M114Notes/01-Preliminaries.Rmd at main · TheElementsMath/M114Notes · GitHub

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#  (PART) Essentials {-}

#  Preliminaries


##  Lecture 1:  The Basics {.unnumbered}

###  Learning Outcomes: {-}

1.  Understand BEDMAS.
2.  Expand and factor expressions.
3.  Solve systems of equations by substitution and elimination.


###  Lecture Notes: {-}

Lecture material for this class come from Sections 1.1 -- 1.5 and can be found below.  This material is considered review material and so it is not covered in depth.

1. **Document:  [Setting up my Calculator](https://www.wikihow.com/Set-Decimal-Places-on-a-TI-BA-II-Plus-Calculator)**
Your TIBA II Plus financial calculator does not do BEDMAS the way you think it should.  You will need to change the settings.<br><br>

2. **Video: [BEDMAS](https://youtu.be/ezGExislcbU)**
Use BEDMAS to help with order of operations.
   1. Brackets
   2. Exponents
   3. Multiplication and Division in order from left to right
   4. Addition and Subtraction in order from left to right <br><br>

3. **Video: [Rational Numbers](https://youtu.be/Yf2jnQHyHdQ)**
Add, subtract, multiply and divide fractions.
   1. $\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}$
	 2. $\frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{cb}{db} =  \frac{ad + cb}{bd}$
	 3. $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
	 4. $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$  <br><br>

4. **Video: [Basic Algebra I](https://youtu.be/i8xFdGZ5zOU)**
**Video: [Basic Algebra II](https://youtu.be/Vb_2Aqq08ok)**
Know how to manipulate equations and expressions.
   1. Solve linear equations for a given variable.
	 2. Evaluate expressions.
	 3. Factor and expand a given algebraic expression.  <br><br>

5. **Video: [Systems of Equations I](https://youtu.be/ooUss4RfLHI)**
**Video: [Systems of Equations II](https://youtu.be/rDDTETJjmyY)**
You can use the methods of substitution and elimination to solve systems of equations.
   1. Use substitution when it is easy to isolate one of the variables in one of the equations and substitute it into the other equation.
	 2. Use elimination if the coefficients in front of one of the variables is the same size in both equations.
	 3.  Either method will work for most problems and you are free to use whichever works best for you.  <br><br>


###  Lecture Problems: {-}

   1.  Simplify: \[15 + \left[ \frac{3\left(8 + (2 - 10)^2 /4 \right)}{12} \right]^2.\]
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/3WmpvL441rA)</div>
   2.  Simplify and collect like terms:  \[\frac{8x}{0.5} + \frac{5.5x}{11} + 0.5\left(4.6x - 17 \right).\]
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/BHPP---sVZw)</div>
   3.  Solve the following:  \[\frac{x}{1.1^2} + 2x(1.1)^3 = \$1000.\]
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/PZqTWdq1wiI)</div>
   4.  Solve the system of equations: \begin{align*}     3x + 4y &= 55\\  10x + 5y &= 100.\end{align*}
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/LvRuXkQ5Mao)</div>
   5.  Solve the system of equations: \begin{align*}     10x - 6y &= -10 \\ 4x + 6y &= 38.\end{align*}
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/YQjb2Z6y51E)</div>
   6.  Solve the system of equations: \begin{align*}     4x - 5y &= 18 \\ 4x + 3y &= 2.\end{align*}
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/c_YPN8Xt0Fc)</div>


###  Additional Problems: {-}

Additional problems that are typically done in class (with video solutions) can be found here:

*  [BEDMAS](https://theelementsmath.github.io/M114/preliminaries.html\#order-of-operations)

*  [Rational Numbers](https://theelementsmath.github.io/M114/preliminaries.html\#rational-numbers)

*  [Expanding and Factoring](https://theelementsmath.github.io/M114/preliminaries.html\#expanding-and-factoring)

*  [Solving Systems of Equations](https://theelementsmath.github.io/M114/preliminaries.html\#solving-systems-of-equations)


##  Lecture 2:  Ratios, Rates, Proportions and Percentages {.unnumbered}

###  Learning Outcomes: {-}

1. Interpret and apply rates.
2. Understand and use percentages.
3. Work with fractions and decimals.
4. Convert from rates/ percentages to ratios to fractions/ decimals.
5. Solve problems Involving ratios, rates, and percentages.

###  Review Problems From Last Lecture: {-}

   1.  Simplify the expression: $\;\;\;\;\; 900(1 + 0.05(15/12))$.\]
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/zgLRtB3JUpE)</div>
   2.  Simplify the expression: $\;\;\;\;\; 300 \left[\dfrac{(1 + 0.08/2)^{30} - 1}{0.08/2} \right]$.
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/Emv6osLWEvE)</div>
   3.  Solve the following equation for $x$:  $\;\;\;\;\; 15 x - \frac{5}{2} = \frac{15}{2}x + 10$.
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/uo48t8kVRkI)</div>
   41.  Solve the following system of equations:  \begin{align*} x + y &= 13\\ 2x - y &=8 \end{align*}
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/57H_RLVgpc8)</div>

###  Lecture Notes: {-}

Lecture material for this class come from Sections 1.6 and can be found below.  This material is considered review material and so it is not covered in depth.


1. **Video: [Percentages](https://www.youtube.com/watch?v=GU3i-S4sxdw)**
**Video: [Ratios](https://www.youtube.com/watch?v=Nj1cYFE7cBA)**
**Video: [Proportions](https://www.youtube.com/watch?v=j4TboDA7cNI)**
Ratios are used to compare multiple quantities.  They can be expressed: (a) using a colon, (b) as a fraction, (c) as a decimal, and (d) as a percentage.
   * **Ratios:** A comparison between two quantities using division.
    Example: A ratio of 3 to 2 can be written as \( 3:2 \), \( \frac{3}{2} \), or "3 to 2".
   *  **Rates:** A specific type of ratio comparing quantities with different units.
    Example: A car travels 60 miles in 2 hours, so the rate is \( \frac{60 \text{ miles}}{2 \text{ hours}} = 30 \text{ miles/hour} \).
   *  **Fractions:** Express part of a whole as a ratio of two numbers (numerator and denominator).
    Example: \( \frac{1}{4} \) means 1 part out of 4 equal parts.
   *  **Percentages:** A fraction with a denominator of 100. Expresses how many parts out of 100.
    Example: \( 25\% = \frac{25}{100} = 0.25 \)
   *  **Decimals:** A way of expressing fractions and percentages using powers of 10.
    Example: \( 0.5 = \frac{1}{2} = 50\% \)<br><br>


2. **Video: [Taxes](https://www.youtube.com/watch?v=O9YKlxowu68)**
**Video: [Exchange Rates](https://www.youtube.com/watch?v=ChgppEgOYc0)**
Ratios, rates and proportions have many different applications. They allow us to compare quantities, scale values, and interpret data in meaningful ways.
   * **Ratios in Simplest Integer Form:**  In many contexts, we prefer to express ratios using whole numbers for simplicity and clarity.
       * Example: A recipe that uses 2 parts sugar to 3 parts flour can be written as the ratio $2:3$.
   * **Ratios with a term of 1:**  It is often helpful to express a ratio with one of the terms as 1 to make comparisons easier.
       * Example: If the ratio of students to teachers is \( 24:1 \), it means there are 24 students per teacher.
   * **Multiple Ratios or Rates in One Situation:**  Some problems involve more than two quantities or rates that must be compared simultaneously.
       * Example: A mixture of paint may require a ratio of red : blue : white = \( 3:2:5 \).
       * In finance, one might compare exchange rates between three currencies: USD to EUR, EUR to GBP, and USD to GBP.
   * **Real-World Applications:** Currency, Tax, and Conversions: Ratios and rates are commonly used in practical calculations such as:
       * Exchange Rates: Converting currencies (e.g., \( 1 \text{ USD} = 0.92 \text{ EUR} \)).
       * Taxes and Discounts: Applying percentage increases or decreases to prices (e.g., adding 15\% tax, taking off 20\% discount).
       * Unit Conversions: Converting between units using rates (e.g., \( 1 \text{ inch} = 2.54 \text{ cm} \)). <br><br>


###  Lecture Problems: {-}

   1. Percentages
       a. Calculate 175\% of \$500.
	     b. 65\% of what amount is \$85?
	     c. The tax rate is 8.25\%.  How much tax is required for a purchase of \$450?
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/C5Ik5tzzrtE)</div>
   2. The accounting, mathematics and economics departments have 10, 14 and 8 employees respectively.  We have a \$10,000 marketing budget that needs to be divided among those departments based on the ratio of the number of employees.  How much should each department receive?
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/2CGmE2oImEk)</div>
   3. The accounting, mathematics and economics departments have 10, 14 and 8 employees respectively.  The departments teach 500, 1000 and 350 students respectively.  Which department is generating the most student revenue compared to their department size?
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/FLtoyc6iuGY)</div>
   4.  Jim and Ben have a partnership with a 40-60 split.  Jim is going to invest an additional \$20,000 into the partnership.  How much should Ben invest?
	  	 <div style="text-align: right;">[Click here for the solution](https://youtu.be/v_de0pgxKco)</div>


###  Additional Problems: {-}

Additional problems that are typically done in class (with video solutions) can be found here:

*  [Ratios, Proportions and Percentages](https://theelementsmath.github.io/M114/preliminaries.html\#ratios-proportions-and-percentages)