Example 6.2.1.
| Audryn | August | |
|---|---|---|
| Initial Value | 68 | 25 |
| Percent Change | 18% | -15% |
| Change Factor | 1.18 | 0.85 |
Section 6.2 Percent Change
In Section 6.1, we saw:
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If Audryn left a $12.24 tip on a $68.00 bill, then she left an 18% tip. She paid $80.24 in total, or 118% of the original bill.
-
If August bought a shirt that originally cost $25 but was 15% off, then he paid $21.25 for the shirt. The price he paid was 85% of the original price.
We often describe increases and decreases in terms of percents.
-
The initial value is the original amount.
-
The percent change or relative change is the percent increase or decrease from the initial value. The percent change is negative if it’s a decrease.
-
The change factor is the new value as a percent of the original value, divided by 100.
The change factor is always
\begin{equation*}
\frac{100 + \text{Percent Change}}{100}\text{.}
\end{equation*}
Activity 6.2.1.
(a)
The city of Denver has a 9.15% sales tax. What is the change factor for the cost of items purchased in Denver?
(b)
A store is having a sale and all of its items are 25% off. What is the change factor for the price of items in this store?
Activity 6.2.2.
(a)
Multiply \(68 \times 1.18\text{.}\) How does your answer compare to the total Audryn paid for her dinner?
(b)
Multiply \(25 \times 0.85\text{.}\) How does your answer compare to what August paid for his shirt?
(c)
Mccoy bought a pair of earbuds in Denver for $79.99.
(i)
Find the amount they paid after sales tax.
(ii)
(iii)
What do you notice about your answers?
In Activity 6.2.2, you should have noticed that
\begin{equation*}
\text{New Value} = \text{Initial Value } \times \text{ Change Factor}
\end{equation*}
Activity 6.2.3.
In 2024, Jacob worked as a fast food manager and had a taxable income of $61,009. He paid 13.9% in federal income tax.
(a)
What are the initial value and percent change?
(b)
What is the change factor?
(c)
Use the change factor to find Jacob’s income after taxes.
Activity 6.2.4.
According to the US Census Bureau, the population of Denver County was 600,158 people in 2010. Denver County’s population increased 19.4% between 2010 and 2023.
(a)
What are the initial value and percent change?
(b)
What is the change factor?
(c)
Use the change factor to find Denver County’s population in 2023.
Activity 6.2.5.
According to a local news report from August 2024, the median rent for a two-bedroom apartment in Denver was $2,470, a 1.6% increase from August 2023.
(a)
What is the change factor for the median rent of a two-bedroom apartment in Denver from 2023 to 2024?
(b)
The news report included the median rent in 2024, but not the median rent in 2023! That is, it gave us the new value, instead of the initial value. Use the new value (given in the problem statement) and the change factor to find the initial value (the median rent in 2023).
Activity 6.2.6.
A store advertises an Apple Watch Series 10 as on sale for $309. The ad says that this is a 28% discount from its original price.
(a)
What is the change factor for the price of the Apple Watch?
(b)
The ad includes the sales price, but not the original price. Use the sales price and the change factor to find the original price.
Activity 6.2.7.
Celeste has $40 to pay for lunch, including tip. She always tips 21%, and wants to figure out the maximum amount that she can spend on her lunch in order to have enough to pay for both the lunch and the tip.
(a)
What is the change factor?
(b)
Is $40 the initial value or the new value?
(c)
Is the max amount that Celeste can spend on lunch the initial value or the new value?
(d)
Use the change factor and your answers to parts (b) and (c) to find the maximum amount that Celeste can spend on lunch.
Activity 6.2.8.
The table below gives the median household income in Colorado, according to the Federal Reserve Bank of St. Louis
| Year | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|
| Median Income ($) | 72,500 | 83,780 | 84,950 | 89,930 | 96,640 |
(a)
Find the change factor from each year to next. Round to three (3) decimal places. The change factor from 2019 to 2020 has been done for you, as an example.
| Years | Change Factor |
|---|---|
| 2019 to 2020 | \(\frac{83,780}{72,500} = 1.156\) |
| 2020 to 2021 | |
| 2021 to 2022 | |
| 2022 to 2023 |
(b)
Find the percent change from each year to the next in Colorado median household income. The percent change from 2019 to 2020 has been done for you, as an example.
| Years | Percent Change |
|---|---|
| 2019 to 2020 | \(1.156 \times 100 - 100 = 15.6%\) |
| 2020 to 2021 | |
| 2021 to 2022 | |
| 2022 to 2023 |
(c)
What do you notice? What do you wonder?
Activity 6.2.9.
Challenge: Kaitlyn is ordering pizza. The screenshot below shows her order in the app, just before placing the order. On the previous screen, Kaitlyn chose to leave a 20% tip, and entered a code for a free pizza.
Subtotal of $14.28, discount of $10.79, $0.30 in taxes, $2.86 tip, total of $6.65.
(a)
Kaitlyn’s friend Brixton tells her not to press the Place Order button, because there’s obviously a scam involved in calculating the tip. Use what you’ve learned about percent change to explain why Brixton is suspicious.
(b)
There isn’t actually a scam here, but:
-
The way the app has displayed the subtotal, discount, taxes, and tip is misleading and
-
The way the app calculated the tip might not align with what Kaitlyn was expecting.
Use what you’ve learned about percent change to figure out what the app has done, and explain it to Kaitlyn and Brixton.
Activity 6.2.10.
In April 2025, the Trump Administration raised tariffs on goods imported to the US. The Office of the US Trade Representative (USTR) initially published a table of new tariffs, but followed up with more details on the calculations. The White House calls the new tariffs “reciprocal tariffs” in the Executive Order authorizing the new rates, and the document explaining the calculations says that “Reciprocal tariffs are calculated as the tariff rate necessary to balance bilateral trade deficits between the U.S. and each of our trading partners.”
4
www.whitehouse.gov/wp-content/uploads/2025/04/Annex-I.pdf5
ustr.gov/sites/default/files/files/Issue_Areas/Presidential%20Tariff%20Action/Reciprocal%20Tariff%20Calculations.pdf6
www.whitehouse.gov/presidential-actions/2025/04/regulating-imports-with-a-reciprocal-tariff-to-rectify-trade-practices-that-contribute-to-large-and-persistent-annual-united-states-goods-trade-deficits/According to the USTR document explaining the calculation, the formula used to calculate the new tariff rates is
\begin{equation*}
\Delta\tau_i = \frac{x_i - m_i}{\epsilon \times \phi \times m_i} \text{.}
\end{equation*}
This looks complicated! The document tells us that the variables here are
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\(\Delta\tau_i = \text{Change in tariff rate (%) on Country } i\) (this is what we’re calculating)
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\(\displaystyle x_i = \text{Total exports from US to Country } i\)
-
\(\displaystyle m_i = \text{Total imports from US to Country } i\)
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\(\displaystyle \epsilon = \text{Elasticity of imports with respect to import prices}\)
-
\(\displaystyle \phi = \text{Passthrough from tariffs to import prices}\)
Although \(x_i\) and \(m_i\) are both numbers that we could look up, \(\epsilon\) and \(\phi\) are numbers whose values aren’t certain. Depending on the assumptions you make, you could reasonably get differnt values for \(\epsilon\) and \(\phi\text{.}\) The document tells us that the Trump Administration assumed that \(\epsilon = -4\) and \(\phi = 0.25\text{.}\) Since \(4 \times 0.25 = -1\text{,}\) all of this information lets us rewrite the very complicated looking formula as
\begin{equation*}
\text{Change in Tariff Rate} = \frac{\text{Exports} - \text{Imports}}{-\text{Imports}}\text{.}
\end{equation*}
(a)
Use the formula to calculate the change in tariff rate for each of the following countries:
(i)
China. The US exports $143.5 billion in goods to China, and imports $438.9 billion in goods.
(ii)
European Union. The US exports $370.2 billion in goods to the EU, and imports $605.8 billion in goods.
(b)
The rewritten formula
\begin{equation*}
\text{Change in Tariff Rate} = \frac{\text{Exports} - \text{Imports}}{-\text{Imports}}
\end{equation*}
looks similar to the percent change formula that we developed in this section!
(i)
Explain what impact using “-Imports” instead of “Imports” in the denominator has on the output of the formula.
(ii)
Reports about the new tariff rates generally gave one of two explanations for the reasoning behind the formula:
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It calculates a fair tariff rate that takes into account the tariffs a country charges on imported US goods, plus other factors that make it hard for the US to export to that country.
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It calculates the percent difference between the amount of goods imported to a country from the US and the amount of goods exported from that country to the US.
Which of these explanations matches the formula more closely? Explain your answer.
(c)
The Trump Administration did two more steps after using the formula (which they claim gives the fair tariff rate):
-
Divide the answer by 2, to get the discounted tariff rate that the US will actually use in order to be nice.
-
If your answer is now less than 10%, ignore it and just use 10% instead.
Use these last two steps to find the actual new tariff rates for each of the following countries/regions:
(i)
China
(ii)
European Union
(iii)
Venezuela. The US exports $4.2 billion in goods to the UK, and imports $6.0 billion in goods.
(iv)
United Kingdom. The US exports $79.9 billion in goods to the UK, and imports $68.1 billion in goods.
(d)
Use what you’ve learned from doing these calculations, and about percent differences, to decide:
(i)
How could the Trump Administration improve their communication about the way the new tariffs were calculated? Explain your answer.
(ii)
Are the new tariffs fair? Explain your answer.
