1.4: Computing and Interpreting Percentages

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INTRODUCTION

In this section, you will learn about adding a tip to a meal bill and determining sale prices. Most students find this process straightforward when they need to calculate the total sale price given the original price and the rate to add. However, many students find it difficult to calculate the original price when they are given the total sale price and the rate added.

The formula you will use in this section is:

Total price = original price + original price * tax rate,

Or, alternatively: Total price = original price * (1 + tax rate)

You should use the distributive property to convince yourself that the two formulae are the same. The second formula will be much more useful when you are trying to determine the original price of an item, given the total price.

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

a percentage could represent a ratio which is made up of both a numerator and denominator.
a percentage is used to measure relative change in a quantity.
a percentage could represent a chance or probability.

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

compute and understand the difference between the absolute and relative change between two quantities (including rates).
correctly compare quantities using both absolute and relative change (including rates).
represent chances or probabilities by both percentages and ratios.

PROBLEM SITUATION 1: COMPUTING WITH PERCENTAGES

(1) The state of Illinois has a 6.25% sales tax.⁵

(a) Calculate the amount of tax on a $120 purchase. Round to the nearest cent.

(b) What is the total sale price of a $120 purchase, including tax? Round to the nearest cent.

(2) Illinois has a 6.25% sales tax, and textbooks are taxable in Illinois.⁶ The amount that a student paid at her school bookstore was $185. What was the price of the books and school supplies before tax? Round to the nearest cent.

(3) In 2022, Jacob worked as a fast food manager and:

had a taxable income of $30,000
was required to pay $3,398 in federal taxes.⁷

What is the federal tax rate as a percentage? Round to the nearest percent.

(4) According to Wikipedia, the information in the table below represents a breakdown of how the U.S. federal government spent money during the 2022 fiscal year:⁸ Compute the percentage of the total amount of money that the U.S. federal government spent in each category and complete the table above. Round to the nearest percentage point.

Category	Billions of Dollars Spent in Fiscal Year 2022	Percentage
Defense Department	751	(i)
Social Security	1,200	(ii)
Healthcare	1,339	(iii)
Net Interest	475	(iv)
Other Mandatory	1,583	(v)
Other Discretionary	910	(vi)

PROBLEM SITUATION 2: MEASURING CHANGE WITH PERCENTAGES

There are several different ways that one can measure the change between an old value and a new value. These values could be, for example, the weight of the same person at two different points in time, or the age of the previous president and the age of the current president.

The absolute change tells us by how much the two values differ. The relative change is the absolute change compared to the reference value (earlier quantity). Relative change is often given as a percentage.

You may recall the following two formulas, where “Old” is the starting value, and “New” is the ending value:

\[ Absolute\:Change = New - Old \nonumber\]

\[ Relative\:Change = \dfrac{New - Old}{Old} *100\% \nonumber\]

When working on Questions 5 and 6, think carefully about what change you are trying to measure. To measure a relative change, one must understand which value constitutes the “starting, reference value or old value” and which constitutes the “ending, final or new value.” The context and the surrounding language in a question help to clarify which is which.

(5) An article about the 2016 presidential election stated: “Trump lost the popular vote by about 2.1 percentage points. Meanwhile, Trump racked up 306 electoral voters—14 percent more than Clinton.”⁹ Answer the following questions about this statement.

Does “2.1 percentage points” refer to an absolute or a relative change?

Does “14 percent more” refer to an absolute or a relative change?

(6) The average annual snowfall for Boise, ID, is 19.8 inches. Moline, IL, receives an average of 33.3 inches. Fill in the blanks:

(a) Moline receives, on average, ______ more inches of snow than Boise.

(b) Moline receives, on average, ______ % more snow than Boise. Round to the nearest tenth of a percent.

(d) In Question 6(a), were you computing absolute or relative change?

(e) In Questions 6(b) and 6(c), were you computing absolute or relative change?

PROBLEM SITUATION 3: PERCENTAGES AS RATES

(7) According to the CDC, 11.5% of Americans smoked tobacco in 2021.¹⁰

(a) What is meant by this statement? What was the numerator? What was the denominator? What assumptions could you make about the population of Americans included in this statement?

(b) Discuss in your group how this percentage (11.5%) was probably computed. Record your group’s thinking below.

(8) The Insurance Institute for Highway Safety (IIHS) is an independent, nonprofit scientific and educational organization dedicated to reducing deaths, injuries and property damage from motor vehicle crashes. In 2021, the IIHS published statistics showing, in part, that almost three out of four motor vehicle crash deaths were males.¹¹

(a) Use the figures from the last sentence (“almost three out of four motor vehicle deaths”). What percentage of motor vehicle crash deaths in the U.S. were males? Round to the nearest percentage point.

(b) What percentage of motor vehicle crash deaths in the U.S. were female? Round to the nearest percentage point.

(c) Do your answers to Questions 8(a) and 8(b) support the statement, “In general, males are more likely to be killed in traffic accidents than females”? Explain your reasoning. Support your answer with some possible examples.

PROBLEM SITUATION 4: PERCENTAGES, PROPORTIONS, AND RISK

(9) A flyer that went to households in the mail, from a company called Life Line Screening,¹² included the following claim:

“Your risk of suffering a stroke is high—it affects 1 in 5 women, and 1 in 6 men.”

Answer the questions below by converting the risk of suffering a stroke from a proportion to a percentage.

(a) What is the likelihood of a woman suffering a stroke? Calculate the percentage.

(b) What is the likelihood of a man suffering a stroke? Calculate the percentage. Round to the nearest percentage point.

(c) There are many different ways to compare the differences in these two risks. Find two or three ways, and write a complete sentence for the ways you chose.

(10) Some research suggests that the risk of developing type 2 diabetes by age 70 is 11%. How could you express this with a meaningful rate or ratio?

PROBLEM SITUATION 5: MEASURING CHANGES IN PERCENTAGES

(11) The unemployment rate is the percentage of people in the labor force who are unemployed and actively seeking work. Going into the Covid pandemic in January 2020, the unemployment rate in the United States was 3.5%. In May of that year, unemployment was 13.2%.¹³ There are many ways that this change in unemployment rate could be described. Fill in the blanks in the following statements so they correctly describe this change.

(a) From January 2020 to May 2020 the unemployment rate increased by _______ percentage points. (Hint: You are computing an absolute change here.)

(b) From January 2020 to May 2020 the unemployment rate increased _______ percent. (Hint: You are computing a relative change here.) Round to the nearest percent.

(c) From January 2020 to May 2020 the unemployment rate ____________ (fill in with “nearly” or “more than”) _____________ (fill in with “tripled” or “quadrupled”).

(12) Question 7 provided information from the CDC about smoking habits in the U.S. The table below presents data from other countries.

Country	Percentage of Smokers
Greece	39.1
Germany	28.0
France	34.6
Sweden	8.0
Canada	17.5

(a) Write a statement comparing the smoking rates of people in Greece and Sweden, using a measure of absolute change.

(b) Write a statement comparing the smoking rates of Canada and Greece, using a measure of relative change.

(c) Describe what you learned by writing a statement comparing the smoking rates of Canada and Greece. Use phrases similar to “almost double,” “more than triple,” and “almost a fourth of.”