# 2.4: Applications of Percents

- Page ID
- 130928

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We've now learned about several ways to describe numbers and their relationships: ratios, rates, proportions, and percents. In this section, we'll explore more applications of these concepts to real-life situations.

- Accurately compute marked down prices using percents
- Accurately compute and interpret the percent change between two quantities

## Computing Markdowns

In this section, we will learn how to compute prices that have been *marked down* by a percent of the original prices. A markdown is a form of discount in which a percent of the original price is subtracted. This is a frequent way in which sale prices in stores are expressed — as a "percent off" the original price. We'll use one main formula:

\[\text{Marked Down Price} = \text{Original Price} - (\text{Markdown Percent} \times \text{Original Price})\]

In the equation above, the Markdown Percent must be expressed as a decimal! That's because the original price is the "whole" and the markdown percent is the "percent," so we are finding the "part" that represents the portion of the original price being subtracted from the original price. In other words, the "part" is the amount a consumer would save as compared to the original price. An example should make this clear:

The original price of a pair of pants is \(\$24.99\). There is a \(30\%\) off sale at the store where the pants are being sold. If you buy these pants, how much will you pay for them? How much do you save as compared to the original price? Round to the nearest cent.

###### Solution

This is a straightforward use of the markdown equation, shown above. The original price is $24.99, and the mark down percent is 30%. As a decimal, this is 0.30, because we move the decimal point two places to the left. Therefore, our mark down equation gives us \[\text{Marked Down Price} = 24.99 - (0.30 \times 24.99)\]

Now, we need to be careful with the order of these operations. The multiplication inside the parentheses happens first, so we can compute \(0.30 \times \$24.99 = \$7.497\) Therefore, we have \[\text{Marked Down Price} = 24.99 - 7.497 = 17.493 \] The instruction is to round to the nearest cent. Recall that cents consist of two places to the right of the decimal point. Therefore, when we round to the cents place, we get that the marked down price is \(\$17.49\), which is what we will pay for these pants during the \(30\%\) off sale. Since \(\$24.99-\$17.49 = \$7.50\), we are saving \(\$7.50\) as compared to the original price of the pants.

This is the essence of a mark down — it takes a specific percentage of the price and subtracts it from the original price to give the discounted price.

You may have noticed a slightly more efficient way to accomplish this. Instead of using the original markdown equation as it was written, we could reason as follows: If the pants are marked down \(30\%\), that means they cost \(70\%\) of their original price, since \(100\% - 30\% = 70\%\). Thus, we could simply compute

\[0.70 \times 24.99 = 17.493\]

and after rounding we get that the sale price of the pants is \(\$17.49\), matching the answer we have in the solution above. As usual, either way of answering the question is fine — you should pick the method that makes the most sense to you. However, this second way of thinking about the problem, in which we subtracted the markdown from \(100\%\), will be helpful in the next example, so it is worth making sure you understand both.

We can ask another type of question with markdowns: what if we want to find the original price given the marked down price? The following example illustrates how to do this.

A coat was marked down by \(25\%\). Its cost after the markdown is \(\$41.24\). What was the original price of the coat? Round to the nearest cent.

###### Solution

This is a bit trickier to figure out — our formula above does not easily lend itself to solving this problem. Instead of using the markdown formula, we'll instead use what we know about percents.

The key intuition for this problem is: if the coat was marked down by \(25\%\), its new cost is \(75\%\) of the original cost, since \(75\% = 100\% - 25\%\). Since the marked down price is calculated by subtracting a percent of the original cost, the original cost can be viewed as \(100\%\), and the cost after a markdown of \(n\%\) is \((100-n)\%\) of the original cost.

In our case, this means that the new price of the coat, \(\$41.24\), is \(75\%\) of the original cost of the coat. Therefore, we can use the Basic Percent Equation with \(\text{percent } =0.75\) and \(\text{part } = \$41.24\). We will solve for "whole" in this equation, and that answer will be the original price of the coat.

Therefore we have: \[.75 \times \text{ whole } = 41.24\] Now we'll use Division undoes Multiplication to find the whole: \[ \text{ whole } = \frac{41.24}{.75} = 41.24 \div .75 = 54.99\] where we've already rounded to the nearest cent. Therefore, the original cost of the coat was \(\$54.99\).

Next time you're at the store, see if you can find mark downs phrased this way. You can test your understanding by computing your savings!

## Percent Change

It's common to see phrases like the "the population of North Plains, Oregon increased by \(58\%\) from \(2016\) to \(2021\)." In this case, there are percents being used, but it's not exactly stating that one number is a certain percent *of *another number. Instead, it is describing a percent by which one quantity changed with respect to its original value. While the previous formulas involving percents can be used to understand problems like this, it's sometimes easier to reframe them slightly into the following formula, specifically designed to deal with situations where a percent change in being described.

Given two quantities \(Q_1\) and \(Q_2\), the **percent change **between the quantities is \[\text{percent change } = \frac{Q_2 - Q_1}{Q_1} \times 100\]

Note that in this definition, we are multiplying by \(100\), so there is no need to convert to a percent afterwards. The answer will automatically be a percent.

Typically, when using this formula, the two quantities involved are ordered. \(Q_1\) represents the "first value," and \(Q_2\) represents the "second value." As usual, it will be necessary to read carefully and think critically to determine what the values of \(Q_1\) and \(Q_2\) are in context. Here is an example to see how this works:

According to climate.gov, between the years \(1990\) and \(2000\), there were \(52\) weather-related natural disasters in the United States that cost more than \(1\) billion dollars to mitigate. Between \(2010\) and \(2020\), there were \(119\) weather-related natural disasters in the United States that cost more than \(1\) billion dollars to mitigate. Comparing these two decades, what is the percent change in weather-related natural disasters from the first decade to the second? Round to the nearest tenth of a percent.

###### Solution

There are a lot of numbers in this question! Let's focus on the important ones. We see that during the earlier decade, from \(1990\) to \(2000\), we have \(52\) disasters. That is our first quantity, since it occurs at an earlier time. That means \(Q_1 = 52\). Likewise, in the second decade from \(2010\) to \(2020\), we have \(119\) disasters, so \(Q_2 = 119\). The years are simply telling us what happens first and second; they do not factor into the equation at all, and neither does the \(1\) billion dollars — that is just part of the description of the facts we are looking at.

Using \(Q_1 = 52\) and \(Q_2 = 119\), we have \[\text{percent change } = \frac{Q_2 - Q_1}{Q_1} \times 100 = \frac{119 -52}{52} \times 100\]

Then \[\text{percent change } = \frac{67}{52} \times 100\]

When working with division and multiplication together, we work left to right, first computing the fraction as a decimal, and then multiplying by \(100\): \[\text{percent change } \approx 1.288461 \times 100 = 128.8461\]

The instructions were to round to the nearest tenth of a percent, so we will state the answer as \(128.5\%\). This means that the number of natural disasters costing more than \(1\) billion dollars rose \(128.8\%\) between the decade from \(1990-2000\) to the decade from \(2010-2020\).

It is important to note what happens if the two quantities, \(Q_1\) and \(Q_2\), represent a decrease, meaning that \(Q_2 \) is smaller than \(Q_1\). In this case, the percent change formula will output a negative number. This simply means that there is a decrease by the given percent. This does not indicate a wrong answer! It is important to make sure that whatever \(Q_1\) is, it's in the denominator of the fraction. That's why it's always important to label your variables and rewrite the original equation before proceeding.

## Exercises

In this section, when there is a percent as an answer, please round to the nearest percent. If your answer is a number of people, round to the nearest whole. And if your answer is an amount of money, round to the nearest cent.

- A dresser has an original price of \(\$120\), but it is marked down \(25\%\). If you buy the dresser with this markdown, how much will you pay for the dresser?
- A TV originally cost \(\$350\). It was marked down \(15\%\) when you went to buy it. You also have a coupon that takes \(10\%\) off your total purchase. When you buy it, how much money do you save off of the original price?
- A pair of shoes currently has a price of \(\$31.44\). You see a sticker that says the current price is a \(15\%\) mark down off of the original price. What was the original price of the shoes?
- The price of Google stock on April 16th, 2020 was \(\$1266.78\). The price of Google stock on April 17th, 2020 was \(\$1257.43\). What is the percent change of the Google stock price between these two days? Round to the nearest hundredth of a percent.
- This question has two parts. Remember to check to that your answers make sense to you.
- The population of Oregon was \(12,093\) in 1850 and \(90,923\) in 1870. What is the percentage change in the population between those two years? Round to the nearest percent.
- The population of Valsetz, Oregon was \(300\) in 1983 and \(5\) in 1984 (it is now a ghost town — this is what the dining hall at WOU is named for). What is the percentage change in population between those two years? Round to the nearest percent.