Calculating and Comparing Tax Rates

Stuart Boersma, Central Washington University

Based on material in "Case Studies for Quantitative Reasoning: A Casebook of Media Articles" by Bernard L. Madison, Stuart Boersma, Caren L. Diefenderfer, and Shannon W Dingman.

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This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

Summary

This example focuses on six letters to the editor. All six letters attempt to describe and compare the amount of taxes paid on two different incomes: $30,000 and $200,000. Tax rates are expressed in absolute dollars, tax per $1,000 of income, $1 of tax per income amount, and as percents of annual income. Students need to be able to organize the relevant information and convert each stated tax rate to a standard form to help make comparisons. Additionally, students need to be aware that letter writers may make their own mistakes!

Mathematics | College Introductory

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Learning Goals

Learning goals include:

Critical reading of source material and
Performing basic calculations with percents and ratios.

Context for Use

The quantitative skills required in this example are quite simple: basic percent calculations. However, since each letter writer has their own method of describing a tax rate, a bit of work is needed to compare the different representations to decide which values represent different tax rates. This example could appear early in any quantitative reasoning course.

Description and Teaching Materials

All six letters appeared in the Arkansas Democrat-Gazette between June 30 and July 15, 2003. In the first letter the author makes some statements concerning the tax rates of a person who makes either $30,000 or $200,000. The five other letter writers take issue with the calculations which appeared in the first letter, some who make their own mistakes! Unfortunately, due to copyright issues the letters can not be reproduced here in their entirety. Thus, the individual flair and tone of the letters do not come across. Phrases such as "Al Gore's funny math," " His heart is in the right place, but his math is not," " I couldn't figure out what he was talking about," "I wish I was one of those unlucky people making $200,000 a year," and "Who taught this man math?" are, unfortunately, not included with the selections. Here are the relevant excerpts:

Letter 1: A person making $30,000 a year who pays $5,000 in taxes pays $6 on $1,000 whereas a person making $200,000 a year who pays $53,000 in taxes pays $3.77 on $1,000.

Letter 2: The person making $30,000 a year pays $1 tax for every $6 of income. The other person pays $1 tax for every $3.7 of income.

Letter 3: The first person is paying $166.67 per $1,000 in taxes and the rich guy is paying $265 per $1,000.

Letter 4: A person making $30,000 who is single and using the standard deduction will pay taxes on $27,000 and should pay $3,754 in taxes. Under similar assumptions, the person making $200,000 will pay taxes on $198,560 and should have paid $56,748. That would be $284 per $1,000 of income.

Letter 5: Five thousand dollars of taxes on $30,000 of income is not $6 per $1,000, it is $166.67 per $1,000. That represents 16.7 percent of that person's income. Fifty-three thousand dollars of taxes on $200,000 is not $3.77 per $1,000, it is $265 per $1,000. That represents 26.5 percent of that person's income.

Letter 6: He says he pays $5,000 taxes on $30,000 income and calls this a $6-per-$1,000 rate. This is only .06 percent. At this rate he should have paid only $150. If one who makes $200,000 were only paying $3.77 per $1,000 in taxes, they should only have paid $754 not $53,000. In actuality, the person making $30,000 is paying 16.67 percent and the person making $200,000 is paying 26.5 percent.

Study Questions for Students:

Create an organized list for how the tax rates for $5,000 taxes on $30,000 income and $53,000 taxes on $200,000 income are stated in the six letters under consideration.
Which of the stated rates are correct and which are incorrect?
What is the mistake that the first letter writer probably made in computing the tax rates?
Which of the five letters responding to the first letter have errors, and what are those errors?
Which, if any, of the letters dispute the amounts of tax cited by the first letter writer: $5,000 on $30,000 and $53,000 on $200,000?
For each of the five responses to the first letter, write 2-3 sentences critiquing the letter as to its accuracy, tone, and effectiveness.

Teaching Notes and Tips

Sample Solutions:

Create an organized list for how the tax rates for $5,000 taxes on $30,000 income and $53,000 taxes on $200,000 income are stated in the six letters under consideration.

Letter Writer 1 states the rates as $6 per $1,000 and $3.77 per $1,000.
Letter Writer 2 states the rates as $1 per $6 and $1 per $3.7.
Letter Writer 3 states the rates as $166.67 per $1,000 and $265 per $1,000.
Letter Writer 4 re-computes the taxes and states the rates of his computed taxes ($3,754 on $30,000 and $56,748 on $200,000) as $125 per $1,000 and $284 per $1,000.
Letter Writer 5 states the rates as $166.67 per $1,000 and $265 per $1,000.
Letter Writer 5 also states the rates as 16.7 percent and 26.5 percent.
Letter Writer 6 states the rates as 16.67 percent and 26.5 percent.
Letter Writer 6 also restates Letter Writer 1's $6 per $1,000 as .06 percent.

Which of the stated rates are correct and which are incorrect?

Both of Letter Writer 1's rates are incorrect.
Both of Letter Writer 2's rates are near correct. The $3.7 is probably a misprint of$3.77. A more accurate rate is $1 per $3.77358.
Letter Writer 3's rates are correct.
Letter Writer 4's rates are correct for his computed taxes.
Letter Writer 5's rates are correct.
Letter Writer 6's rates are correct, but his restatement of Massery's $6 per $1000 as .06 percent is incorrect; it is 0.6 percent.

What is the mistake that the first letter writer probably made in computing the tax rates?

Which of the five letters responding to the first letter have errors, and what are those errors?

Which, if any, of the letters dispute the amounts of tax cited by the first letter writer: $5,000 on $30,000 and $53,000 on $200,000?
For each of the five responses to the first letter, write 2-3 sentences critiquing the letter as to its accuracy, tone, and effectiveness.

Assessment

Students should complete the reading assignment and prepare the list from item 1 before coming to class. The instructor can quickly check to see that students have come prepared with their lists. Small group discussions should focus on the accuracy of each of the letters (items 2, 3, 4, and 5). The instructor can answer "tax" questions as they arise and clarify any common misconceptions that appear to be forming. For homework, students should re-write responses, individually, for all six study questions to turn in for formal assessment.