These are my course notes for Unit 1. All notes in this course are available in html (and pdf or docx on request).
Your first decision for this course is to decide what calculator you will use in this course. Fluent use of a calculator is essential for this course. Two recommended (and free) options for calculators are outlined below.
For calculations, I recommend using the free online calculator, Desmos:
https://www.desmos.com/scientific
Please familiarize yourself with using this calculator. If you haven’t used a scientific calculator recently, I recommend practicing using with this tutorial.
If you choose to use Desmos, I recommend bookmarking it for easy access as you will be using it constantly throughout the course.
Alternatively, If you are interested in learning a little programming for calculating your work in this course, you may download the following software:
R is a free software environment for statistical computing and graphics. It compiles and runs on a wide variety of UNIX platforms, Windows, and macOS. RStudio is a desktop environment we will be using to run statistical computations.
These notes were generated using R. In the examples below, R code computations are shown in grey boxes. The output of the code is below following the “## [1].”
#Example computation
2+3## [1] 5If you choose to use R, please let me know if you have any questions on how to do anything as we proceed in this course!
The Latin root “cent” means one “one hundred.” You may be familiar with this root in words like:
So, the word percent literally means “per hundred.”
Example: Seventy percent, or 70%, is equivalent to \(\frac{70}{100}=0.7\).
Question: Write each of the following as a percent.
0.24
3/4
2
Example: Sales tax in Chickasha is 9.5%. What sales tax do you pay for on a $140 purchase?
In calculating with percentages, we first must consider the percent as a fraction of a hundred or as a decimal: \[9.5\%=\frac{9.5}{100}=0.095.\]
Next, we can consider the calculation of a percentage as a proportion: \[\text{percent}=\frac{\text{part}}{\text{whole}}.\] Thus,
\[\frac{9.5}{100}=\frac{\text{tax paid}}{\$140}.\] So the total tax paid is calculated, \[\text{tax paid}=\$140*\frac{9.5}{100}=\$140*0.095\]
140*0.095## [1] 13.3Notice how we rearranged this formula to solve for tax paid. In general, we can find the part of a whole value by the formula: \[\text{part}=\text{whole}*\text{percent}.\]
Written Homework:
Some purchases have a certain tax charged by the government that is particular to that good or service. This is called an excise tax. In Oklahoma, the cars are charged a sales tax of only 1.25% and an excise tax of 3.25%.
Example: Tuition has been rising by about 6% annually. Based on this projection, what will tuition be next academic year?
For, this academic year (2020-2021):
If tuition will increase 6%, in-state tuition will increase \(\$3,195*0.06=\$191.70\), so total in-state tuition would be \(\$3,195 + \$191.70 = \$3386.70\).
3195 + 3195*0.06## [1] 3386.7Question:
Example: The value of a car dropped from $7400 to $6800 over the last year. What is the percent change (or percent decrease) in the value of this car?
To compute the percent change, we first need to find the dollar value change: \(\$6800-\$7400 = -\$600\). Often we will take the absolute value of this amount, which is called the absolute change: \(\lvert -\$600 \rvert =\$600\).
Using the absolute change, the percent change of a quantity can be found by modifying the above formula:
\[\text{absolute change} = \text{original value} *\text{percent change}.\] This can be re-written as follows: \[\text{percent change} = \frac{\text{absolute change}}{\text{original value}} = \frac{ \lvert\text{new value}-\text{original value}\rvert}{\text{original value}}.\] The original or starting value in calculating a change is called the base of a percent change. Note that if we were to remove the absolute value from this equation, we would still get the percent change, but possibly a negative, telling us that the change was a decrease.
Thus, solving \(\text{percent change} = \frac{\text{absolute change}}{\text{original value}}=\frac{\$600}{\$7400}=0.08108\) gives a percent change of about \(8.11\%\).
600/7400## [1] 0.08108108Written Homework:
Jim Hines won the 1968 Olympic gold with a time of 9.95 seconds. In the 2012 Olympics, Usain Bolt ran the 100m dash in 9.63 seconds.
Part a. To compute this part, Bolt’s time was \(9.95-9.63=0.32\) seconds faster than Hines’. Since we are comparing against Hines’ time, this is the base, or starting value. Since we are comparing Bolt’s time against Hines’, we will use Hine’s time as the base.
\[\text{percent change} = \frac{\text{absolute change}}{\text{base value}}=\frac{0.32}{9.95} \approx 0.0322\]
percentChange = (9.95-9.63)/9.95
percentChange## [1] 0.0321608Thus, Bolt’s time was 3.22% faster than Hines’.
Part c: To say “Bolt’s time was \(x\%\) of Hines’ time” literally means, “\((\text{Bolt's time}) = x\% * (\text{Hines' time})\),” or \(9.63 = x*9.95\), giving \(x=0.9678\).
x = 9.63/9.95
x## [1] 0.9678392Thus, Bolt’s time was 96.78% of Hines’. In other words, Bolt’s time was \(100\% - 96.78\% = 3.22\%\) less than Hines’ time, as found in part a!
A company wants to decrease its energy use by at least 15%. If their electric bill is currently $2,200 a month, what will their bill be if they decrease their energy use by 15%?
If their next bill is $1,700 a month, were they successful? Why or why not?
A proportion equation is an equation showing the equivalence of two rates or ratios.
Questions:
Solve the following proportion for \(x\): \(\frac{x}{10}=\frac{13}{5}\)
Solve the following proportion for \(x\): \(\frac{4}{5}=\frac{6}{x}\)
Example: A crepe recipe calls for 2 eggs, 1 cup of flour, and 1 cup of milk. How much flour would you need if you decided to use 5 eggs?
We can model this as “If 2 eggs are proportional to 1 cup flour, 5 eggs are proportional to x cups,” i.e. \[\frac{2 \text{ eggs}}{ 1 \text{ cup}} = \frac{5 \text{ eggs}}{ x \text{ cups}}.\] Thus, \(x=\frac{5}{2}=2.5\) cups.
Written Homework:
Some problems involving rates involve multiple proportions that can be computed directly using a process called unit analysis.
Example: If a bicycle is traveling at 15 miles per hour, how fast is that in feet per second?
We know:
Thus: \(\frac{5280 \text{ feet}}{1 \text{ mile}} = 1\), \(\frac{1 \text{ hour}}{60 \text{ minutes}} = 1\), and \(\frac{1 \text{ minute}}{60 \text{ seconds}} = 1\). (Why is this true?)
Using these facts carefully, we convert the units of the rate “15 miles per hour” as follows:
\[\frac{15 \text{ miles}}{1 \text{ hour}}* \frac{5280 \text{ feet}}{1 \text{ mile}}* \frac{1 \text{ hour}}{60 \text{ minutes}}* \frac{1 \text{ minute}}{60 \text{ seconds}} = 22 \frac{\text{feet}}{\text{sec}}\]
(Notice how the units on the top and bottom of the left-hand side of this calculation cancel leaving us with \(\frac{\text{feet}}{\text{sec}}\) as the only remaining units!)
(15*5280)/(60*60)## [1] 22Using this we can ask, how many feet will the bike cover in cover in 10 seconds?
\[10\text{ seconds} * 22 \frac{\text{feet}}{\text{seccond}}=220 \text{ feet}\] (Again the units cancel in such a way that we’ve gotten our desired answer in feet!)
Question:
Written Homework:
There are three main types of taxes:
Suppose you make $30,000. A flat tax of 10% would be \(\$30,000*0.10=\$3,000\).
The United States income tax on earned wages is an example of a progressive tax.
In 2019, a taxpayer filing as single was charged the following tax rates on their taxable income (we’ll define this in a bit).
| Income Range | Income Tax Rate |
|---|---|
| $0 to $9,700 | 10% |
| $9,701 to $39,475 | 12% |
| $39,476 to $84,200 | 22% |
| $84,201 to $160,725 | 24% |
| $160,726 to $204,100 | 32% |
| $204,101 to $510,300 | 35% |
| $510,301+ | 37% |
Example: Suppose you have $30,000 of taxable income.
9700*0.10 + (30000-9700)*0.12## [1] 3406This means you pay $3406 of income.
According to the IRS tax tables, a single person with an income from $30,000 to $30,050 taxable income would pay $3409. Why?
Consider the following computations, and use this to infer how these tables are constructed.
9700*0.10 + (30000-9700)*0.12## [1] 34069700*0.10 + (30025-9700)*0.12## [1] 34099700*0.10 + (30050-9700)*0.12## [1] 3412Question:
| Income Range | Income Tax Rate |
|---|---|
| $0 to $9,700 | 10% |
| $9,701 to $39,475 | 12% |
| $39,476 to $84,200 | 22% |
| $84,201 to $160,725 | 24% |
| $160,726 to $204,100 | 32% |
| $204,101 to $510,300 | 35% |
| $510,301+ | 37% |
When calculating your taxes, not all of your income is taxed. This part of your income is called a deduction. Below are the standard deductions for several filing statuses: single, married, and head of household (unmarried and pay for more than half of the household expenses for you and at least one qualifying child or dependent). These are called standard deductions because if you have a lot of certain kinds of expenses, you might be able to do more work to show you qualify for a larger deduction.
| Filing status | 2019 tax year |
|---|---|
| Single | $12,200 |
| Married, filing jointly | $24,400 |
| Head of household | $18,350 |
Suppose you are single and make $30,000 of annual income. Only \(\$30,000-\$12,200=\$17,800\) of your income will be taxed. This portion is called your taxable income.
Then you pay 10% tax on $9,700 of this income, and 12% tax on the remaining \(\$17,800-\$9,700=\$8,100\).
9700*.10 + 8100*.12## [1] 1942Your total income taxes would be $1,943.
Written Homework:
| Income Range | Income Tax Rate |
|---|---|
| $0 to $19,400 | 10% |
| $19,401 to $78,950 | 12% |
| $78,951 to $168,400 | 22% |
| $168,401 to $321,450 | 24% |
| $321,451 to $408,200 | 32% |
| $408,201 to $612,350 | 35% |
| $612,351+ | 37% |
One more thing that can affect how much you pay are tax credits, which are simply reductions to the amount of taxes you pay. One tax credit you likely qualify for is the American Opportunity Tax Credit (AOTC). (Check and see if you qualify!)
Suppose, in the situation above, you owe $1943 in taxes but qualify for a $2000 tax credit. Then you only pay \(\$1,943-\$2000=-\$57\) taxes. That is, you would receive \(\$57\) back! Actually, according to the rules for the AOTC, you would have 40% of this refunded to you, or \(\$57*0.4=\$22.80\).
(2000-1943)*.4## [1] 22.8Written Homework:
| Income Range | Income Tax Rate |
|---|---|
| $0 to $9,700 | 10% |
| $9,701 to $39,475 | 12% |
| $39,476 to $84,200 | 22% |
| $84,201 to $160,725 | 24% |
| $160,726 to $204,100 | 32% |
| $204,101 to $510,300 | 35% |
| $510,301+ | 37% |
| Filing status | 2019 tax year |
|---|---|
| Single | $12,200 |
| Married, filing jointly | $24,400 |
| Head of household | $18,350 |
| Income Range | Income Tax Rate |
|---|---|
| $0 to $23,200 | 10% |
| $23,200 to $94,300 | 12% |
| $94,300 to $201,050 | 22% |
| $201,050 to $383,900 | 24% |
| $383,900 to $487,450 | 32% |
| $487,450 to $731,200 | 35% |
| $731,200 or more | 37% |
| Filing Status | Deduction Amount |
|---|---|
| Single | $14,600 |
| Married Filing Jointly | $29,200 |
| Head of Household | $21,900 |
Most problems you encounter in life will not have obvious solutions. Here is one outline of a problem-solving process you may want to use if you are stuck solving a problem in this class or life.
In your group, select one of the four problems below you want to work on together. Work together answering the provided questions and/or related questions you find interesting. Then, as a group, write a 1-2 (or more if you choose) page Report presenting:
Make sure you edit your Report to ensure it is readable with no grammar or spelling errors.
Each group will also make a short video presentation of their work, so keep in mind, that your work will be made public to other students to view and study.
Possible Group Member Roles:
ALL group members should participate in solving the problem!
Estimate each of the following, explaining the assumptions and conversions you use to arrive at your final estimate:
Based on this, where could you conserve the most water?
Consider the following four sets of statements. Choose three of the statements that are most interesting to your group. After you have each statement, explain what these numbers mean and whether the two values can be directly compared or not. Pay special attention to what the base quantities might be.
“Guns have murdered more Americans here at home in recent years than have died on the battlefields of Iraq and Afghanistan. In support of the two wars, more than 6,500 American soldiers have lost their lives. During the same period, however, guns have been used to murder about 100,000 people on American soil”
In the 2012 presidential elections, one candidate argued that “the president’s plan will cut $716 billion from Medicare, leading to fewer services for seniors,” while the other candidate rebuts that “Our plan does not cut current spending and actually expands benefits for seniors, while implementing cost-saving measures.”
In 2012, “16.3% of Americans are without health insurance.” (http://www.cnn.com/2012/06/27/politics/btn-health-care/index.html) At the same time, “only 55.9% of adults receive employer-provided health insurance.” (http://www.politico.com/news/stories/0712/78134.html)
In 2016, GOP lawmakers claimed that about “75 percent of the federal health care law’s taxes would be paid by those earning less than $120,000 a year,” citing the Congressional Budget Office (CBO) a statement that “76 percent of those who would pay the penalty [health care law’s taxes] for not having insurance in 2016 would earn under $120,000.” (http://factcheck.org/2012/07/twisting-health-care-taxes/) Are these the same, or is the GOP lawmakers’ claim inaccurate?
Search articles discussing the impact of gun ownership on public safety. Read ones that seem informative and trustworthy, taking note of any statistics that involve percentages until your group finds ten or more statistics describing the impact of gun ownership on public safety. Explain the significance of each statistic as well as what picture these statistics paint as a whole. Pay special attention to what the base quantities might be. Check the source of each statistic and then cite the original study the statistic came from. Below are some articles to start with, but feel free to use others as well.
Imagine the country is made up of 100 households. The federal government needs to collect $800,000 in income taxes to be able to function. The population consists of 6 groups:
This scenario is roughly proportional to the actual United States population and tax needs. You will determine new income tax rates for three proposals:
For Proposals 1 and 2, determine the tax rate needed to collect $800,000 (plus or minus $100). For Proposal 3, choose your own progressive tax rate system so that they add up to close to $800,000. In this case, you may decide to charge each household group the same rate on all their income, or in tax brackets as prescribed by the US tax system.
Discuss the advantages and disadvantages of each proposal, and present an argument for the advantages of your plan.
Hint: If you can figure out a way to use a spreadsheet (Excel or Google Sheets), to calculate and add up your taxes, this can save you some time in figuring this out and presenting your work.
Calculate what your 2024 federal income tax would be for the following possible situations:
Scenario 1: You earn $60,000 in standard wage income (W-2 income), file as single, and claim the standard deduction.
Scenario 2: You earn $20,000 in standard wage income, $40,000 in qualified dividends, file as single, and claim the standard deduction.
Scenario 3: You earn $60,000 in small business income, file as single, and claim the standard deduction.
Scenario 4: You are:
Use the 2024 tax tables, as well as any of the hints below.
After calculating taxes, answer the following questions: From these four scenarios, what are your impressions of how the income tax system treats these different forms of income and expenses? How does this inform how you will do taxes in the future?
| Filing Status | Deduction Amount |
|---|---|
| Single | $14,600 |
| Married Filing Jointly | $29,200 |
| Head of Household | $21,900 |
| Income Range | Income Tax Rate | Long-Term Capital Gains Rate |
|---|---|---|
| $0 to $11,600 | 10% | 0% |
| $11,601 to $47,150 | 12% | 0%; 15% at $47,025 |
| $47,151 to $100,525 | 22% | 15% |
| $100,526 to $191,950 | 24% | 15% |
| $191,951 to $243,725 | 32% | 15% |
| $243,726 to $609,350 | 35% | 15%; 20% at $518,900 |
| $609,351 or more | 37% | 15% |
| Income Range | Income Tax Rate | Long-Term Capital Gains Rate |
|---|---|---|
| $0 to $23,200 | 10% | 0% |
| $23,200 to $94,300 | 12% | 0%; 15% at $94,050 |
| $94,300 to $201,050 | 22% | 15% |
| $201,050 to $383,900 | 24% | 15% |
| $383,900 to $487,450 | 32% | 15% |
| $487,450 to $731,200 | 35% | 15%; 20% at $583,750 |
| $731,200 or more | 37% | 15% |
Hints:
Economic disparity in the United States has been continually increasing in the United States over the last 50 years. In other words, the rich are getting richer, the poor are getting poorer. One proposed way of addressing income inequality is through a wealth tax. (Currently, our tax system only taxes income, not wealth, or money that people already have.)
Read at least five articles discussing the proposed benefits or potential problems with implementing a wealth tax in the United States. Describe your findings, discussing specific plans and at least five statistics cited in these articles that involve percentages. Describe the significance of each statistic as well as what picture these statistics paint as a whole. Pay special attention to what the base quantities of comparative statistics might be. Be sure to check the source of each statistic and cite what study it came from. Below are some articles to start with, but feel free to use others as well.
Finally, based on your findings, explain what (if any) modifications to our current tax system your group thinks is best and why.