 3/5

1. Convert both mixed numbers into improper fractions 3 1/3 = 10/3 and 2 3/5 = 13/5 2. Multiply across 10 x 13= 130 and 3 x 5 = 15 3. Simplify 130/15 simplified = 26/3 4. Convert to mixed fraction 26/ 3 into a mixed number = 8 2/3

Dividing mixed numbers

1. Convert mixed number into improper fraction 2. Divided fractions (keep change flip) 3. Simplify 4. Convert to mixed number.

Solving Proportional Ratio example: 3:4 = 9:X

1. Cross multiply 3 x X= 3x and 4 x 9= 36 2. divide 36/3x -> 12x

Rounding Decimals

1. Identify what place value you want to round to (ex. tenths, hundredths, thousandths, tens thousandths etc.) 2. Find the digit to the right of the place value you are rounding to 3. If it is 5 or greater, add 1 to the place value you are rounding to 4. if it is lesser than 5 remove it completely.

Dividing Decimals

1. Ignore decimal and set up like a normal division problem 2. If the divisor (the number that divides) is a decimal, multiply both the divisor and dividend by 10 3. Solve

Multiplying Decimals

1. Ignore decimal points and set up like columns like they are whole numbers 2. multiply 3. count how many digits are to the right of the decimal point 4. for every digit to the right of the decimal point it is placed further to the left of the product.

Linear equation with unknown coefficient

1. Isolate all identical coefficients to one side. 2. Add all non identical variables to one side 3. rewrite in

Adding and Subtracting Decimals

1. Line up decimal points of both numbers 2. Add zeros if necessary so both have equal number of digits 3. add or subtract using column method

Adding Fractions Example 3/4 + 2/3

1. Make denominators like, HCP of denominators 4 and 3 is 12. 2. Multiply Numerators, 4x 3=12 so 3x3= 9. first fraction then equals 9/12. 3 x 4=12 so 2x 4 = 8, the second fraction is 8/12. 3. Add across both fractions numerators 9/12 + 8/12 = 17/12 4. Simplify or convert to mixed fraction

Adding and Subtracting Fractions

1. Make sure both fractions are "like" meaning they both have the same denominator 2. To make fractions like find the highest common product (number they both go into equally) then make that both denominators. 3. Multiply their numerators by what the original denominator was multiplied by to make it its highest common product 4. Add/Subtract both numerators and the denominator stays the the same. 5. Convert to mixed fraction or simplify. (if the denominators are already the same skip steps 2-3)

Multiplication of Exponents example: 3x^2 x 4x^7

1. Multiply Integers 3 x 4 = 12 2. Add Exponents 2 + 7= 9 Answer= 12x^9

Multiplication of Exponents

1. Multiply Integers 2. Add exponents like simple addition.

Multiplying Fractions

1. Multiply across numerators 2. Multiply across denominators 3. simplify

Converting to improper fraction example 4 3/9

1. Multiply denominator by whole number 9 x 4 = 36 2. Add numerator 36 + 3 = 39 3. put over denominator 39/9 2. simplify

Multiplying fractions example 6/5 x 2/3

1. Multiply numerators 6 x 2 = 12 2. Multiply denominators 5 x 3= 15 3. simplify 12/15 = 4/5

When Exponent are outside of parenthesis (a^b)^3

1. Multiply the exponent outside of the parenthesis by each individual exponent 2. bring all integers up to the power of the exponent outside of the parenthesis.

To find the discount

1. Regular price x discount percent

Raising a fraction to a negative power

1. Reverse the numerator and denominator( this makes the exponent positive) 2. bring both numerator and denominator up to the exponents power

Combining Like Terms

1. Solve for Parenthesis 2. Combine like terms, this means adding numbers with the same variable together, and adding numbers with no variables together. 3. Rewrite with combine terms, the numbers with variables are usually written first

Dividing Decimal example: 1.50 divided by 0.5

1. Solve like normal division problem 0.5 is a decimal so we divided both 0.5 and 1.50 by 10 0.5 x 10 = 5 and 1.50 x 10 = 15 15 divided by 5 = 3

Multiplying decimals example: 0.53 x 0.32

1. Solve like whole number 53 x32= 1,696 2. There are 4 digits to the right of the decimal point ( 0.53 and 032) 3. Add decimal point 1,696 = 0.1696

Dividing Mixed Number Example: 2 2/3 divided by 1 1/2

1. convert into improper fractions 2 2/3 = 8/3 and 1 1/2= 3/2 2. Divide fractions 8/3 divided by 3/2 = 16/9 3. simplify ( not possible here) 4. convert to mixed number 16/9 into a mixed number = 1 7/9

Converting Improper fraction to mixed number

1. divide numerator by denominator (The product becomes the whole number) 2. The remainder becomes the numerator 3. The denominator stays the same

Converting improper fraction to mixed number example: 21/8

1. divided numerator by denominator 21 divided by = 16 and goes in twice so the whole number is 2 2. The remainder = 5 Answer = 2 5/8

Rounding Decimals example: round 3.374839 to the thousandths place

1. the digit to the right of the thousandths place is 8 2. 5 is greater than five so it is removed along with all the digits to the right of it and one is added to the thousandths place. Answer: 3.375

Simplifying Fractions

1.Find LCM (Number that can divide/go into both numerator and denominator) 2. Put those products into fraction form 3. Make sure its in lowest terms.

Combing Like Terms Example

2(-14 + r) -(-3r - 5) 1. Deal with parenthesis Multiply the outside number by all the terms within the parenthesis 2 x -14 = -28 | 2 x r = 2r | -1 x -3r = 3r | -1 x -5 = 5 | 2. Combine like terms -28 + 5 = -23 | 2r + 3r = 5r | 3. Rewrite 5r - 23

General Rule of percents

20 is 5% of 400 20= The part 5%= The percent 400= The base

Converting mixed number to fraction

3 2/3 is a mixed number, (the numerator is greater than the denominator) 1. Multiply denominator by whole number 2. add the numerator 3. put some over denominator (denominator stays the same)

Dividing Fractions Example 4/5 divided by 3/8

4/5 divided by 3/8 1. keep the first fraction 4/5 the same 2. convert division into multiplication 4/5 x 3/8 3. Flip right fraction 4/5 x 8/3 4. Multiply across 4/5 x 8/3 = 32/15 5. Convert

Finding the Base

Base = Part divided by the percent

Dividing Exponents example: 12x^2 y/ 3xy^3

Don't worry, it is much simpler than it seems. 1. Cancel out common factors. 3 is a factor of the integer 12, so 3x cancels out, 12 divided by 3 = 4 so it becomes 4 Now we are left with 4x^2 y/ xy^3 2. Subtract like exponents from each other x^2 and x are "like" because their variable is x The dividends (x^2) exponent is greater than x so we use this formula X^ 2 -1 = X (because x is equal to 1) Now we are left with 4x y/ y^3 3. Continue to solve by simplifying expression through elimination The divisor (y^3) is greater than y so we use the formula 1/y^ 3-1 = y^2 So our answer is 4x/y^2

When exponents are outside of the parenthesis example 1: (x^2 y^4)^3 Example 2: -x^4 y^3 x (-5x^6 y^2)

Example 1: 1. Multiply each inner exponent by the outer exponent separately 2 x 3 =6 and 4 x 3= 12, these are the new exponents Answer: x^6 y^12 Example 2. 1. Simply add the exponents to their their respective variables (x variables go with x variables, y variable go with y variables) 6 + 4 = 10 and 2 + 3 = 5, these are the new exponents, Answer= -5x^10 y ^5 everything else to the left of the multiplication symbol cancels out unless they were integers, than they would be multiplied by their respective integers within the parenthesis)

Simplifying Ratios

Ratios are used to make comparisons between two numbers 1. Divide both sides of the ration by the same number to simplify it

General Rule 1.

For fractions the final answer usually must be in its most simplified form or as a mixed number unless directly specified.

How to add, subtract, multiply or divide a whole number by a fraction

General Rule. Convert whole number into a fraction by adding 1 as its denominator then follow the usually steps based for adding, subtracting, dividing or multiplying two fractions. 4/7 x/-/+/divided by/ 3 = 4/7 +/-/x/ divided by/ 3/1.

Finding missing side of a similar triangles

If side

Dividing Fractions

Keep, Change, Flip. 1. keep the first fraction unaltered 2.convert the division symbol to multiplication 3. flip the right fraction upside down 4. Multiply across denominators and numerators 5. Convert to mixed fraction if needed

Order of Operations

P: Parenthesis E: Exponents M: Multiplication D: Division A: Addition S: Subtraction

Finding the Part

Part = Percent x Base

Finding a percentage

Percent = Part divided by the Base x 100

Percent Increase and decrease

Percent of change is a concept that represents a degree of change over time. 1. New number - Original Number = sum 2. (Sum divided by original number) x 100 If the answer is positive, it is an increase, if the answer is negative, it is a decrease

linear equations with an unknown coefficient example

Py + 7 = 6y + q 1. add all identical coefficients to one side subtract 6y from

Find the tax amount

Tax= Percent of tax (tax rate) x taxable amount (income, price of item etc.)

Subtracting exponents example 2: 3^2 x 3^3- 3^3 x 3

This looks confusing but is actually quite simple. When two integers are both divisors/dividends (on the same side) One of the cancels out, all you need to so is add its exponent to the other exponent So 3^2 x 3^3 -> 3^ 2 + 3 = 3^5 and 3^3 x 3 -> 3^ 3 + 1 = 3^4 so now we have 3^5/3^4 and can solve like normal 3^ 5- 4= 3^1 or simply 3.

Proportional Ratios

Two Ratios are proportional if they represent the same relationship. (are equal) 1. To solve for x cross multiply ratios 2. Divide

General Rule of Exponents

Variables are equal to 1. ex. x = 1 Exponents are shortened forms of repeated multiplication of a number by itself ex. 4^5 = 4 x 4 x 4 x4 x 4 = 1,024, x^2 = square root x^3 = cubic root 5x^7 (5 = integer, X = variable, 7 = exponent) Our goal when multiplying and dividing exponents is to get rid of (cancel out) as many integers as possible.