# M: General Math # General Math

The sections below are designed to help Lethbridge College students learn and practice general math skills! You can find information on arithmetic, fractions, factoring, and more below!

## Arithmetic

Arithmetic is a branch of mathematics that deals with the study of basic mathematical operations such as addition, subtraction, multiplication, and division. It is concerned with the manipulation and calculation of numbers. Arithmetic is one of the most fundamental branches of mathematics, and it forms the basis for more advanced mathematical topics such as algebra, calculus, and number theory.

## Fractions

A fraction is a way of representing a part of a whole or a ratio between two quantities. Fractions are typically written as two numbers separated by a horizontal line, with the number on top called the numerator and the number on the bottom called the denominator.

A simple way to remember this is the mnemonic device: nUmerator Up, Denominator Down.

Alternatively, think “Notre Dame” (N before D, Numerator before Denominator).

The numerator represents the number of equal parts being considered, while the denominator represents the total number of equal parts in the whole. For example, the fraction 3/4 represents three equal parts out of a total of four equal parts. Fractions can be added, subtracted, multiplied, and divided, but to do so, they must have a common denominator. A Common Denominator is a multiple of both denominators, which allows the fractions to be combined. See the examples below.

To add fractions with the same denominator (the number at the bottom of the fraction), you simply add the numerators (the numbers at the top) and keep the denominator the same. For example:

To add fractions with different denominators, you need to first find a common denominator. This is the smallest multiple that both denominators share. Once you have the common denominator, you convert each fraction so that they have the same denominator, and then you add the numerators. For example:

Here is another example where we can use equivalent (which means “the same as”) fractions to add fractions that do not have the same denominator:

Now we have:

To subtract fractions with the same denominator, you simply subtract the numerators and keep the denominator the same. For example:

To subtract fractions with different denominators, you need to first find a common denominator, just like when adding fractions (see the above example). Once you have the common denominator, you convert each fraction so that they have the same denominator, and then you subtract the numerators. For example:

#### Multiplying Fractions

To multiply fractions, you simply multiply the numerators and then multiply the denominators.

For example:

#### Dividing Fractions

To divide fractions, you invert the second fraction (that is, swap the numerator and denominator) and then multiply the first fraction by the inverted second fraction. For example:

• Visual Fractions help you to understand fractions by manipulating visual pieces. You can learn about identifying, renaming, comparing, adding, subtracting, multiplying, and dividing fractions.
• Math Drills offers fraction printouts and practice question worksheets.

#### Simplifying Fractions

Simplifying a fraction means reducing it to its lowest terms. In other words, you want to express the fraction as a whole number or as a fraction with the smallest possible numerator and denominator.

To simplify a fraction, you need to find the greatest common factor (GCF) of the numerator and denominator and then divide both of them by this number. The GCF is the largest number that divides into both the numerator and denominator evenly.

For example, let’s say we want to simplify the fraction

First, we need to find the GCF of 12 and 18. To do this, we can list the factors of both numbers:

The largest number that appears on both lists is 6, so the GCF of 12 and 18 is 6.

Next, we divide both the numerator and denominator by the GCF, which we found to be 6:

Therefore, the simplified fraction is

#### Mixed Fractions

A mixed fraction is a combination of a whole number and a proper fraction. It is also known as a mixed number.

In a mixed fraction, the whole number represents a whole quantity, while the proper fraction represents a part of that whole.

For example, let’s consider the mixed fraction  3 1/4

In this case:

The whole number is 3, which represents three whole units. The proper fraction is 1/4, which represents one-fourth of a unit. Together, represent three whole units and one-fourth of an additional unit.

To convert a mixed fraction into an improper fraction, you can multiply the whole number by the denominator of the fraction, and then add the numerator. Finally, this sum becomes the numerator of the improper fraction, and the denominator remains the same.

Let’s convert  to an improper fraction:

1. Multiply 3 (the whole number) by 4 (the denominator): 3 x 4 = 12.
2. Add the numerator (1) to the product: 12 + 1 = 13.
3. Therefore, 3 1/4 as an improper fraction is 13/4.

It’s important to note that mixed fractions and improper fractions represent the same value, but they are just different ways of expressing it.

#### Improper Fractions

An improper fraction is a fraction where the numerator is equal to or greater than the denominator. Unlike a mixed fraction, an improper fraction does not have a separate whole number component. Instead, it represents a fraction that is equal to or greater than one whole unit.

Here’s an example to help illustrate it:

Let’s consider the fraction 7/3

In this case:

The numerator is 7, which represents seven equal parts. The denominator is 3, which divides the whole into three equal parts. The fraction 7/3 as an improper fraction means that we have seven equal parts, each of which is one-third of a whole.

To convert an improper fraction into a mixed fraction, you divide the numerator by the denominator. The quotient becomes the whole number part, and the remainder becomes the numerator of the proper fraction. The denominator remains the same.

Using our previous example, 7/3 to convert  into a mixed fraction:

Divide 7 (the numerator) by 3 (the denominator): 7 ÷ 3 = 2 remainder 1.

Therefore, 7/3 as a mixed fraction is 2 1/3 . This means we have two whole units and one-third of an additional unit.

Remember, improper fractions and mixed fractions represent the same value, but they are just different ways of expressing it.

## Long Division

The steps involved in Long Division can be remembered through the simple mnemonic device: Dad, Mom, Sister, Brother, and Rover.

2. Multiply (Mother)
3. Subtract (Sister)
4. Bring Down (Brother)
5. Repeat (Rover)

This short tutorial will take you through the necessary steps of Long Division. To learn about Long Division with Decimals, check out this Khan academy tutorial.

• Math is Fun has several long division instructions with animated videos and practice questions.
• Soft Schools offers practice questions with step-by-step instructions.

## Order of Operations

The Order of Operations, also known as the “BEMDAS” rule, is important because it provides a standardized way to evaluate mathematical expressions. If you do not follow the correct order of operations, you can end up with an incorrect result!

To help you remember the order in which components of an equation should be calculated, consider using the mnemonic device: BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, and Subtraction).

The order of operations is important because it tells you which calculations to perform first when solving a problem with multiple operations. For example, in the expression 3 + 4 x 2, you would use BEDMAS to determine that you need to perform the multiplication before the addition. So, you would first multiply 4 x 2 to get 8, and then add 3 to get 11.

Here’s an example of a more complex problem:

Using BEDMAS, we would start by calculating anything in brackets first, so we would simplify (2 + 5) to get 7 and (1+3) to get 4:

Next, we would calculate any exponents, which in this case is 4² (4 x 4), which equals 16:

Next, we would perform any division or multiplication in order from left to right. The first multiplication is 7 x 3, which equals 21. The first division is 16 ÷ 4, which equals 4:

Finally, we perform any addition or subtraction in order from left to right:

So, the final answer to the problem is 17.

• Math Is Fun offers instructions with examples, practice questions, and an Order of Operations calculator.
• MathFrog, from the University of Waterloo, offers a skill-testing order of operations game.

## Algebra

Algebra is used to solve problems and to find solutions to equations. We use letters or symbols to represent unknown variables, such as x or y, and then use different functions to solve for them.

• The Algebra Refresher provides links to a series of algebra-related videos from Khan Academy
• Desmos is a user-friendly graphing tool that will graph one or more functions to help you visualize the graph and/or compare functions
• Arizona State University has an algebra crash course
• GeoGebra offers tons of free interactive resources

## Geometry

Geometry is a branch of mathematics that deals with the study of shapes, sizes, positions, and properties of objects in space. We also use concepts such as symmetry, congruence, similarity, and transformation to compare and classify these figures.

The Geometry Refresher website provides links to a series of videos from Khan Academy that will help reacquaint you with geometry-related topics.

You can also check out the “Area and Perimeter” chart in the Trades Math section for a quick reference guide.

## Factoring

Factoring refers to the process of breaking down a mathematical expression or number into smaller factors that, when multiplied together, produce the original expression or number. Factoring is an important concept in algebra and arithmetic, as it allows us to simplify and solve equations, identify patterns, and work with large numbers more easily.

Solving quadratic equations by factoring is a method used to find the values of the variable that make a quadratic equation true. A quadratic equation is an equation of the form

Where a, b, and c are constants, and x is the variable.

The process of solving quadratic equations by factoring involves the following steps:

1. Write the quadratic equation in the standard form: ax2 + bx + c = 0
2.  Factor the quadratic expression on the left side of the equation, if possible, into two binomial factors
3. Set each factor equal to zero. This step involves equating each binomial factor to zero and solving for x. This will give you two separate linear equations.
4. Solve each linear equation to find the values of x.

It’s important to note that not all quadratic equations can be factored easily. Some quadratic equations may have complex solutions or require alternative methods, such as completing the square or using the quadratic formula.

## Polynomials

Adding or subtracting polynomials involves combining like terms to simplify the expression. To add polynomials, follow these steps:

1.  Identify the like terms. Like terms are terms that have the same variable(s) raised to the same power(s). For example, in the expression 3x2 + 2x2 – 5x + 7, the terms 3x2 and 2x2 are like terms because they have the same variable (x) raised to the same power (2).
2. Group the like terms together by placing them next to each other. In the example above, you would group the 3x2 and 2x2 terms together.
3.  Add or subtract the coefficients of the like terms. The coefficient is the number in front of the variable. In the example, you would add 3 and 2 to get 5, resulting in 5x2. The -5x term remains as it is, and the constant term 7 remains unchanged, as there are no other variables to combine these with.
4. Combine the simplified terms to form the final polynomial expression. In the example, after simplifying the like terms, the expression becomes 5x2 – 5x + 7.

Here’s another example:

In this example, the like terms are the terms with the same variable and exponent. We have 2x³ and -x³, -4x² and 2x², 3x and -5x.

Group the like terms together:

Add or subtract the coefficients of the like terms to get:

Combine the simplified terms to get the final expression:

So, the sum of the two polynomials is – 2x² – 2x + 1.

#### Subtracting Polynomials

Here’s how to subtract polynomials step-by-step:

1. Write down the two polynomials that you want to subtract. Let’s look at the example (2x² – 3x + 5) – (4x² + 2x – 1)
2.  To subtract the second polynomial, change the sign of each term in the second polynomial by distributing the negative (-). For example, if you have (2x²– 3x + 5) – (4x² + 2x – 1), change the sign of each term in the second polynomial to get -4x² – 2x + 1.
3. Combine the like terms by adding or subtracting the coefficients of the corresponding terms. Like terms have the same variables with the same exponents.
4. Write down the simplified polynomial expression by combining the coefficients of the like terms. Include any terms that were not combined.

For example,

simplifies to

In summary, to subtract polynomials, you change the sign of the second polynomial and then combine like terms by adding or subtracting the coefficients. Finally, write the simplified polynomial expression.

#### Multiplying Monomials

Multiplying monomials involves multiplying the coefficients and the variables and then adding the exponents of the variables. Here is a step-by-step description of how to multiply monomials:

1. Identify the coefficients and variables. In a monomial, the coefficient is the numerical factor, and the variable is the symbol that represents an unknown value. For example, in the monomial 3x², the coefficient is 3 and the variable is x.
2. Multiply the coefficients together. For example, if you have 3x² and 4x³, multiply the coefficients 3 and 4 to get 12.
3. Multiply the variables together. In this step, apply the rule of adding the exponents when multiplying variables with the same base. For example, if you have and x³, multiply and x³ to get x⁽²⁺³⁾ = x⁵.
4. Combine the results from steps 2 and 3 to form the product of the monomials. For example, if you had 3x² and 4x³, the product would be 12x⁵.

Here’s another example:

1. Identify the coefficients and variables. The coefficients are 2 and 3, respectively. The variable is x.
2. Multiply the coefficients 2 and 3 to get 6.
3. Multiply the variables x⁴ and x². Apply the exponent addition rule: x * x² = x(4+2) = x⁶.
4. Combine the results from steps 2 and 3 to form the product: 6x⁶.

Therefore, the product of

Remember to multiply the coefficients and add the exponents of the variables when multiplying monomials.

#### Multiplying Polynomials

Multiplying two polynomials involves applying the distributive property and combining like terms. Here’s a step-by-step description of how to multiply two polynomials:

1. Write down the two polynomials that you want to multiply. For example, let’s use (2x + 3)(5x – 8).
2. To multiply two polynomials, apply the distributive property, which states that for any numbers or variables a, b, and c: (a + b) c = ac + bc. See the image below.
1. Using the distributive property, multiply each term of the first polynomial by each term of the second polynomial. For example: (2x + 3)(5x – 8) = (2x * 5x) + (2x * (-8)) + (3 * 5x) + (3 * (-8)). Often this is called the FOIL method (F=First, O=Outer, I=Inner, L=Last), which is depicted in the image below.
1. Multiply the coefficients and add the exponents for each term to simplify the expression. Combine like terms, if any. For the example we get: 10x² – 16x + 15x – 24.
2. If necessary, combine the like terms in the resulting expression by adding or subtracting the coefficients of the like terms. In this example, combine the -15x and 4x terms to get the final expression: 10x² – x – 24.

The final expression after multiplying the two polynomials (2x + 3)(5x – 8) = 10x² – x – 24.

Remember to use the distributive property to multiply each term of one polynomial by each term of the other polynomial, and then simplify the resulting expression by combining like terms if necessary.

## Exponent Laws

Below you will find a brief overview, with examples, of the most common exponent laws.

1. Product Rule: When multiplying two terms with the same base, you can add their exponents.
• It is represented as am x an = a(m+n). For example, 23 x 24 = 2(3+4) = 27.
2. Quotient Rule: When dividing two terms with the same base, you can subtract their exponents.
• It is represented as am / an = a(m-n).  For example, 57 / 54 = 5(7-4)= 53.
3. Power Rule: When raising a power to another power, you can multiply the exponents.
• It is represented as (am)n = a(m x n). For example, (32)4 = 3(2 x 4) = 38.
4. Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1.
• It is represented as a0 = 1, where a ≠ 0. For example, 50 = 1.
5. Negative Exponent Rule: When a number is raised to a negative exponent, it can be rewritten as the reciprocal of the positive exponent.
• It is represented as a-m = 1 / am, where a ≠ 0. For example, 2-3 = 1 / 23 = 1/8.
6. Product of Powers Rule: When a product of two or more variables (x and y in this case) is raised to an exponent (n), you can distribute the exponent to each individual variable within the parentheses.
• It can be represented as: (xy)n = xnyn. For example, (xy)3 = (x3y3).
7. Quotient of Powers Rule
• It is represented as (x/y)n = (xn / yn). For example, (x/y)2 = (x2/y2).

• Jeff Hamilton – YouTube: Our instructors at Lethbridge College post tons of great videos, so check out Jeff’s YouTube channel for lots of great information.
• The PrepSTEP database contains excellent tutorials and practice activities for the topics below and more. These resources are free for Lethbridge College students, but not for everyone, so you will have to login using your LC email address and password.
• Math-Drills.com is an online resource that provides printable worksheets on several fundamental math topics.
• Mathplanet offers robust educational videos, written resources, examples, and practice questions for various levels of math.
• Math Interactives is a multimedia resource that covers the topics of numbers, patterns and relations, shape and space, and statistics and probability. The website includes games and instructions that relate mathematics to real world situations.
• Math Is Fun offers a range of math-related instructions, practice questions, worksheets, and games. This website also includes calculus and physics material.
• Mathwords: Terms and Formulas from Beginning Algebra to Calculus is a website that provides formulas and definitions for various levels of math.
• Khan Academy is an online video resource that provides informative educational videos pertaining to Developmental math, algebra, and geometry.

Some content from the General Math webpage was generated using ChatGPT. OpenAI. (2023). ChatGPT (Mar 14-June 21 versions) [Large language model]. https://chat.openai.com/chat

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