Opening the book…
Figure “The federal budget describes how money is spent (Total Outlays) and how money is earned (Total Revenues)”.. (credit: [1])- Convert between percent, decimal, and fractional values.
- Calculate the total, percent, or part.
- Solve application problems involving percent.
- Calculate the percent increase and percent decrease.
- Compute discounts, markups, and sales tax.
- Solve application problems involving discounts, markups, and sales tax.
In \(2024,\) the U.S. federal government budgeted $3.8 billion for the National Park Service(opens in new window), which appears to be a very large number (and is!) and a large portion of the total federal budget. However, the total outlays from the U.S. federal government in 2024 were $6.8 trillion(opens in new window). So, the amount budgeted for the National Park Service was less than one-tenth of \(1\) percent, or \(\frac{1}{10}\%,\) of the total outlays. This percentage describes a specific number. Understanding that ratio puts the \($3.8\) billion budgeted to the National Park Service in perspective.
This chapter \(4\) focuses on percent as a primary tool for understanding money management. The interest paid on debt, the interest earned through investments, and even taxes are entirely determined using percentages. This section introduces the basics of working with this invaluable tool.
The word percent comes from the Latin phrase per centum, which means "by the hundred." So any percent is a number divided by \(100\). Changing a percent to a fraction means writing the percent as a fraction. To write \(n\%\) in its fractional form is to write the percent as the fraction \(\frac{n}{100}\).
Convert Between Percent, Decimal, and Fractional Values
The fraction form of \(n\%\) is found by dividing \(\frac{n}{100}\) and calculating the decimal value of the fraction.
Or you move the decimal to the left and remove the percent sign.
To convert the number \(x\) from decimal form (or fraction form) to percent, multiply \(x\) by \(100\) and place a percent sign, \(\%,\) after the number.
Or to convert from decimal form to percent form, you can simply move the decimal two places to the right without performing the multiplication.
A percent need not be an integer and does not have to be less than \(100.\)
Example \(\PageIndex{1}\): Rewriting a Percent as a Fraction or Decimal Form
Rewrite the following as fractions and decimal form:
- \(8\%\)
- \(38.7\%\)
- \(\frac{1}{4}\%\)
- \(213\%\)
- Answer
-
- Using the definition, \(n=8.\) Decimal form = \(0.08\) and fractional form \[\frac{8}{100}=\frac{2}{25}\nonumber\]
- Using the definition, \(n=38.7.\) Decimal form = \(0.387\) and fractional form \[\frac{387}{100}\nonumber\]
- Using the definition, \(n=\frac{1}{4}=0.25.\) Decimal form = \(0.0025\) and fractional form \[\frac{0.25}{100}=\frac{\frac{1}{4}}{100}=\frac{1}{400}\nonumber\]
- Using the definition, \(n=213.\) Decimal form = \(2.13\) and fractional form \[\frac{213}{100}\nonumber\]
When any calculation with a percent is to be performed, the form of the percent must be changed to its fractional form or its decimal form. We can convert a percent to a decimal by dividing the percent by \(100\) and representing the result as a decimal.
Example \(\PageIndex{2}\): Converting the Decimal Form or Fraction to Percent
Convert each of the following to a percent:
- \(0.34\)
- \(4.15\)
- \(0.0391\)
- \(\frac{3}{8}\)
- Answer
-
- Using the formula, \(x=0.34,\) we calculate, \(0.34\times100\), which gives us \(34\%.\)
- Using the formula, \(x\) = \(4.15,\) we calculate, \(4.15\times100\), which gives us \(415\%.\)
- Using the formula, \(x\) = \(0.0391,\) we calculate, \(0.0391\times100\), which gives us \(3.91\%.\)
- Using the formula, \(x\) = \(\frac{3}{8},\) we calculate, (\(\frac{3}{8}\times100\%\)), which gives us \(37.5\%.\)
Calculate the Total (Whole), Percent, or Part
The word “of” indicates multiplication using fractions, as in “one-fourth of \(56.\)” To find “one-fourth of \(56\)” we would multiply \(56\) by one-fourth. We can think of percents as fractions with a specific denominator: \(100.\)
So, to calculate “\(25\%\) of \(52,\)” we multiply \(52\) by \(25\%.\) But first, we need to convert the percent to decimal form, \(0.25.\) Using the decimal form of \(25\%,\) we have \(0.25\) × \(52,\) which equals \(13.\)
In this problem, \(52\) is the total or base, \(25\) is the percentage, and \(13\) is the percentage of \(52\) or the part (amount) of \(52.\)
If we have a part that is some percent of a whole (Total), then
\begin{align*}
\text{percent}& = \dfrac{\text{part}}{\text{total}}\\
\text{part} &= \text{percent} \cdot \text{total}\\
\text{total} &= \frac{\text{part}}{\text{percent}}\\
\end{align*}
We can find any missing component if the other two are known by using the above formula.
To perform the calculations, we convert the percent to a decimal. Recall, to rewrite the percent as a decimal, we move the decimal over two places to the left, or, equivalently, divide the percent number by \(100\).
- In all calculations, the percent is expressed as a decimal.
- In Calculation, the word "of" implies multiplication.
- Any number after the word "of" is total or whole.
Knowing any two of the values in our formula allows us to calculate the third value. In the following example, we know the total and the percent, and are asked to find the percentage of the total.
Example \(\PageIndex{3}\): Finding the Percent of a Total
Determine \(156\%\) of \(720.\)
- Answer
-
The total is \(720,\) and the percent is \(156\%.\) The decimal form of \(156\%\) is \(1.56.\) To find the part, or percent of the total, substitute those values into the formula and calculate.
\begin{align*}
\text { part } & =\text { percent } \times \text { total } \\
& =1.56 \times 720 \\
& =1,123.2
\end{align*}From this, we say that \(156 \%\) of \(720\) is \(1,123.20.\)
In the previous example, we knew the total and the percent and found the part using our formula. We may instead know the percent and the part, but not the total. We can use our formula again to solve for the total.
Example \(\PageIndex{4}\): Finding the Total from the Percent and the Part
What is the total if \(10\%\) of the total is \(4,000?\)
- Answer
-
The percent is \(10,\) which in decimal form is \(0.10.\) We were given that \(10 \%\) of the total is \(4,000,\) so the part is \(4,000.\) Substituting into the formula, we have
\begin{align*}
\text { part } & =\text { percent } \times \text { total } \\
4,000 & =0.10 \times \text { total }
\end{align*}To find the total, we solve the equation for the total.
\begin{align*}
4,000 & =0.10 \times \text { total } \\
\frac{4,000}{0.10} & =\frac{\cancel{0.10} \times \text { total }}{\cancel{0.10}} \\
40,000 & =\text { total }\\
&=40,000\\
\end{align*}From this we see that \(40,000\) is the total, or that \(10\%\) of \(40,000\) is \(4,000.\)
Or you can use the formula directly
\begin{align*}
\text{total} &= \frac{\text{part}}{\text{percent}}\\
&=\frac{4,000}{0.10} \\
&=40,000
\end{align*}
Similarly, the percent can be found if the total and the percent of the total (the part) are known. This will result in the decimal form of the percent, so it must be converted to percent form.
Example \(\PageIndex{5}\): Finding the Percent from the Total and the Part
What percent of \(500\) is \(175?\)
- Answer
-
The total is \(500,\) the percent of the total is \(175.\) Substituting into the formula, we have
\begin{align*}
\text { part } & =\text { percent } \times \text { total } \\
175 & =\text { percent } \times 500
\end{align*}
To find the percent, we solve the equation for the percent.
\begin{align*}
175 & =\text { percent } \times 500 \\
\frac{175}{500} & =\frac{\text { percent } \times \cancel{500}}{\cancel{500}} \\
0.35 & =\text { percent }
\end{align*}We see the percent in decimal form is \(0.35.\) Converting from the decimal form yields \(35\%.\) We say that \(175\) is \(35 \%\) of \(500.\)
Or we can use a direct formula:
\begin{align*}
\text{percent} &= \frac{\text{part}}{\text{total}}\\
&=\frac{175}{500} \\
&=0.35 \\
&=35\%
\end{align*}
Solve Application Problems Involving Percent
Percentages are frequently used in finance, research, science experiments, and casual conversations. Understanding these values helps when consuming media or discussing finances, for instance. Effectively working with and interpreting numbers and percentages will help you become an informed consumer of this information.
In most cases, working through what is presented requires you to identify that you are indeed working with a question of percent, which two of the three values that are related through percent are known, and which of the three values you need to find.
Example \(\PageIndex{6}\): Retention Rate at College
Justine applied to a medium-sized university outside her hometown and found out that the retention rate (percent of students who return for their sophomore year) for the \(2021\) academic year at the university was \(84\%.\) During a visit to the registrar’s office, she found out that \(1,350\) people had enrolled in the academic year \(2021.\) How many students from the academic year \(2021\) are returning for the \(2022\) academic year?
- Answer
-
The percentage of students who will return for the \(2022\) academic year (the retention rate) is \(84\%.\) The total number of students enrolled in the \(2021\) academic year was \(1,350.\) This means the percent is known, and the total is known. From this, we can determine the number of students who will return (percent of the total) for the \(2022\) academic year using the formula.
\begin{align*}
\text{part} & =\text { percent } \times\text{total} \\
\text{part} & =0.84 \times1,350\\
\text{part} & =1,134\\
\end{align*}
So, \(1,134\) students will return for the \(2022\) academic year.
Example \(\PageIndex{7}\): Percent of Chemistry Majors
Cameron enrolls in a calculus class. In this class of \(45\) students, there are \(18\) chemistry majors. What percent of the class are chemistry majors?
- Answer
-
In this situation, the percentage is to be determined. We know the total number of students is \(45,\) and the part of the students who are chemistry majors is \(18.\) Using that information and the formula.
\begin{align*}
\text{percent}& = \dfrac{\text{part}}{\text{total}}\\
&=\frac{18}{45}\\
&=0.4\\
&=40\%
\end{align*}Converting the \(0.4\) from decimal form, we find that \(40\%\) of the students in the calculus class are chemistry majors.
Example \(\PageIndex{8}\): Total Sales and Commission
Mariel makes a \(20\%\) commission on every sale she makes. One week, her commission check is for \(\$153.00\). What were her total sales that week?
- Answer
-
In this problem, Mariel's total sales are to be determined. We know the percent she earns is \(20\%\). We also know that her sales commission was \(\$ 153.00 \), which is a part of the total. Using this information and the formula
\[
\begin{aligned}
\text{total}& = \dfrac{\text{part}}{\text{percent}}\\
&=\frac{153}{0.2}\\
&=765
\end{aligned}
\nonumber \]Mariel’s total sales were $\(765.00.\)
Who Knew?: LED Lightbulbs
According to the U.S. government's energy website, LED lightbulbs use at least \(75\%\) less energy than incandescent bulbs. They also last up to \(25\) times as long as an incandescent bulb. If lighting accounts for a significant percentage of your electricity use, replacing all incandescent bulbs with LED bulbs can significantly reduce your electricity bill.
Percent increase and percent decrease are ways to express a change in a quantity as a percentage of its original value. To calculate the percent decrease, you use the following formula:
\[\text{Change in Value} =\text {New Value - Old Value}\nonumber \]
The value of the Increase (Decrease) is also called the absolute change. If the value decreases, the absolute change will be negative.
\[\text{Percent Increase (Decrease)} = \frac{\text {Change in Value}}{\text{Old Value}}× 100\%\nonumber\]
The percent Increase (or Decrease) is also called a relative change.
Example \(\PageIndex{9}\): Percent Increase or Decrease
- If the price of a shirt increases from \($50\) to $\(70 \), what is the percent increase?
- If the price of a shirt decreases from \($80\) to $\(60 \), what is the percent decrease?
- Answer
-
Old value = $\(50\) AND New value = $\(70\)
\[\begin{aligned}
\text{Change in Value} &=$70 - $50= $20\\
\text{Percent Increase} &= \frac{\text {Change in Value}}{\text{Old Value}}× 100\%\\
&=\frac{\$20}{\$50}× 100\%= 40\%
\end{aligned}\] - Answer
-
Old value = \($80\) AND New value = \($60\)
\[\begin{aligned}
\text{Change in Value} &=$60- $80= -$20\\
\text{Percent Decrease} &= \frac{\text {Change in Value}}{\text{Old Value}}× 100\% \\
&=\frac{-\$20}{\$80}× 100\%= -25\%
\end{aligned}\]So, the percent decrease is \(25\%.\) Negative signs indicate a decrease.
Many people first encounter percentages during a retail transaction such as a percent discount (SALE! \(25\%\) off!!), or through sales tax ("Wait, I thought this was \($1.99?\)"), a report that something has increased by some percentage of the previous value (NOW! \(20%\) more!!). These are examples of percent decreases and percent increases. In this section, we discuss the decrease, increase, and then the case of sales tax.
Calculating Discounts, Mark Up, and Sales Tax
Retailers frequently hold sales to help move merchandise. The sale price is almost always expressed as some amount off the original price. These are discounts, a reduction in the price of something. The price after the discount is sometimes referred to as the reduced price or the sale price.
When a reduction is a percent discount, it is an application of percent, which was introduced in Understanding Percent. The formula used was "." In the formula, it would be helpful to remember that the percent is expressed as a decimal. For \(25\%\) discount, use \(0.25\) as percent discount.In a discount application, the discount plays the role of the part, the percent discount is the percentage, and the original price plays the role of the total.
If the final value (or amount) is \(p\%\) more than the initial
\[\begin{align*}
\text{Final} &= (100+p)\% \times \text{Initial}\\
\text{Initial} &= \dfrac{\text{final}}{(100+p)\%}\\
\end{align*}\]
If the final value (or amount) is \(p\%\) less than the initial
\[\begin{align*}
\text{Final} &= (100-p)\% \times \text{Initial}\\
\text{Initial} &= \dfrac{\text{final}}{(100-p)\%}\\
\end{align*}\]
Note: \(p\) in the above formula could be the sales tax rate, markup percent, or discount percent.
If we have a part that is some percent of a whole (Total), then
\[\begin{align*}
\text{percent}& = \dfrac{\text{part}}{\text{total}}\\
\text{part} &= \text{percent} \cdot \text{total}\\
\text{total} &= \frac{\text{part}}{\text{percent}}\\
\end{align*}\]
If we solve the sales tax and markup problem, the final price is always less than the initial price.
Note: When you solve a percent problem, you always change percent to decimal either by dividing by \(100\) or by moving the decimal two places to the left.
Example \(\PageIndex{10}\): Calculating Discount for a Percent Discount
Calculate the discount for the given price and discount percentage. Then calculate the sale price after calculating the discount.
original price = \($75.80;\) percent discount is \(25\%.\)
- Answer
-
Here, the discount is part of the original price (the whole or total).
\[
\begin{aligned}
\text{discount (part)} &=\text{percent }\times\text {total}\\
&=\text{0.25}\times75.80\\
&=18.95\\
\end{aligned}
\nonumber \]The sale price of the item is then
\[ \text{Sale price (final price)} =$75.80-$18.95=$56.85\nonumber\]
The sale price is \($56.85.\)
Or you can find the sale price (final price) directly using the following formula. Note that the Final price is \(25\% \) less than the original price.
\[\begin{align*}
\text{Final} &= (100-p)\% \times \text{Initial}\\
&= (100-25)\%\times 75.80\\
&=75\%\times 75.80\\
&= 0.75\times 75.80\\
&=56.85
\end{align*}\]
Example \(\PageIndex{11}\): Calculating the Percent Discount from the Original and Sale Prices
Determine the percent discount based on the given original and sale prices.
original price = \($1,200.00;\) sale price = \($900.00\)
- Answer
-
Find the discount, which is part of the \($1,200.00 \).
To find the discount percent, substitute the part \($300.00\) and the original price (total)\($1,200.00 \) into the formula
\[
\begin{aligned}
\text{percent} &=\frac{\text{part}}{\text{total}}\\
&=\frac{300}{1200} \\
&=0.25=25\% \\
\end{aligned}
\nonumber \]\(25\%\) discount of \($1,200\) is \($300\).
Example \(\PageIndex{12}\): Calculating the Original Price from the Percent Discount and Sale Price
Determine the original price based on the percent discount and sale price.
percent discount \(10\%;\) sale price = \($450.00\)
- Answer
-
Using the percent discount and the final price (sale price), we can find the original price (initial price) with the formula
\[
\begin{aligned}
\text {Initial (origianl price)} &=\frac{\text {Final (sale price) }}{ \text {(100-p)\%}}\\
&=\frac{450}{(100- 10)\%}\\
&=\frac{450}{ 90\%}\\
&=\frac{450}{0.90}\\
&= 500\\
\end{aligned}
\nonumber \]Original Price is \($500\). And \(10\%\) discount of it is \($50.00\) and you pay \($450\).
There are cases where retailers allow multiple discounts to be applied. However, it is rare that the discount percentages are added together. For example, if you have a \(15\%\) coupon and qualify for a \(20\%\) price reduction, the retailer typically does not add those two percentages together to determine the new price. The retailer instead applies one discount and then applies the second discount to the price obtained after the first discount was deducted.
Research the original prices of two different laptops offered by one retail outlet. Assume you will receive a student discount of \(12\%\) and your outlet of choice is having a \(15\%\) off sale on all laptops.
For each laptop:
\(1.\) List the original price and calculate the price after applying the student discount \(12\%\) only.
\(2.\) Then find the price after applying the sale discount (\(15\%\) off) to the price found in Step \(1.\)
\(3.\) Determine the total saved on the laptop and what percent discount the total savings represent.
\(4.\) Now, apply the discounts in reverse order (first the sale discount, then the student discount).
\(5.\) Note anything interesting about your findings.
Calculate Markups
When retailers purchase goods to sell, they pay a certain price, called the cost. The retailer then charges more than that amount for the goods. This increase is called the markup. This selling price, or retail price, is what the retailer charges the consumer to cover their own costs and make a profit. Markup, then, is very similar to discount, except we add the markup rather than subtract the discount.
Markup is also similar to a discount, but instead of subtracting the percentage, we add it. In markup, the final price is greater than the initial price.
The selling price is the retail price, and the cost is the original price before the markup is added. Sometimes it is also called wholesale price. The markup amount is the part that is added to the cost. When the markup is added to the cost, the result is the retail (selling) price.
Example \(\PageIndex{13}\): Determining the Retail Price Based on the Cost and the Percent Markup
For the given cost and markup percentage, calculate the retail price.
cost = \($62.00;\) percent markup is \(15\%.\)
- Answer
-
Here, markup is part of the cost (whole or total).
\[
\begin{aligned}
\text{Mark up (part)} &=\text{percent }\times\text {total}\\
&=\text{0.15}\times62\\
&=9.3\\
\end{aligned}
\nonumber \]The sale price of the item is then
\[ \text{Sale price (final price)} =$62+$9.30=$71.30\nonumber\]
So retail price (after markup added) is \($71.30\)
Or you can find the retail price (final price) directly using the following formula. Note that the final price is \(15\% \)more than the Initial price.
\[
\begin{aligned}
\text{Final (retail)} &=\text{Initial}\times (100+p)\%\\
&= 62\times (100+15)\%\\
&= 62\times (1.15)\\
&=71.30\\
\end{aligned}
\nonumber \]So retail price of item is \( $71.30.\)
Example \(\PageIndex{14}\): Calculating the Percent Markup from the Cost and Retail Price
Determine the percent markup based on the given cost and retail price. Round percentages to two decimal places.
cost = \($5.20;\) retail price = \($9.90\)
- Answer
-
Using the cost (total) and the retail price, we can find the markup is \($9.90-$5.20\), which is \($4.70 \), which is part.
After substituting the part and whole into the formula,\[
\begin{aligned}
\text{Percent}&=\frac{\text{part}} {\text{total (whole)}}\\
&=\frac{4.70}{5.20}\\
&=0.9038 \\
&=90.38\%\\
\end{aligned}
\nonumber \]
Converting to percent form, the percent markup is \(90.38\%.\)
So the percent markup is \(90.38\%.\)
Example \(\PageIndex{15}\): Calculating the Cost from the Percent Markup and Retail Price
Determine the cost based on the percent markup and retail price.
percent markup \(225\%;\) retail price =\( $26.55\)
- Answer
-
Note that the retail price is \(225\%;\) more than the initial price (cost).
\[
\begin{aligned}
\text {Initial (origianl price)} &=\frac{\text {Final (retail price) }}{ \text {(100+p)\%}}\\
&=\frac{26.55}{(100+ 225)\%}\\
&=\frac{26.55}{ 325\%}\\
&=\frac{26.55}{3.25}\\
&= 8.17\\
\end{aligned}
\nonumber \]So cost is \($8.17\). And \(225\%\) markup in this price will be \($26.55\).
Compute Sales Tax
Sales tax is applied to the sale or lease of some goods and services in the United States but is not determined by the federal government. It is most often set, collected, and spent by individual states, counties, parishes, and municipalities. None of these sales tax revenues go to the federal government.
For example, North Carolina has a state sales tax of \(4.75\%\) while New Mexico has a state sales tax of \(5\%.\) Additionally, many counties in North Carolina charge an additional \(2\%\) sales tax, bringing the total sales tax for most \(72\) of the \(100\) counties in North Carolinians to \(6.75\%.\) However, in Durham, the county sales tax is \(2.25\%\) plus an additional \(0.5\%\) tax used to fund public transportation, bringing Durham County’s sales tax to \(7\%.\) To find the sales tax in a particular place, add other locality sales taxes to the base state sales tax rate.
How much we pay in sales tax depends on where we are and what we are buying.
To determine the amount of sales tax on a taxable purchase, we need to find the purchase price and add the sales tax for that locality
The formula for sales tax is similar to markup because both involve adding a percentage to the original amount. In sales tax, the amount before tax is the total or whole (original price), and the tax is the part added to it. To find the final amount, you add the sales tax to the original price. The final amount is always greater than the initial amount because the tax increases the total cost.
Rounding to the nearest penny or rounding to the nearest cent means rounding to two decimal places.
You should notice that this is the same as markup, except using a different term. Sales tax plays the role of markup, the purchase price plays the role of cost, and the tax rate plays the role of percent markup. This means all the strategies developed for markups apply to this situation, with the changes indicated.
Example \(\PageIndex{16}\): Calculating the Sales Tax from the Purchase Price
The sales tax in Kankakee, Illinois, is \(8.25\%.\) Find the sales tax and final price of items based on the purchase price listed. Purchase price = \($428.99\)
- Answer
-
Here, sales tax is part of the original price (whole or total).
\[
\begin{aligned}
\text{Sale tax (part)} &=\text{percent }\times\text {total}\\
&=\text{0.0825}\times428.99\\
&=35.39\\
\end{aligned}
\nonumber \]The final price of the item is then
\[ \text{Final price} =$428.99+$35.39=$464.38\nonumber\]
The final price including tax is \($464.38.\)
Or you can find the final price directly using the following formula. Note that the final price is \(25\% \) less than the original price (purchased price).
\[
\begin{aligned}
\text{Fina price } &=\text{Initial price}\times (100+p)\%\\
&=428.99 \times (100+8.25)\%\\
&=428.99 \times (1.0825)\\
&=464.38
\end{aligned}
\nonumber \]
Be aware that almost all sales tax rates are structured as full percentages, half percentages, one-quarter percentages, or three-quarter percentages. This means the decimal value of the sales tax rate, written as a percent, will be either \(0,\) as in \(5.0\%,\) \(5\) as in \(7.5\%,\) \(25\) as in \(3.25\%,\) or \(75\) as in \(4.75\%.\) When rounding for the sales tax percentage, be sure to use this guideline.
Example \(\PageIndex{17}\): Calculating the Purchase Price from the Sales Tax and Total Price
Determine the purchase price for the specified sales tax rate and total cost.
sales tax rate = \(5.75\%;\) price after tax = \($36.56\)
- Answer
-
\begin{align*}
\text {Initial} & =\frac{\text {Final}}{( 100+p)\%} \\
& =\frac{\text {36.56}}{(100+5.75)\%} \\
& =\frac{\text {36.56}}{105.75\%} \\
& =\frac{\text {36.56}}{ \text { 1.0575}} \\
& =34.57\\
\end{align*}
West Virginia was the first state to impose a sales tax. This happened on May \(3,\) \(1921.\)
4.7: Understanding Percents- Discounts, Markups, and Sales Tax is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.
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