1.2: Percentages

Percentages

In 2020, the U.S. federal government budgeted $3.5 billion for the National Park Service, 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 2020 was $6.6 trillion. So, the amount budgeted for the National Park Service was less than one-tenth of 1 percent, or 1/10%, of the total outlays. This percent describes a specific number by relating the size of a part to a whole. Understanding that ratio puts the $3.5 billion budgeted to the National Park Service in perspective.

After reviewing the fundamentals of percentages we will examine applications including calculating flat taxes, tips, discounts based on percentages for a purchase, and progressive income tax.

The word percent comes from the Latin phrase per centum, which means “by the hundred.” It is really two words, “per cent,” and means per one hundred. The percentage of something is a ratio that compares the relative size of a part to a whole.

Changing a percent to a fraction is to write the percent in its fractional form. To write n% in its fractional form is to write the percent as the fraction $\frac{n}{100}$.

When we write 40%, this is equivalent to the fraction $\frac{40}{100}$ or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since $\frac{80}{200}=\frac{10}{25}=\frac{40}{100}$.

We will always do calculations with a percent after it has been converted to a decimal or fractional form. In the definition below when you see the equation the percent has been converted to a fractional or decimal form.

It is important that we convert a percent to a decimal or fractional form in many of the equations we use.

In this text we will mainly be using percentages that are written in a decimal form.

Example 2

Convert the percentage in the table below to decimal and fractional form.

Percent	Decimal	Fraction
6%
78%
135%
12.5%

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Solution

Figure 1 shows how the given percents are converted to decimals by moving the decimal point two places to the left. Notice that we may need to add zeros in front of the number when moving the decimal to the left.

The table below shows the results from the conversions.

Percent	Decimal
6%	0.06
78%	0.78
135%	1.35
12.5%	0.125

To convert each of the percentages to fractional form we will divide each percent by 100 and remove the percent symbol.

Percent	Decimal	Fraction
6%	0.06	$\frac{6}{100}$
78%	0.78	$\frac{78}{100}$
135%	1.35	$\frac{135}{100}$
12.5%	0.125	$\frac{12.5}{100}$

Calculate the total, percent, or part.

Now lets revisit that definition we gave on a percent and look at how we go between the percent, whole and part.

Take a look at an example of how we use this definition to find the percent of some number (i.e. find the part related to a certain percent).

Solve Application Problems Involving Percents

Percents are frequently used in finance, purchases, taxes, research, science experiments, politics, and even casual conversation. Understanding these types of values helps when consuming media or discussing finances, for instance. Effectively working with and interpreting numbers and percents 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 percents, which two of the three values that are related through percents are known, and which of the three values you need to find.

Sales tax is something we all have to deal with when purchasing a product from a business (in most states in the U.S.). Although many states do collect sales tax the rate itself is composed of a state rate and local rate. We will only look at an overal sales tax rate in our applications to keep the discussion easier to follow. Even states without an official sales tax for consumers may have some sort of sales tax at a local level or at the business level.

When the sales tax calculation results in a fraction of a penny, then normal rounding rules apply, round up for half a penny or more, but round down for less than half a penny.

Tips work much like sales tax. They take a percent of the total bill and are typically calculated before the sales tax has been applied (pre-tax). You then add the tip and sales tax to the purchase price to find the total price.

Time to practice applying what you have seen on a bit more challenging problem. See how you do on the Try It Now problem below

Percent Increase and Decrease Applications

The work we have done on applications with sales tax can lead us to a general strategy for finding the new amount after a percent increase or decrease.

We are often interested in the amount of an increase or decrease based on a percentage. To find the percent increase, first we find the amount of increase, which is the difference between the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.

Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference between the original amount and the final amount. Then we find what percent the amount of decrease is of the original amount.

We can represent these calculations below.

When we compute percent increase or decrease we always compare how much a quantity has changed to the original amount.

We can also use the formulas to solve questions about finding the original amount after a percent increase or decrease is observed.

In the example above it is far too tempting to take 30% of $22.40 and add that to the sales price to get the original amount. The problem is that if you try that approach you are not actually discounting the original price by 30% to get the sales price. Try it yourself and see what happens (you will get a different answer plus that answer you get if you discount it by 30% will not get you the sales price).

When the original value is not known, but the new amount and percent increase or decrease is given we can rewrite the original equations in a new form using a little algebra. Those new forms are given below.

Flat and Progressive Taxes

In some states the amount of income tax calculated is done with a flat rate tax. A flat rate tax is one in which regardless of income everyone pays the same percent on that income. It does not mean that everyone pays the same amount in taxes. Sales tax is not considered to be a flat tax rate system as the proportion of your income going to sales tax goes up as your income decrease (we call sales tax a type of regressive tax system as it has a larger impact on those whose income is lower).

In 2021 the following states had a flat income tax rate system: Colorado, Illinois, Indiana, Kentucky, Massachusetts, Michigan, New Hampshire (interest and dividend income only). North Carolina, Pennsylvania, and Utah.

Example 15

A resident in Illinois has found that they have a taxable net income of $39,500 for line 11 on 2020 Form
IL-1040. Find the result for line 12.

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Solution

The instructions for line 12 state to multiply line 11 by 4.95%. The value in line 11 was given to us as
$39,500, so we have line 12 as:

$$\$39,500\cdot \left(0.0495\right)=\$1,995.25$$

In most states the type of income tax paid is similar to the federal income tax rate where the tax rate changes to higher values as the taxable income amount increases. This type of tax system is referred to as a Graduated-rate or progressive income tax. In Arizona the state income tax is caclulate with a graduated-rate schedule.

Exercises

1. Convert each number to a percentage
   1. 0.731
   2. 0.0025
   3. 2.94
   4. 0.057
   5. $\frac{26}{100}$
   Answer (click to Show/Hide)

   1. 73.1%
   2. 0.25%
   3. 294%
   4. 5.7%
   5. 26%
2. What number is 18% of 75?
   
   Answer (click to Show/Hide)

   Convert 18% to decimal: $18\%=0.18$

   Multiply: $75\times 0.18=13.5$

3. What number is 110% of 50?
   
   Answer (click to Show/Hide)

   Convert 110% to decimal: $110\%=1.10$

   Multiply: $50\times 1.10=55$

4. 30 is 75% of what number?
   
   Answer (click to Show/Hide)

   Let $x$ be the number we’re looking for

   Write the equation: $30=0.75x$

   Solve for $x$:

   $x=30÷0.75=\frac{30}{0.75}=40$

5. 10 is 15% of what number?
   
   Answer (click to Show/Hide)

   Let $x$ be the number we’re looking for

   Write the equation: $10=0.15x$

   Solve for $x$:

   $x=10÷0.15=\frac{10}{0.15}=\frac{200}{3}=66\frac{2}{3}$

6. The sales tax is 8.7%. How much is the sales tax on a purchase of $250? What is the total cost of the purchase including sales tax?
   
   Answer (click to Show/Hide)

   First calculate the tax amount: $250\times 0.087=21.75$

   Next calculate the total cost: $250+21.75=271.75$

7. The sales tax is 6.2%. How much is the sales tax on a purchase of $95? What is the total cost of the purchase including sales tax?
   
   Answer (click to Show/Hide)

   First calculate the tax amount: $95\times 0.062=5.89$

   Next calculate the total cost: $95+5.89=100.89$

8. A restaurant bill for a family of four is $75 before taxes and tip. If the sales tax is 8.7% and they wish to leave a tip of 18% on the original bill before taxes, then what is the total cost for the meal.
   
   Answer (click to Show/Hide)

   First calculate the tax amount: $75\times 0.087=6.53$

   Next calculate the tip amount (18% of original bill): $75\times 0.18=13.50$

   Lastly find the total of the bill with tax and tip: $75+6.53+13.50=95.03$

9. You receive a bill for a meal at a restaurant for $45.90, but it only lists the total after tax. How much was the original amount of the meal before taxes if the sales tax rate is 5.1%?
   
   Answer (click to Show/Hide)

   Let $x$ be the original amount

   Write the equation to represent the unknown original amount increased by 5.1% tax to give the bill total of $45.90:

   $x+0.051x=45.90$

   Now simplify the equation: $1.051x=45.90$

   Solve for $x$:

   $x=45.90÷1.051=43.67$

10. You receive a bill for a meal at a restaurant for $89.24, but it only lists the total after tax. How much was the original amount of the meal before taxes if the sales tax rate is 8.7%?
    
    Answer (click to Show/Hide)

    First, define a variable for the unknown original amount of the meal. Let’s call it $x$.

    The total bill is the sum of the original meal cost ($x$) and the sales tax. The sales tax is 8.7% of the original amount, which can be written as $0.087x$. This gives us the equation:

    $$\text{Original Amount}+\text{Sales Tax}=\text{Total Bill}$$
    $$x+0.087x=89.24$$

    Combine the terms with $x$:

    $$1.087x=89.24$$

    To solve for $x$, divide both sides of the equation by 1.087:

    $$x=\frac{89.24}{1.087}$$

    Calculate the value and round to two decimal places for currency:

    $$x\approx \mathrm{82.0975\dots }$$
    $$x\approx 82.10$$

    Answer: The original amount of the meal before tax was $82.10.

11. Out of 230 racers who started the marathon, 212 completed the race, 14 gave up, and 4 were disqualified.  What percentage did not complete the marathon? If you are disqualified from a race you can not be considered a completer.
    
    Answer (click to Show/Hide)

    First, find the total number of racers who did not complete the marathon. This includes those who gave up and those who were disqualified.

    $$14+4=18$$

    Next, calculate the fraction of racers who did not complete. This is the number of non-completers divided by the total number of racers who started.

    $$\text{Fraction}=\frac{\text{Did Not Complete}}{\text{Total Starters}}=\frac{18}{230}$$

    To express this fraction as a percentage, multiply the fraction by 100.

    $$\text{Percentage}=\frac{18}{230}\times 100$$

    Calculate the final value and round to two decimal places.

    $$\text{Percentage}\approx 7.83\%$$

    Answer: Approximately 7.83% of the racers did not complete the marathon.

12. Patrick left an $8 tip on a $50 restaurant bill.  What percent tip is that?
    
    Answer (click to Show/Hide)

    To find the tip percentage, you need to determine what part of the total bill the tip represents. This can be set up as a fraction.

    $$\text{Fraction}=\frac{\text{Tip Amount}}{\text{Bill Amount}}=\frac{8}{50}$$

    To convert this fraction into a percentage, multiply it by 100.

    $$\text{Percentage}=\frac{8}{50}\times 100$$

    Calculate the final value.

    $$\text{Percentage}=0.16\times 100=16\%$$

    Answer: Patrick left a 16% tip.

13. Ireland has a 23% VAT (value-added tax, similar to a sales tax).  How much will the VAT be on a purchase of a €250 (250 Euros) item?
    
    Answer (click to Show/Hide)

    To calculate the VAT amount, you multiply the original price of the item by the VAT rate. First, convert the percentage to a decimal.

    $$23\%=0.23$$

    Now, set up the equation to find the VAT.

    $$\text{VAT}=\text{Original Price}\times \text{VAT Rate}$$
    $$\text{VAT}=250\times 0.23$$

    Calculate the final value.

    $$\text{VAT}=57.50$$

    Answer: The VAT on the item will be €57.50.

14. Employees in 2012 paid 4.2% of their gross wages towards social security (FICA tax), while employers paid another 6.2%.  How much would someone earning $45,000 a year have paid towards social security out of their gross wages in 2012?
    
    Answer (click to Show/Hide)

    The problem asks for the amount the employee paid, so we only need to use the employee’s tax rate of 4.2%. The employer’s rate is extra information. First, convert the employee’s percentage to a decimal.

    $$4.2\%=0.042$$

    Now, set up the equation to find the tax paid.

    $$\text{Tax Paid}=\text{Gross Wages}\times \text{Tax Rate}$$
    $$\text{Tax Paid}=45000\times 0.042$$

    Calculate the final value.

    $$\text{Tax Paid}=1890$$

    Answer: The employee would have paid $1,890 towards social security in 2012.

15. A project on Kickstarter.com was aiming to raise $15,000 for a precision coffee press.  They ended up with 714 supporters, raising 557% of their goal.  How much did they raise?
    
    Answer (click to Show/Hide)

    The problem asks for the total amount of money raised. The number of supporters is extra information. We need to calculate 557% of the $15,000 goal. First, convert the percentage to a decimal.

    $$557\%=\frac{557}{100}=5.57$$

    Now, set up the equation to find the total amount raised.

    $$\text{Amount Raised}=\text{Goal}\times \text{Percentage Raised}$$
    $$\text{Amount Raised}=15000\times 5.57$$

    Calculate the final value.

    $$\text{Amount Raised}=83550$$

    Answer: They raised $83,550.

16. Another project on Kickstarter for an iPad stylus raised 1,253% of their goal, raising a total of $313,490 from 7,511 supporters.  What was their original goal? Round your answer to nearest thousand dollars.
    
    Answer (click to Show/Hide)

    Let the original goal be represented by the variable $G$. The number of supporters is extra information that is not needed to solve for the goal.

    The amount raised is the original goal multiplied by the percentage. First, convert the percentage raised into a decimal.
    $$1253\%=\frac{1253}{100}=12.53$$

    Set up the equation using the information from the problem.
    $$\text{Goal}\times \text{Percentage Raised}=\text{Amount Raised}$$
    $$G\times 12.53=313490$$

    To solve for $G$, divide the total amount raised by the decimal percentage.
    $$G=\frac{313490}{12.53}$$

    Calculate the final value.
    $$G=25000$$

    Answer: The original goal was $25,000.

17. A company wants to decrease their energy use by 15%.
    1. If their electric bill is currently $2,200 a month, what will their bill be if they’re successful?
    2. If their next bill is $1,700 a month, were they successful?  Why or why not?
    Answer (click to Show/Hide)

    First, calculate the target bill amount for a 15% decrease. The decrease is 15% of the original amount. Convert the percentage to a decimal.
    $$15\%=0.15$$

    Calculate the monetary value of the decrease.
    $$\text{Decrease Amount}=2200\times 0.15=330$$

    Subtract the decrease from the original bill to find the new target bill.
    $$\text{Target Bill}=2200–330=1870$$

    The second part of the problem asks if a new bill of $1,700 represents success. To be successful, the new bill must be less than or equal to the target bill.
    $$1700<1870$$

    Answer: Their target bill to achieve a 15% reduction is $1,870. They were successful, because their new bill of $1,700 is lower than their target bill of $1,870.

18. A store is hoping an advertising campaign will increase their number of customers by 30%. They currently have about 80 customers a day.
    1. How many customers will they have if their campaign is successful?
    2. If they increase to 120 customers a day, were they successful?  Why or why not?
    Answer (click to Show/Hide)

    First, calculate the target number of customers for a 30% increase. The increase is 30% of the original number of customers. Convert the percentage to a decimal.
    $$30\%=0.30$$

    Calculate the value of the increase in customers.
    $$\text{Increase Amount}=80\times 0.30=24$$

    Add the increase to the original number of customers to find the new target.
    $$\text{Target Customers}=80+24=104$$

    The second part of the problem asks if an increase to 120 customers represents success. To be successful, the new number of customers must be greater than or equal to the target.
    $$120>104$$

    Answer: Their target is 104 customers a day to achieve a 30% increase. They were successful, because their new daily customer count of 120 is higher than their target of 104.

19. An article reports “attendance dropped 6% this year, to 300.”  What was the attendance before the drop?
    
    Answer (click to Show/Hide)

    Let the original attendance be represented by the variable $A$. The new attendance is the result of a 6% drop from this original number.

    A 6% drop means that 94% of the original attendance remains (100% – 6% = 94%). Convert this remaining percentage to a decimal.
    $$94\%=0.94$$

    Set up the equation. The original attendance multiplied by the remaining percentage equals the new attendance.
    $$\text{Original Attendance}\times \text{Remaining Percentage}=\text{New Attendance}$$
    $$A\times 0.94=300$$

    To solve for $A$, divide the new attendance by the decimal percentage.
    $$A=\frac{300}{0.94}$$

    Calculate the value and round to the nearest whole number, since attendance involves people.
    $$A\approx 319.15$$

    Answer: The attendance before the drop was approximately 319.

20. An article reports “sales have grown by 30% this year, to $200 million.”  What were sales before the growth?
    
    Answer (click to Show/Hide)

    Let the original sales be represented by the variable $S$. The new sales figure is the result of a 30% growth from this original amount.

    A 30% growth means that the new sales are 130% of the original sales (100% + 30% = 130%). Convert this total percentage to a decimal.
    $$130\%=1.30$$

    Set up the equation. The original sales multiplied by the new percentage equals the new sales figure.
    $$\text{Original Sales}\times \text{New Percentage}=\text{New Sales}$$
    $$S\times 1.30=\mathrm{200,000,000}$$

    To solve for $S$, divide the new sales figure by the decimal percentage.
    $$S=\frac{\mathrm{200,000,000}}{1.30}$$

    Calculate the value and round to the nearest dollar.
    $$S\approx \mathrm{153,846,153.85}$$

    Answer: The sales before the growth were approximately $153,846,154.

21. A store has clearance items that have been marked down by 60%.  They are having a sale, advertising an additional 30% off clearance items.  What percent of the original price do you end up paying?
    
    Answer (click to Show/Hide)

    Let the original price of the item be represented by the variable $P$. The first markdown is 60%. This means you pay 40% of the original price.
    $$100\%–60\%=40\%$$

    The price after the first markdown (the clearance price) is 40% of the original price, $P$.
    $$\text{Clearance Price}=P\times 0.40$$

    The second discount is an additional 30% off the clearance price. This means you pay 70% of the clearance price.
    $$100\%–30\%=70\%$$

    Calculate the final price by taking 70% of the clearance price.
    $$\text{Final Price}=(P\times 0.40)\times 0.70$$

    Multiply the decimals to find the final percentage of the original price.
    $$0.40\times 0.70=0.28$$

    This means the final price is $P\times 0.28$, or 28% of the original price.

    Answer: You end up paying 28% of the original price.

22. Which is better:  having a stock that goes up 30% on Monday then drops 30% on Tuesday, or a stock that drops 30% on Monday and goes up 30% on Tuesday?  In each case, what is the net percent gain or loss?
    
    Answer (click to Show/Hide)

    Let the original price of the stock be represented by the variable $P$. We will analyze both scenarios.

    Scenario 1: Up 30% on Monday, then down 30% on Tuesday

    1. On Monday, the price increases by 30%. The new price is 130% of the original.
       $$\text{Monday Price}=P\times (1+0.30)=P\times 1.30$$
    2. On Tuesday, this new price drops by 30%. This means the final price is 70% of Monday’s price.
       $$\text{Final Price}=(P\times 1.30)\times (1–0.30)=(P\times 1.30)\times 0.70$$
    3. Multiply the decimals to find the final percentage of the original price.
       $$1.30\times 0.70=0.91$$
    4. The final price is 91% of the original price, which represents a 9% loss.
       $$1–0.91=0.09=9\%$$

    Scenario 2: Down 30% on Monday, then up 30% on Tuesday

    1. On Monday, the price drops by 30%. The new price is 70% of the original.
       $$\text{Monday Price}=P\times (1–0.30)=P\times 0.70$$
    2. On Tuesday, this new price increases by 30%. The final price is 130% of Monday’s price.
       $$\text{Final Price}=(P\times 0.70)\times (1+0.30)=(P\times 0.70)\times 1.30$$
    3. Multiply the decimals to find the final percentage of the original price.
       $$0.70\times 1.30=0.91$$
    4. The final price is also 91% of the original price, which represents a 9% loss.
       $$1–0.91=0.09=9\%$$

    Answer: Neither is better as you end up with the same loss of 9%.

23. The Inconsistent Books store changes the normal prices of books on Mondays and Tuesdays. On Monday the price of a book is 10% above the normal price. On Tuesday the price is 10% below the normal price. Sarah bought a book on Monday for $16.50. What would be the price of that book on a Tuesday?
    
    Answer (click to Show/Hide)

    First, determine the book’s normal price based on the Monday price. Let the normal price be $N$. The Monday price is 10% above normal, or 110% of the normal price, which is $16.50.
    $$N\times 1.10=16.50$$

    Solve for the normal price by dividing the Monday price by 1.10. The normal price is $15.00.
    $$N=\frac{16.50}{1.10}=15.00$$

    Now, calculate the Tuesday price. The Tuesday price is 10% below the normal price, which means it is 90% of the normal price.
    $$\text{Tuesday Price}=15.00\times 0.90$$

    Calculate the final price.
    $$15.00\times 0.90=13.50$$

    Answer: The price of the book on a Tuesday would be $13.50.

24. Determine the amount of taxes paid if you are a resident in Illinois with a calculated taxable net income of $102,500.
    
    Answer (click to Show/Hide)

    Identify the net income and the tax rate.
    Net Income = $102,500
    Tax Rate = 4.95%

    Convert the tax rate from a percentage to a decimal by dividing by 100.
    $$4.95\%=\frac{4.95}{100}=0.0495$$

    Multiply the net income by the decimal tax rate to find the total tax paid.
    $$\mathrm{102,500}\times 0.0495=5073.75$$

    Answer: The amount of taxes paid would be $5,073.75.

25. If you were a resident of Illinois and your reported taxable net income doubled from $102,500 to $205,000 would your amount of taxes owe double on line 12?
    
    Answer (click to Show/Hide)

    First, calculate the tax on the original income of $102,500 with the flat tax rate of 4.95% (or 0.0495).
    $$\mathrm{102,500}\times 0.0495=5073.75$$

    Next, calculate the tax on the doubled income of $205,000 using the same tax rate.
    $$\mathrm{205,000}\times 0.0495=10147.50$$

    Compare the two tax amounts by dividing the new tax amount by the original tax amount.
    $$\frac{10147.50}{5073.75}=2$$

    Answer: Yes, the amount of taxes owed would double. The new tax amount ($10,147.50) is exactly twice the original tax amount ($5,073.75). This happens because Illinois uses a flat tax rate. Since the tax rate is a constant percentage, any change in income results in a directly proportional change in the tax owed.

26. Answer the following about a resident filing single in Arizona for their income taxes in 2020. Use the table provided in example 16 to answer this question.
    1. Find the income tax for a person with a taxable income of $50,000.
    2. Find the income tax for a person with a taxable income of $20,000.
    3. Would the income tax for a person in Arizona double if their taxable income doubled?
    4. If a person had a taxable income of $30,000 would the income tax double if we double the taxable income to $60,000. Why or why not?
    Answer (click to Show/Hide)

    1. Find the income tax for a person with a taxable income of $50,000.
       
       This requires a marginal tax calculation.
       
       The first $27,272 is taxed at 2.59%.
       
       The rest of the income ($50,000 – $27,272 = $22,728) is taxed at 3.34%.
       $$(\mathrm{27,272}\times 0.0259)+(\mathrm{22,728}\times 0.0334)=706.34+759.25=1465.59$$
       The total tax would be $1,465.59.
    2. Find the income tax for a person with a taxable income of $20,000.
       
       A taxable income of $20,000 falls entirely within the first bracket, so it is all taxed at 2.59%.
       $$\mathrm{20,000}\times 0.0259=518$$
       The tax would be $518.
    3. Would the income tax for a person in Arizona double if their taxable income doubled?
       
       No. Because of the marginal tax brackets, as income increases, a larger portion of it is taxed at a higher rate. This means the total tax will increase by more than double if the income doubles and crosses into a new tax bracket.
    4. If a person had a taxable income of $30,000, would the income tax double if the taxable income was doubled to $60,000? Why or why not?
       
       First, calculate the tax for each income level using a marginal tax table.
       
       Tax on $30,000:

       Income Segment	Amount	Rate	Tax
       $0 – $27,272	$27,272	2.59%	$706.34
       $27,272 – $30,000	$2,728	3.34%	$91.12
       Total	$30,000		$797.46

       Tax on $60,000:

       Income Segment	Amount	Rate	Tax
       $0 – $27,272	$27,272	2.59%	$706.34
       $27,272 – $54,544	$27,272	3.34%	$910.88
       $54,544 – $60,000	$5,456	4.17%	$227.50
       Total	$60,000		$1,844.72

       To check if the tax doubled, multiply the first tax amount by 2:
       $$797.46\times 2=1594.92$$
       Since $1,844.72 is not equal to $1,594.92, the tax did not double. The tax on $30,000 is $797.46, while the tax on $60,000 is $1,844.72. The tax more than doubled because doubling the income pushed a portion of that income into a higher tax bracket (4.17%), leading to a higher overall effective tax rate.

Attributions

• This page contains modified content from David Lippman, “Math In Society, 2nd Edition.” Licensed under CC BY-SA 4.0.
• This page contains modified content and images from “OpenStax PreAlgebra” by Lynn Marecek, MaryAnne Anthony-Smith. Licensed under CC BY-NC 4.0 / A derivative from the original work
• This page contains modified content and images from “Contemporary Mathematics” by Donna Kirk and is licensed under CC BY-NC 4.0 / A derivative from
  the original work.
• This page contains content by Robert Foth, Math Faculty, Pima Community College, 2021. Licensed under CC BY 4.0.
• Portions of the exercise solutions in this answer key were generated with the assistance of Gemini, a large language model from Google.