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Quantway College with Corequisite (Carnegie Math Pathways)
4: Corequisite Module N
4.5: Corequisite- Percentages and Taxes, Ratios, Proportions and Dimensional Analysis

4.5: Corequisite- Percentages and Taxes, Ratios, Proportions and Dimensional Analysis

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SPECIFIC OBJECTIVES

By the end of this lesson, you should understand that

units can be used in dimensional analysis to set up calculations.
precision should be based on several factors, including the size of the numbers used and the precision of the original values. Rounding can produce large differences in results.

By the end of this lesson, you should be able to

solve a complex problem with multiple pieces of information and steps.
use dimensional analysis.
investigate how changing certain values affects the result of a calculation.

PROBLEM SITUATION 1: FICA TAXES

The United States government requires that businesses pay into two national insurance programs—Social Security and Medicare—which help senior citizens pay for their personal expenses (things that a person or people spend money on) and health care. Businesses take money out of their employees’ paychecks in order to pay the government. In other words, if you work for a business, your employer deducts Social Security and Medicare taxes from your paycheck. Also, the business pays a portion of the taxes for you. These taxes are called Federal Insurance Contributions Act (FICA) taxes.

People who own their own businesses are self-employed. They have to pay their own taxes. This
is called the self-employment tax. In this problem situation, you will calculate how much three self-employed individuals owe in self-employment tax. You will do this by using a tax worksheet called the Schedule SE. This is an Internal Revenue Service (IRS) tax form. The IRS is the part of the government that collects taxes. It has many different types of forms that help individuals figure out how much they owe in taxes.

(1) Below is a copy of a blank schedule SE.

Marianne Lopez has a part-time job as a math tutor. She offers tutoring services to students both in person and online. Last year, she earned $11,385 in revenue, or the amount of money she received from selling her service. Her expenses totaled $3,862. You are going to calculate how much self-employment tax Marianne owes. That is, you are going to determine what value should be put into the cell on line 5. Take a moment to do this on your own before sharing your calculations in your group.

Note: Marianne does not have a farm, and she does not receive social security retirement or disability benefits. She is also not a minister or a member of a religious order and does not have other income. You do not need to know which schedule or form Marianne’s net profit (the actual amount of money made after expenses) comes from. These cells have been zeroed out to avoid confusion. You also do not need to calculate line 13.

$Screenshot of Self-Employment tax form. At the top, it notes: If your only income subject to self-employment tax is church employee income, see instructions for how to report your income and the definition of church employee income. Part A: If you are a minister, member of a religious order, or Christian science practitioner and you filed Form 4361, but you had $400 or more other net earnings from self-employment, check here and continue with Part I. Skip lines 1a and 1b if you use the farm optional method in part II. See instructions. 1a: Net farm profit or (loss) from Schedule F, line 34, and farm partnerships, Schedule K-1 (Form 1065), box 14, code A. b: If you received social security retirement or disability benefits. enter the mount of conservation Reserve program payment included on Schedule F, line 4b, or listed on Schedule K-1 (Form 1065), box 20, code AH. Skip line 2 if you use the non farm optional method in Part II. See instructions. 2: Net profit or (loss) from Schedule C, line 31, and Schedule K (Form 1065), box 14, code A (other than farming). See instructions for other income to report or if you are a minister or member of a religious order. 3: Combine lines 1a a, 1b, and 2. 4a: If line 3 is more than zero, multiply line 3 by 92.35% (0.9235). Otherwise, enter amount from line 3. Note: If line 4a is less than $400 due to Conservation Reserve Program payments on line 1b, see instructions. b: If you elect one or both of the optional methods, enter the total of line 15 and 17 here. c: Combine lines 4a and 4b. If less than $400, stop. You don;t ow self-employment tax. Exception: if less than $400 and you had church employee income, enter 0 and continue. 5a: Enter your church employee income from Form W-2. See instructions. 5b: Multiply line 5a by 92.35% (0.9235). If less than $100, enter 0. 6: Add line 4c and 4b. 7: Max amount of combined wages and self-employment earnings subject to social security tax or the 6.2% portion of the 7.56% railroad retirement (tier 1) for 2021. 8a: Total social security wa$

(a) What value should be put into the cell on line 2?

(b) What value should be put into the cell on line 3?

(d) What value should be put into the cell on line 6?

(e) What value should be put into the cell on line 9?

(f) What value should be put into the cell on line 10? Round your answer to two decimal places.

(g) What value should be put into the cell on line 11? Round your answer to two decimal places.

(h) What value should be put into the cell on line 12?

(2) Leigh Olson started a small bakery last year. She has no income to report on line 1a or line 1b. She earned $1,050 in revenue selling cookies and cupcakes. Her expenses totaled $630. In your group, work through the questions below to figure out how much self-employment tax Leigh Olson owes.

Note: Leigh does not have a farm, and she does not receive social security retirement or disability benefits. She is also not a minister or a member of a religious order. You do not need to know which schedule or form Leigh’s net profit comes from. These cells have been zeroed out to avoid confusion. You also do not need to calculate line 13.

(a) What value should be put into the cell on line 2?

(b) What value should be put into the cell on line 3?

(d) How much tax does Leigh owe? Be careful here!

(3) In Question 1, you learned about Marianne Lopez. You used the form to calculate how much self-employment tax Marianne owes. Now, write the calculation you completed to answer Question 1 as a single expression. An example of a single expression is shown below.

Example: The following calculations,

19 - 4 = 15

5.75*15 = 86.25

0.3*86.25 = 25.875

Can be written as a single expression: 0.3(5.75)(19 - 4) = 25.875

(4) Look back at the single expression you wrote for Question 3. Imagine you have to explain the expression to Marianne to help her understand how you calculated the amount of self-employment tax that she owes. Answer these questions about the expression:

(a) What does the operation $11,385 − $3,862 represent for Marianne?

(b) What does the operation of multiplying by 0.9235 represent for Marianne?

Self-Employment and the Medicare Tax

The Affordable Care Act was passed by Congress and signed into law in 2010. The law reforms the health care system in the United States by protecting consumers from losing their health insurance, bringing down insurance costs, expanding access to health care services, strengthening Medicare with added benefits, and improving the quality of health care for Medicare patients. Medicare is a federal (government-run) health insurance program for seniors, people who are 65 and older, in the United States. All taxpayers pay into the Medicare program.

In order to raise revenue to support the expansion of care under the Affordable Care Act, the U.S. government created an additional Medicare tax for individuals with higher incomes. People who are self-employed use Part II of Form 8959 to determine how much additional Medicare tax they have to pay (if any). Only individuals and couples with incomes over certain amounts have to pay this tax. These amounts appear in line 9 of Form 8959.

$Screen capture showing Part II - Additional Medicare Tax on Self-Employment Income, indicating the following values: 8 = $215,500 9 = $200,000 10 = $0 11 = $200,000$

(5) Andie Henson is a single, self-employed psychiatrist. Her self-employment income from Schedule SE (Form 1040) Section A was $215,500. She did not earn wages from another job during this year. This information has already been entered in lines 8–11 of Form 8959. Use mental math to find the exact value of line 12 and add it to the form. Then, estimate the value of line 13 and record your estimate in the space below. Do not use a pencil and paper or a calculator. Briefly explain your estimation strategy.

PROBLEM SITUATION 2: PROPORTIONS IN ARTWORK

Many professionals such as graphic artists, architects, and engineers work with objects that are enlarged or shrunk. It is usually important that the objects have the same appearance despite the change in size. For example, a business logo on a billboard needs to look the same as a logo on a coffee mug.

In this collaboration, you will explore the mathematics behind these changes in size.

Imagine that this graphic is our original image. It has the following dimensions:

Width: 1.9 inches

Height: 1 inch

$Cloud with sun.$

Let’s say we make a change to the width only. This new graphic has the following dimensions:

Width: 2.45 inches

Height: 1 inch

$Cloud with sun.$

Now, let’s say we take the original image and we make changes to both the width and the height, but we do not maintain proportionality. This new graphic has the following dimensions:

Width: 2.2 inches

Height: 1.19 inches

$Cloud with sun.$

And lastly, let’s imagine we take the original image and make changes to both the width and the height, maintaining proportionality. This new graphic has the following dimensions:

Width: 2.55 inches

Height: 1.29 inches

$Cloud with sun.$

Notice that it can be difficult to spot just by looking at the new image if it is proportional to the original image. In the next question, you’re going to explore what can be done to check for proportionality.

(6) Suppose you were given three different kinds of dimensional changes done to the original graphic without actually seeing the new graphics. The dimensional changes are shown in the table below.

Original graphic dimensions:

Width: 1.9 inches

Height: 1 inch

Three Different Kinds of Changes
Change One		Change Two		Change Three
Width	Height	Width	Height	Width	Height
7.45 in	1 in	4 in	1.19 in	0.95 in	0.5 in

How could you tell which changes were proportional and which were not?

(7) You are a graphic artist hired to make a billboard for a college. The original logo is 2-1/4 inches (width) by 3-3/8 inches (length). You need to enlarge the logo so it has a length of six feet. How wide will the enlarged version be? Hint: Think back to the work about equal ratios in Question 1 and to the idea of using a variable to represent an unknown. What could we do if we use the variable x to stand in for the width of the enlarged version?

(8) In Question 7, you could have used the following proportion to represent the relationship between the original and enlarged objects:

\[\dfrac{2.25}{3.375} = \dfrac{4}{6} \nonumber\]

Could this proportion be written in other ways? Explain.

(9) Suppose you had set up the following proportion to solve the original problem in Question 7. What steps would you use to solve the equation?

\[\dfrac{3.375}{2.25} = \dfrac{6}{x} \nonumber\]

(10) Solve the equation. Round to the nearest tenth.

\[\dfrac{12.7}{x} = \dfrac{0.2}{3} \nonumber\]

(11) Solve the equation. Round to the nearest tenth.

\[\dfrac{8,500}{4,200} = \dfrac{x}{5} \nonumber\]

(12) Many small engines for saws, motorcycles, and utility tractors require a mixture of oil and gas. If an engine requires 20 ounces of oil for five gallons of gas, how much oil would be needed for eight gallons of gas?

PROBLEM SITUATION 3: ESTIMATING COSTS FOR CAR OR ACTIVITY

(13) If you drive a car or plan to get a car, complete (a) below. If you do not have a car, complete (b) below. Show your work.

(a) Estimate the cost per mile of driving your car based on what you actually pay for insurance and gas mileage. Don’t forget about the cost of maintenance and how often you drive.

(b) If you do not have a car, estimate the cost per year of some activity or item that you pay for at least twice a week on average. For example, buying a cup of coffee or energy drink, downloading music, going to a movie, paying a babysitter, riding a bus, etc.

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