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== 3.2#h(0.6em)Solve Percent Applications

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Convert 4.5% to a decimal. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "0.045")[$0.045$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert 0.6 to a percent. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "60 %")[$60 %$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round 0.875 to the nearest hundredth. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "0.88")[$0.88$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply (4.5)(2.38). #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "10.71")[$10.71$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve #math.equation(block: false, alt: "3.5 equals 0.7 n .")[$3.5 = 0.7 n .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "n equals 5")[$n = 5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract #math.equation(block: false, alt: "50 minus 37.45 .")[$50 − 37.45 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "12.55")[$12.55$]
]
]


=== Translate and Solve Basic Percent Equations

We will solve percent equations using the methods we used to solve equations with fractions or decimals. Without the tools of algebra, the best method available to solve percent problems was by setting them up as proportions. Now as an algebra student, you can just translate English sentences into algebraic equations and then solve the equations.

We can use any letter you like as a variable, but it is a good idea to choose a letter that will remind us of what you are looking for. We must be sure to change the given percent to a decimal when we put it in the equation.

#examplebox("Example 1")[][
Translate and solve: What number is 35% of 90?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical problem asking 'What number is 35% of 90?' with light blue brackets underlining different parts of the question to highlight them.], alt: "A mathematical problem asking 'What number is 35% of 90?' with light blue brackets underlining different parts of the question to highlight them.", caption: none)]),
  [Translate into algebra. Let #math.equation(block: false, alt: "n")[$n$]= the number.], [#figure(figph[A mathematical equation shows 'n = 0.35 . 90' displayed on a white background, representing a calculation where 'n' is the product of 0.35 and 90.], alt: "A mathematical equation shows 'n = 0.35 . 90' displayed on a white background, representing a calculation where 'n' is the product of 0.35 and 90.", caption: none)],
  [Remember "of" means multiply, "is" means equals.], [],
  [Multiply.], [#figure(figph[The image displays the equation 'eta = 31.5' or 'n = 31.5' in a simple, clear font against a white background.], alt: "The image displays the equation 'eta = 31.5' or 'n = 31.5' in a simple, clear font against a white background.", caption: none)],
  [], [#math.equation(block: false, alt: "31.5")[$31.5$] is #math.equation(block: false, alt: "35%")[$35%$] of #math.equation(block: false, alt: "90")[$90$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

What number is 45% of 80?

#solutionbox[
36
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

What number is 55% of 60?

#solutionbox[
33
]
]

We must be very careful when we translate the words in the next example. The unknown quantity will not be isolated at first, like it was in . We will again use direct translation to write the equation.

#examplebox("Example 2")[][
Translate and solve: 6.5% of what number is \$1.17?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical word problem asks: '6.5% of what number is \$1.17?' The text is written in black on a white background, with light blue brackets segmenting parts of the question.], alt: "A mathematical word problem asks: '6.5% of what number is $1.17?' The text is written in black on a white background, with light blue brackets segmenting parts of the question.", caption: none)]),
  [Translate. Let #math.equation(block: false, alt: "n equals")[$n =$] the number.], [#figure(figph[A mathematical equation on a white background reads '0.065   n = 1.17', showing a multiplication problem with a variable 'n' and a decimal result.], alt: "A mathematical equation on a white background reads '0.065 n = 1.17', showing a multiplication problem with a variable 'n' and a decimal result.", caption: none)],
  [Multiply.], [#figure(figph[The mathematical equation 0.065n = 1.17 is clearly displayed on a plain white background.], alt: "The mathematical equation 0.065n = 1.17 is clearly displayed on a plain white background.", caption: none)],
  [Divide both sides by 0.065 and simplify.], [#figure(figph[The text 'n = 18' is displayed in the upper right portion of a plain white background.], alt: "The text 'n = 18' is displayed in the upper right portion of a plain white background.", caption: none)],
  [], [#math.equation(block: false, alt: "6.5%")[$6.5%$] of#math.equation(block: false, alt: "$18")[$\$18$] is #math.equation(block: false, alt: "$1.17")[$\$1.17$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

7.5% of what number is \$1.95?

#solutionbox[
\$26
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

8.5% of what number is \$3.06?

#solutionbox[
\$36
]
]

In the next example, we are looking for the percent.

#examplebox("Example 3")[][
Translate and solve: 144 is what percent of 96?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical problem asks: 144 is what percent of 96? The numbers and text are underlined with light blue brackets to indicate distinct parts of the question.], alt: "A mathematical problem asks: 144 is what percent of 96? The numbers and text are underlined with light blue brackets to indicate distinct parts of the question.", caption: none)]),
  [Translate into algebra. Let #math.equation(block: false, alt: "p equals")[$p =$] the percent.], [#figure(figph[A mathematical equation is displayed, showing '144 = p ×. 96' with the variable 'p' and a multiplication dot before '96' on a white background.], alt: "A mathematical equation is displayed, showing '144 = p ×. 96' with the variable 'p' and a multiplication dot before '96' on a white background.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical equation is displayed, showing '144 = 96p' in gray text on a white background.], alt: "A mathematical equation is displayed, showing '144 = 96p' in gray text on a white background.", caption: none)],
  [Divide by 96 and simplify.], [#figure(figph[The image displays the equation '1.5 = p' on a white background, suggesting a mathematical problem or definition.], alt: "The image displays the equation '1.5 = p' on a white background, suggesting a mathematical problem or definition.", caption: none)],
  [Convert to percent.], [#figure(figph[The image displays the equation '150% = p' in black text on a white background, representing a mathematical relationship where 'p' is equivalent to 150 percent.], alt: "The image displays the equation '150% = p' in black text on a white background, representing a mathematical relationship where 'p' is equivalent to 150 percent.", caption: none)],
  [], [#math.equation(block: false, alt: "144")[$144$] is #math.equation(block: false, alt: "150%")[$150%$] of #math.equation(block: false, alt: "96")[$96$]],
))

Note that we are asked to find percent, so we must have our final result in percent form.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

110 is what percent of 88?

#solutionbox[
125%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

126 is what percent of 72?

#solutionbox[
175%
]
]


=== Solve Applications of Percent

Many applications of #strong[percent]—such as tips, sales tax, discounts, and interest—occur in our daily lives. To solve these applications we’ll translate to a basic percent equation, just like those we solved in previous examples. Once we translate the sentence into a percent equation, we know how to solve it.

We will restate the problem solving strategy we used earlier for easy reference.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Use a Problem-Solving Strategy to Solve an Application.]

+ #strong[Read] the problem. Make sure all the words and ideas are understood.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for. Choose a variable to represent that quantity.
+ #strong[Translate] into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.
]

Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications will involve everyday situations, you can rely on your own experience.

#examplebox("Example 4")[][
Dezohn and his girlfriend enjoyed a nice dinner at a restaurant and his bill was \$68.50. He wants to leave an 18% tip. If the tip will be 18% of the total bill, how much tip should he leave?

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [the amount of tip should Dezohn leave],
  [#emph[Step 3. Name] what we are looking for.], [], [],
  [Choose a variable to represent it.], [], [Let #emph[t] = amount of tip.],
  [#emph[Step 4. Translate] into an equation.], [], [#figure(figph[The tip is 18% of the total bill.], alt: "The tip is 18% of the total bill.", caption: none)],
  [Write a sentence that gives the information to find it.], [], [#figure(figph[The image displays the phrase 'The tip is 18% of \$68.50' with light blue brackets underlining each word or number, indicating a breakdown or emphasis on the components of the statement.], alt: "The image displays the phrase 'The tip is 18% of $68.50' with light blue brackets underlining each word or number, indicating a breakdown or emphasis on the components of the statement.", caption: none)],
  [Translate the sentence into an equation.], [], [#figure(figph[A mathematical equation is displayed with the variable 't' set equal to the product of 0.18 and 68.50, demonstrating a calculation likely for a percentage or fraction of a given value.], alt: "A mathematical equation is displayed with the variable 't' set equal to the product of 0.18 and 68.50, demonstrating a calculation likely for a percentage or fraction of a given value.", caption: none)],
  [#emph[Step 5. Solve] the equation. Multiply.], [], [#figure(figph[The image displays the text 't = 12.33' in black on a white background, likely representing a time or numerical value in a scientific or technical context.], alt: "The image displays the text 't = 12.33' in black on a white background, likely representing a time or numerical value in a scientific or technical context.", caption: none)],
  [#emph[Step 6. Check.] Does this make sense?], [], [],
  [Yes, 20% of \$70 is \$14.], [], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [], [Dezohn should leave a tip of \$12.33.],
))

Notice that we used #emph[t] to represent the unknown tip.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Cierra and her sister enjoyed a dinner in a restaurant and the bill was \$81.50. If she wants to leave 18% of the total bill as her tip, how much should she leave?

#solutionbox[
\$14.67
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Kimngoc had lunch at her favorite restaurant. She wants to leave 15% of the total bill as her tip. If her bill was \$14.40, how much will she leave for the tip?

#solutionbox[
\$2.16
]
]

#examplebox("Example 5")[][
The label on Masao’s breakfast cereal said that one serving of cereal provides 85 milligrams (mg) of potassium, which is 2% of the recommended daily amount. What is the total recommended daily amount of potassium?

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [the total amount of potassium that is recommended],
  [#emph[Step 3. Name] what we are looking for.], [], [],
  [Choose a variable to represent it.], [], [Let #math.equation(block: false, alt: "a equals")[$a =$] total amount of potassium.],
  [#emph[Step 4. Translate.] Write a sentence that gives the information to find it.], [], [#figure(figph[The image shows the text '85 mg is 2% of the total amount', with light blue brackets under each phrase: '85 mg', 'is', '2%', 'of the', and 'total amount'.], alt: "The image shows the text '85 mg is 2% of the total amount', with light blue brackets under each phrase: '85 mg', 'is', '2%', 'of the', and 'total amount'.", caption: none)],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation is displayed, showing '85 = 0.02 \* q'. This represents a basic algebraic problem where the variable 'q' is being multiplied by 0.02, resulting in 85.], alt: "A mathematical equation is displayed, showing '85 = 0.02 * q'. This represents a basic algebraic problem where the variable 'q' is being multiplied by 0.02, resulting in 85.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[The image displays the equation 4,250 = a, rendered in a simple white background with dark gray text.], alt: "The image displays the equation 4,250 = a, rendered in a simple white background with dark gray text.", caption: none)],
  [#emph[Step 6. Check.] Does this make sense?], [], [],
  [Yes, 2% is a small percent and 85 is a small part of 4,250.], [], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [], [The amount of potassium that is recommended is 4,250 mg.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One serving of wheat square cereal has seven grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber?

#solutionbox[
25 grams
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?

#solutionbox[
2,375 mg
]
]

#examplebox("Example 6")[][
Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat?

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [the percent of the total calories from fat],
  [#emph[Step 3. Name] what we are looking for.], [], [],
  [Choose a variable to represent it.], [], [Let #math.equation(block: false, alt: "p equals")[$p =$] percent of fat.],
  [#emph[Step 4. Translate.] Write a sentence that gives the information to find it.], [], [#figure(figph[A math problem on a white background asks, 'What percent of 480 is 240?' Light blue brackets underline individual words and numbers in the question.], alt: "A math problem on a white background asks, 'What percent of 480 is 240?' Light blue brackets underline individual words and numbers in the question.", caption: none)],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation shows 'p' multiplied by 480 equals 240, where 'p' is likely intended to be 0.5.], alt: "A mathematical equation shows 'p' multiplied by 480 equals 240, where 'p' is likely intended to be 0.5.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[A cropped image showing the text '480 p = 240' on a white background.], alt: "A cropped image showing the text '480 p = 240' on a white background.", caption: none)],
  [Divide by 480.], [], [#figure(figph[The image displays the text 'p = 0.5' in the top right corner against a white background.], alt: "The image displays the text 'p = 0.5' in the top right corner against a white background.", caption: none)],
  [Put in a percent form.], [], [#figure(figph[The image features the text 'p = 50%' in black, sans-serif font, against a plain white background, indicating a probability or percentage value.], alt: "The image features the text 'p = 50%' in black, sans-serif font, against a plain white background, indicating a probability or percentage value.", caption: none)],
  [#emph[Step 6. Check.] Does this make sense?], [], [],
  [Yes, 240 is half of 480, so 50% makes sense.], [], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [], [Of the total calories in each brownie, 50% is fat.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve. Round to the nearest whole percent.

Veronica is planning to make muffins from a mix. The package says each muffin will be 230 calories and 60 calories will be from fat. What percent of the total calories is from fat?

#solutionbox[
26%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve. Round to the nearest whole percent.

The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat?

#solutionbox[
40%
]
]


=== Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.

To find the #strong[percent increase], first we find the amount of increase, the difference of the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Find the Percent Increase.]

+ Find the amount of increase. #linebreak() 
#math.equation(block: false, alt: "new amount minus original amount equals increase")[$"new amount" − "original amount" = "increase"$]
+ Find the percent increase. #linebreak() 
The increase is what percent of the original amount?
]

#examplebox("Example 7")[][
In 2011, the California governor proposed raising community college fees from \$26 a unit to \$36 a unit. Find the percent increase. (Round to the nearest tenth of a percent.)

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [the percent increase],
  [#emph[Step 3. Name] what we are looking for.], [], [],
  [Choose a variable to represent it.], [], [Let #math.equation(block: false, alt: "p equals")[$p =$] the percent.],
  [#emph[Step 4. Translate.] Write a sentence that gives the information to find it.], [], [],
  [First find the amount of increase.], [], [new amount − original amount = increase],
  [], [], [#math.equation(block: false, alt: "36 minus 26 equals 10")[$36 − 26 = 10$]],
  [Find the percent.], [], [Increase is what percent of the original amount?],
  [], [], [#figure(figph[A mathematical problem is displayed on a white background, asking: '10 is what percent of 26?' Light blue brackets are placed beneath each word and number, seemingly to highlight or group them.], alt: "A mathematical problem is displayed on a white background, asking: '10 is what percent of 26?' Light blue brackets are placed beneath each word and number, seemingly to highlight or group them.", caption: none)],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation is displayed, showing '10 = p . 26' against a plain white background. The numbers and symbols are clearly visible.], alt: "A mathematical equation is displayed, showing '10 = p . 26' against a plain white background. The numbers and symbols are clearly visible.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[A mathematical equation on a white background states '10 = 26p'.], alt: "A mathematical equation on a white background states '10 = 26p'.", caption: none)],
  [Divide by 26.], [], [#figure(figph[The equation 0.385 = p is displayed, indicating that the variable p is equal to the decimal value 0.385.], alt: "The equation 0.385 = p is displayed, indicating that the variable p is equal to the decimal value 0.385.", caption: none)],
  [Change to percent form; round to the nearest tenth.], [], [#figure(figph[A mathematical equation displays '38.5% = p' in black text on a white background.], alt: "A mathematical equation displays '38.5% = p' in black text on a white background.", caption: none)],
  [#emph[Step 6. Check.] Does this make sense?], [], [],
  [Yes, 38.4% is close to #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$], and 10 is close to #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] of 26.], [], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [], [The new fees represent a 38.5% increase over the old fees.],
))

Notice that we rounded the division to the nearest thousandth in order to round the percent to the nearest tenth.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the percent increase. (Round to the nearest tenth of a percent.)

In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents.

#solutionbox[
8.8%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the percent increase.

In 1995, the standard bus fare in Chicago was \$1.50. In 2008, the standard bus fare was \$2.25.

#solutionbox[
50%
]
]

Finding the #strong[percent decrease] is very similar to finding the percent increase, but now the amount of decrease is the difference of the original amount and the new amount. Then we find what percent the amount of decrease is of the original amount.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Find the Percent Decrease.]

+ Find the amount of decrease. #linebreak() 
#math.equation(block: false, alt: "original amount minus new amount equals decrease")[$"original amount" − "new amount" = "decrease"$]
+ Find the percent decrease. #linebreak() 
Decrease is what percent of the original amount?
]

#examplebox("Example 8")[][
The average price of a gallon of gas in one city in June 2014 was \$3.71. The average price in that city in July was \$3.64. Find the percent decrease.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [the percent decrease],
  [#emph[Step 3. Name] what we are looking for.], [], [],
  [Choose a variable to represent that quantity.], [], [Let #math.equation(block: false, alt: "p equals")[$p =$] the percent decrease.],
  [#emph[Step 4. Translate.] Write a sentence that gives the information to find it.], [], [],
  [First find the amount of decrease.], [], [#math.equation(block: false, alt: "3.71 minus 3.64 equals 0.07")[$3.71 − 3.64 = 0.07$]],
  [Find the percent.], [], [Decrease is what percent of the original amaount?],
  [], [], [#figure(figph[A mathematical question written in black text on a white background asks: '0.07 is what percent of 3.71?' Light blue brackets underline different parts of the question.], alt: "A mathematical question written in black text on a white background asks: '0.07 is what percent of 3.71?' Light blue brackets underline different parts of the question.", caption: none)],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation is displayed on a white background: 0.07 = p ××3.71.], alt: "A mathematical equation is displayed on a white background: 0.07 = p ××3.71.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[The image shows a mathematical equation on a white background. The equation presented is '0.07 = 3.71 p'.], alt: "The image shows a mathematical equation on a white background. The equation presented is '0.07 = 3.71 p'.", caption: none)],
  [Divide by 3.71.], [], [#figure(figph[The image displays the equation '0.019 = p' in black text against a plain white background.], alt: "The image displays the equation '0.019 = p' in black text against a plain white background.", caption: none)],
  [Change to percent form; round to the nearest tenth.], [], [#figure(figph[The image displays the mathematical expression '1.9% = p' in black text against a plain white background, indicating a percentage value assigned to the variable p.], alt: "The image displays the mathematical expression '1.9% = p' in black text against a plain white background, indicating a percentage value assigned to the variable p.", caption: none)],
  [#emph[Step 6. Check.] Does this make sense?], [], [],
  [Yes, if the original price was \$4, a 2% decrease would be 8 cents.], [], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [], [The price of gas decreased 1.9%.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the percent decrease. (Round to the nearest tenth of a percent.)

The population of North Dakota was about 672,000 in 2010. The population is projected to be about 630,000 in 2020.

#solutionbox[
6.3%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the percent decrease.

Last year, Sheila’s salary was \$42,000. Because of furlough days, this year, her salary was \$37,800.

#solutionbox[
10%
]
]


=== Solve Simple Interest Applications

Do you know that banks pay you to keep your money? The money a customer puts in the bank is called the #strong[principal], #emph[P], and the money the bank pays the customer is called the #strong[interest]. The interest is computed as a certain percent of the principal; called the #strong[rate of interest], #emph[r]. We usually express rate of interest as a percent per year, and we calculate it by using the decimal equivalent of the percent. The variable #emph[t], (for #emph[time]) represents the number of years the money is in the account.

To find the interest we use the simple interest formula, #math.equation(block: false, alt: "I equals P r t .")[$I = P r t .$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simple Interest]

If an amount of money, #emph[P], called the principal, is invested for a period of #emph[t] years at an annual interest rate #emph[r], the amount of interest, #emph[I], earned is

#math.equation(block: true, alt: "I equals P r t, where, I, equals, interest; P, equals, principal; r, equals, rate; t, equals, time")[$I = P r t &  &  & "where" &  &  & I & = & "interest" \ P & = & "principal" \ r & = & "rate" \ t & = & "time"$]Interest earned according to this formula is called #strong[simple interest].
]

Interest may also be calculated another way, called compound interest. This type of interest will be covered in later math classes.

The formula we use to calculate simple interest is #math.equation(block: false, alt: "I equals P r t .")[$I = P r t .$] To use the formula, we substitute in the values the problem gives us for the variables, and then solve for the unknown variable. It may be helpful to organize the information in a chart.

#examplebox("Example 9")[][
Nathaly deposited \$12,500 in her bank account where it will earn 4% interest. How much interest will Nathaly earn in 5 years?

#math.equation(block: true, alt: "I, equals, ?; P, equals, $12,500; r, equals, 4 %; t, equals, 5 years")[$I & = & ? \ P & = & \$12,500 \ r & = & 4 "%" \ t & = & 5 #h(0.2em) "years"$]#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what we are looking for.], [the amount of interest earned],
  [#emph[Step 3. Name] what we are looking for.  #linebreak() Choose a variable to represent that quantity.], [Let #math.equation(block: false, alt: "I equals")[$I =$] the amount of interest.],
  [#emph[Step 4. Translate] into an equation.
     #linebreak() Write the formula.
	 #linebreak() Substitute in the given information.], [#math.equation(block: false, alt: "I equals P r t; I equals open parenthesis 12,500 close parenthesis open parenthesis .04 close parenthesis open parenthesis 5 close parenthesis")[$I = P r t \ I = ( 12,500 ) ( .04 ) ( 5 )$]],
  [#emph[Step 5. Solve] the equation.], [#math.equation(block: false, alt: "I equals 2,500")[$I = 2,500$]],
  [#emph[Step 6. Check]: Does this make sense?
 	 #linebreak() Is \$2,500 a reasonable interest on \$12,500? Yes.], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [The interest is \$2,500.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Areli invested a principal of \$950 in her bank account with interest rate 3%. How much interest did she earn in 5 years?

#solutionbox[
\$142.50
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Susana invested a principal of \$36,000 in her bank account with interest rate 6.5%. How much interest did she earn in 3 years?

#solutionbox[
\$7,020
]
]

There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we don’t know the rate. To find the rate, we use the simple interest formula, substitute in the given values for the principal and time, and then solve for the rate.

#examplebox("Example 10")[][
Loren loaned his brother \$3,000 to help him buy a car. In 4 years his brother paid him back the \$3,000 plus \$660 in interest. What was the rate of interest?

#math.equation(block: true, alt: "I, equals, $660; P, equals, $3,000; r, equals, ?; t, equals, 4 years")[$I & = & \$660 \ P & = & \$3,000 \ r & = & ? \ t & = & 4 #h(0.2em) "years"$]#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what we are looking for.], [the rate of interest],
  [#emph[Step 3. Name] what we are looking for.  #linebreak() Choose a variable to represent that quantity.], [Let #math.equation(block: false, alt: "r equals")[$r =$] the rate of interest.],
  [#emph[Step 4. Translate] into an equation.
     #linebreak() Write the formula.
	 #linebreak() Substitute in the given information.], [#math.equation(block: false, alt: "I equals P r t; 660 equals open parenthesis 3,000 close parenthesis r open parenthesis 4 close parenthesis")[$I = P r t \ 660 = ( 3,000 ) r ( 4 )$]],
  [#emph[Step 5. Solve] the equation.
     #linebreak() Divide.
	 #linebreak() Change to percent form.], [#math.equation(block: false, alt: "660 equals open parenthesis 12,000 close parenthesis r; 0.055 equals r; 5.5% equals r")[$660 = ( 12,000 ) r \ 0.055 = r \ 5.5% = r$]],
  [#emph[Step 6. Check]: Does this make sense?
 	 #linebreak() #math.equation(block: false, alt: "I equals P r t; 660 equals ? open parenthesis 3,000 close parenthesis open parenthesis 0.055 close parenthesis open parenthesis 4 close parenthesis; 660 equals 660 ✓")[$I = P r t \ 660 limits(=)^(?) ( 3,000 ) ( 0.055 ) ( 4 ) \ 660 = 660 ✓$]], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [The rate of interest was 5.5%.],
))
]
]

Notice that in this example, Loren’s brother paid Loren interest, just like a bank would have paid interest if Loren invested his money there.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Jim loaned his sister \$5,000 to help her buy a house. In 3 years, she paid him the \$5,000, plus \$900 interest. What was the rate of interest?

#solutionbox[
6%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Hang borrowed \$7,500 from her parents to pay her tuition. In 5 years, she paid them \$1,500 interest in addition to the \$7,500 she borrowed. What was the rate of interest?

#solutionbox[
4%
]
]

#examplebox("Example 11")[][
Eduardo noticed that his new car loan papers stated that with a 7.5% interest rate, he would pay \$6,596.25 in interest over 5 years. How much did he borrow to pay for his car?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what we are looking for.], [the amount borrowed (the principal)],
  [#emph[Step 3. Name] what we are looking for.  #linebreak() Choose a variable to represent that quantity.], [Let #math.equation(block: false, alt: "P equals")[$P =$] principal borrowed.],
  [#emph[Step 4. Translate] into an equation.
     #linebreak() Write the formula.
	 #linebreak() Substitute in the given information.], [#math.equation(block: false, alt: "I, equals, P r t; 6,596.25, equals, P open parenthesis 0.075 close parenthesis open parenthesis 5 close parenthesis")[$I & = & P r t \ 6,596.25 & = & P ( 0.075 ) ( 5 )$]],
  [#emph[Step 5. Solve] the equation.
     #linebreak() Divide.], [#math.equation(block: false, alt: "6,596.25, equals, 0.375 P; 17,590, equals, P")[$6,596.25 & = & 0.375 P \ 17,590 & = & P$]],
  [#emph[Step 6. Check]: Does this make sense?
 	 #linebreak() #math.equation(block: false, alt: "I, equals, P r t; 6,596.25, equals ?, open parenthesis 17,590 close parenthesis open parenthesis 0.075 close parenthesis open parenthesis 5 close parenthesis; 6,596.25, equals, 6,596.25 ✓")[$#h(2.5em) I & = & P r t \ #h(2.5em) 6,596.25 & limits(=)^(?) & ( 17,590 ) ( 0.075 ) ( 5 ) \ #h(2.5em) 6,596.25 & = & 6,596.25 ✓$]], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [The principal was \$17,590.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Sean’s new car loan statement said he would pay \$4,866.25 in interest from an interest rate of 8.5% over 5 years. How much did he borrow to buy his new car?

#solutionbox[
\$11,450
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
In 5 years, Gloria’s bank account earned \$2,400 interest at 5%. How much had she deposited in the account?

#solutionbox[
\$9,600
]
]


=== Solve Applications with Discount or Mark-up

Applications of discount are very common in retail settings. When you buy an item on sale, the original price has been discounted by some dollar amount. The #strong[discount rate], usually given as a percent, is used to determine the amount of the discount. To determine the #strong[amount of discount], we multiply the discount rate by the original price.

We summarize the discount model in the box below.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Discount]

#math.equation(block: true, alt: "amount of discount equals discount rate times original price; sale price equals original price minus amount of discount")[$"amount of discount" = "discount rate" #h(0.2em) × #h(0.2em) "original price" \ "sale price" = "original price" − "amount of discount"$]Keep in mind that the sale price should always be less than the original price.
]

#examplebox("Example 12")[][
Elise bought a dress that was discounted 35% off of the original price of \$140. What was ⓐ the amount of discount and ⓑ the sale price of the dress?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ 
		 #linebreak()  #math.equation(block: false, alt: "Original price, equals, $ 140; Discount rate, equals, 35 %; Discount, equals, ?")[$"Original price" & = & "$" 140 \ "Discount rate" & = & 35 "%" \ "Discount" & = & ?$]], []),
  [#emph[Step 1. Read] the problem.], [],
  [#emph[Step 2. Identify] what we are looking for.], [the amount of discount],
  [#emph[Step 3. Name] what we are looking for.  #linebreak() Choose a variable to represent that quantity.], [Let #math.equation(block: false, alt: "d equals")[$d =$] the amount of discount.],
  [#emph[Step 4. Translate] into an equation.
     #linebreak() Write a sentence that gives the information to find it.
     #linebreak() Translate into an equation.], [#linebreak() The discount is 35% of \$140.
    	 #linebreak() #math.equation(block: false, alt: "d equals 0.35 open parenthesis 140 close parenthesis")[$d = 0.35 ( 140 )$]],
  [#emph[Step 5. Solve] the equation.], [#math.equation(block: false, alt: "d equals 49")[$d = 49$]],
  [#emph[Step 6. Check]: Does this make sense?
 	 #linebreak() Is a \$49 discount reasonable for a \$140 dress? Yes.], [],
  [#emph[Step 7. Write] a complete sentence to answer the question.], [The amount of discount was \$49.],
))

ⓑ  #linebreak() 
Read the problem again. #linebreak()  #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Identify] what we are looking for.], [], [the sale price of the dress]),
  [#emph[Step 2. Name] what we are looking for.], [], [],
  [Choose a variable to represent that quantity.], [], [Let #math.equation(block: false, alt: "s equals")[$s =$] the sale price.],
  [#emph[Step 3. Translate] into an equation.], [], [],
  [Write a sentence that gives the information to find it.], [], [#figure(figph[The image shows the phrase 'The sale price is the \$140 minus the \$49 discount' with light blue brackets under different parts of the sentence, breaking down the equation.], alt: "The image shows the phrase 'The sale price is the $140 minus the $49 discount' with light blue brackets under different parts of the sentence, breaking down the equation.", caption: none)],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation displaying 'S = 140 - 49' on a white background.], alt: "A mathematical equation displaying 'S = 140 - 49' on a white background.", caption: none)],
  [#emph[Step 4. Solve] the equation.], [], [#figure(figph[The image displays the mathematical expression 'S = 91' centered on a plain white background, indicating a variable 'S' is assigned the value 91.], alt: "The image displays the mathematical expression 'S = 91' centered on a plain white background, indicating a variable 'S' is assigned the value 91.", caption: none)],
  [#emph[Step 5. Check.] Does this make sense?], [], [],
  [Is the sale price less than the original price?], [], [],
  [Yes, \$91 is less than \$140.], [], [],
  [#emph[Step 6. Answer] the question with a complete sentence.], [], [The sale price of the dress was \$91.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find ⓐ the amount of discount and ⓑ the sale price:

Sergio bought a belt that was discounted 40% from an original price of \$29.

#solutionbox[
ⓐ \$11.60 ⓑ \$17.40
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find ⓐ the amount of discount and ⓑ the sale price:

Oscar bought a barbecue that was discounted 65% from an original price of \$395.

#solutionbox[
ⓐ \$256.75 ⓑ \$138.25
]
]

There may be times when we know the original price and the sale price, and we want to know the #strong[discount rate]. To find the discount rate, first we will find the amount of discount and then use it to compute the rate as a percent of the original price.  will show this case.

#examplebox("Example 13")[][
Jeannette bought a swimsuit at a sale price of \$13.95. The original price of the swimsuit was \$31. Find the ⓐ amount of discount and ⓑ discount rate.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ 
		 #linebreak()  #math.equation(block: false, alt: "Original price, equals, $ 31; Discount, equals, ?; Sale Price, equals, $ 13.95")[$"Original price" & = & "$" 31 \ "Discount" & = & ? \ "Sale Price" & = & "$" 13.95$]], []),
  [#emph[Step 1. Read] the problem.], [],
  [#emph[Step 2. Identify] what we are looking for.], [the amount of discount],
  [#emph[Step 3. Name] what we are looking for.  #linebreak() Choose a variable to represent that quantity.], [Let #math.equation(block: false, alt: "d equals")[$d =$] the amount of discount.],
  [#emph[Step 4. Translate] into an equation.
     #linebreak() Write a sentence that gives the information to find it.
     #linebreak() Translate into an equation.], [#linebreak() The discount is the difference between the original price and the sale price.
    	 #linebreak() #math.equation(block: false, alt: "d equals 31 minus 13.95")[$d = 31 − 13.95$]],
  [#emph[Step 5. Solve] the equation.], [#math.equation(block: false, alt: "d equals 17.05")[$d = 17.05$]],
  [#emph[Step 6. Check]: Does this make sense?
 	 #linebreak() Is 17.05 less than 31? Yes.], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [The amount of discount was \$17.05.],
))

ⓑ  #linebreak() 
Read the problem again. #linebreak()  #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Identify] what we are looking for.], [], [the discount rate]),
  [#emph[Step 2. Name] what we are looking for.], [], [],
  [Choose a variable to represent it.], [], [Let #math.equation(block: false, alt: "r equals")[$r =$] the discount rate.],
  [#emph[Step 3. Translate] into an equation.], [], [],
  [Write a sentence that gives the information to find it.], [], [#figure(figph[A mathematical question asks: 'The discount of \$17.05 is what percent of \$31?'], alt: "A mathematical question asks: 'The discount of $17.05 is what percent of $31?'", caption: none)],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation shows '17.05 = r \* 31' in black text on a white background.], alt: "A mathematical equation shows '17.05 = r * 31' in black text on a white background.", caption: none)],
  [#emph[Step 4. Solve] the equation.], [], [#figure(figph[A mathematical equation is displayed on a white background, reading '17.05 = 31r'.], alt: "A mathematical equation is displayed on a white background, reading '17.05 = 31r'.", caption: none)],
  [Divide both sides by 31.], [], [#figure(figph[The image displays mathematical text, showing the equation '0.55 = r' in black font against a plain white background, centered in the frame.], alt: "The image displays mathematical text, showing the equation '0.55 = r' in black font against a plain white background, centered in the frame.", caption: none)],
  [Change to percent form.], [], [#figure(figph[The image shows the text 'r = 55%' centered on a plain white background, indicating a numerical value or percentage.], alt: "The image shows the text 'r = 55%' centered on a plain white background, indicating a numerical value or percentage.", caption: none)],
  [#emph[Step 5. Check.] Does this make sense?], [], [],
  [Is \$17.05 equal to 55% of \$31?], [], [],
  [#math.equation(block: false, alt: "17.05 ≟ 0.55 open parenthesis 31 close parenthesis")[$17.05 ≟ 0.55 ( 31 )$]], [], [],
  [#math.equation(block: false, alt: "17.05 equals 17.05 ✓")[$17.05 = 17.05 ✓$]], [], [],
  [#emph[Step 6. Answer] the question with a complete sentence.], [], [The rate of discount was 55%.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find ⓐ the amount of discount and ⓑ the discount rate.

Lena bought a kitchen table at the sale price of \$375.20. The original price of the table was \$560.

#solutionbox[
ⓐ \$184.80 ⓑ 33%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find ⓐ the amount of discount and ⓑ the discount rate.

Nick bought a multi-room air conditioner at a sale price of \$340. The original price of the air conditioner was \$400.

#solutionbox[
ⓐ \$60 ⓑ 15%
]
]

Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the #strong[original cost]. The retailer then adds a #strong[mark-up] to the original cost to get the #strong[list price], the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.

We summarize the mark-up model in the box below.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Mark-Up]

#math.equation(block: true, alt: "amount of mark-up equals mark-up rate times original cost; list price equals original cost plus amount of mark up")[$"amount of mark-up" = "mark-up rate" #h(0.2em) × #h(0.2em) "original cost" \ "list price" = "original cost" + "amount of mark up"$]Keep in mind that the list price should always be more than the original cost.
]

#examplebox("Example 14")[][
Adam’s art gallery bought a photograph at original cost \$250. Adam marked the price up 40%. Find the ⓐ amount of mark-up and ⓑ the list price of the photograph.

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [the amount of mark-up],
  [#emph[Step 3. Name] what we are looking for.], [], [],
  [Choose a variable to represent it.], [], [Let #math.equation(block: false, alt: "m equals")[$m =$] the amount of markup.],
  [#emph[Step 4. Translate] into an equation.], [], [],
  [Write a sentence that gives the information to find it.], [], [#figure(figph[The text states, 'The mark-up is 40% of the \$250 original cost,' with sections of the sentence underlined to indicate different components of a calculation or definition.], alt: "The text states, 'The mark-up is 40% of the $250 original cost,' with sections of the sentence underlined to indicate different components of a calculation or definition.", caption: none)],
  [Translate into an equation.], [], [#figure(figph[The image shows the equation m = 0.40 \* 250, likely calculating a mass or a value based on a percentage or factor.], alt: "The image shows the equation m = 0.40 * 250, likely calculating a mass or a value based on a percentage or factor.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[The image displays the simple equation 'm = 100' in a clear, dark grey font against a plain white background, likely representing a variable assignment or a value in a mathematical or programming context.], alt: "The image displays the simple equation 'm = 100' in a clear, dark grey font against a plain white background, likely representing a variable assignment or a value in a mathematical or programming context.", caption: none)],
  [#emph[Step 6. Check.] Does this make sense?], [], [],
  [Yes, 40% is less than one-half and 100 is less than half of 250.], [], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [], [The mark-up on the phtograph was \$100.],
))

ⓑ  #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem again.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [the list price],
  [#emph[Step 3. Name] what we are looking for.], [], [],
  [Choose a variable to represent it.], [], [Let #math.equation(block: false, alt: "p equals")[$p =$] the list price.],
  [#emph[Step 4. Translate] into an equation.], [], [],
  [Write a sentence that gives the information to find it.], [], [#figure(figph[The list price is original cost plus the mark-up.], alt: "The list price is original cost plus the mark-up.", caption: none)],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation shows 'p = 250 + 100' in black text on a white background.], alt: "A mathematical equation shows 'p = 250 + 100' in black text on a white background.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[The image displays the text 'p = 350' centered on a plain white background, indicating a mathematical or scientific variable assignment.], alt: "The image displays the text 'p = 350' centered on a plain white background, indicating a mathematical or scientific variable assignment.", caption: none)],
  [#emph[Step 6. Check.] Does this make sense?], [], [],
  [Is the list price more than the net price? #linebreak() Is \$350 more than \$250? Yes.], [], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [], [The list price of the photograph was \$350.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find ⓐ the amount of mark-up and ⓑ the list price.

Jim’s music store bought a guitar at original cost \$1,200. Jim marked the price up 50%.

#solutionbox[
ⓐ \$600 ⓑ \$1,800
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find ⓐ the amount of mark-up and ⓑ the list price.

The Auto Resale Store bought Pablo’s Toyota for \$8,500. They marked the price up 35%.

#solutionbox[
ⓐ \$2,975 ⓑ \$11,475
]
]


=== Key Concepts

- #strong[Percent Increase] To find the percent increase:
+ Find the amount of increase. #math.equation(block: false, alt: "increase equals new amount minus original amount")[$"increase" = "new amount" − "original" #h(0.2em) "amount"$]
+ Find the percent increase. Increase is what percent of the original amount?
- #strong[Percent Decrease] To find the percent decrease:
+ Find the amount of decrease. #math.equation(block: false, alt: "decrease equals original amount minus new amount")[$"decrease" = "original amount" − "new" #h(0.2em) "amount"$]
+ Find the percent decrease. Decrease is what percent of the original amount?
- #strong[Simple Interest] If an amount of money, #emph[P], called the principal, is invested for a period of #emph[t] years at an annual interest rate #emph[r], the amount of interest, #emph[I,] earned is #linebreak() 
#math.equation(block: true, alt: "I, equals, P r t; where I, equals, interest; P, equals, principal; r, equals, rate; t, equals, time")[$I & = & P r t \ "where" #h(1em) I & = & "interest" \ P & = & "principal" \ r & = & "rate" \ t & = & "time"$]
- #strong[Discount]
- amount of discount is discount rate #math.equation(block: false, alt: "times")[$·$] original price
- sale price is original price – discount
- #strong[Mark-up]
- amount of mark-up is mark-up rate #math.equation(block: false, alt: "times")[$·$] original cost
- list price is original cost + mark up


==== Practice Makes Perfect

#strong[Translate and Solve Basic Percent Equations]

In the following exercises, translate and solve.

What number is 45% of 120?

#solutionbox[
54
]

What number is 65% of 100?

What number is 24% of 112?

#solutionbox[
26.88
]

What number is 36% of 124?

250% of 65 is what number?

#solutionbox[
162.5
]

150% of 90 is what number?

800% of 2250 is what number?

#solutionbox[
18,000
]

600% of 1740 is what number?

28 is 25% of what number?

#solutionbox[
112
]

36 is 25% of what number?

81 is 75% of what number?

#solutionbox[
108
]

93 is 75% of what number?

8.2% of what number is \$2.87?

#solutionbox[
\$35
]

6.4% of what number is \$2.88?

11.5% of what number is \$108.10?

#solutionbox[
\$940
]

12.3% of what number is \$92.25?

What percent of 260 is 78?

#solutionbox[
30%
]

What percent of 215 is 86?

What percent of 1500 is 540?

#solutionbox[
36%
]

What percent of 1800 is 846?

30 is what percent of 20?

#solutionbox[
150%
]

50 is what percent of 40?

840 is what percent of 480?

#solutionbox[
175%
]

790 is what percent of 395?

#strong[Solve Percent Applications]

In the following exercises, solve.

Geneva treated her parents to dinner at their favorite restaurant. The bill was \$74.25. Geneva wants to leave 16% of the total bill as a tip. How much should the tip be?

#solutionbox[
\$11.88
]

When Hiro and his co-workers had lunch at a restaurant near their work, the bill was \$90.50. They want to leave 18% of the total bill as a tip. How much should the tip be?

Trong has 12% of each paycheck automatically deposited to his savings account. His last paycheck was \$2165. How much money was deposited to Trong’s savings account?

#solutionbox[
\$259.80
]

Cherise deposits 8% of each paycheck into her retirement account. Her last paycheck was \$1,485. How much did Cherise deposit into her retirement account?

One serving of oatmeal has eight grams of fiber, which is 33% of the recommended daily amount. What is the total recommended daily amount of fiber?

#solutionbox[
24.2 g
]

One serving of trail mix has 67 grams of carbohydrates, which is 22% of the recommended daily amount. What is the total recommended daily amount of carbohydrates?

A bacon cheeseburger at a popular fast food restaurant contains 2070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?

#solutionbox[
2407 mg
]

A grilled chicken salad at a popular fast food restaurant contains 650 milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount of sodium?

After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa’s original weight?

#solutionbox[
175 lb.
]

Tricia got a 6% raise on her weekly salary. The raise was \$30 per week. What was her original salary?

Yuki bought a dress on sale for \$72. The sale price was 60% of the original price. What was the original price of the dress?

#solutionbox[
\$120
]

Kim bought a pair of shoes on sale for \$40.50. The sale price was 45% of the original price. What was the original price of the shoes?

Tim left a \$9 tip for a \$50 restaurant bill. What percent tip did he leave?

#solutionbox[
18%
]

Rashid left a \$15 tip for a \$75 restaurant bill. What percent tip did he leave?

The nutrition fact sheet at a fast food restaurant says the fish sandwich has 380 calories, and 171 calories are from fat. What percent of the total calories is from fat?

#solutionbox[
45%
]

The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has 190 calories, and 114 calories are from fat. What percent of the total calories is from fat?

Emma gets paid \$3,000 per month. She pays \$750 a month for rent. What percent of her monthly pay goes to rent?

#solutionbox[
25%
]

Dimple gets paid \$3,200 per month. She pays \$960 a month for rent. What percent of her monthly pay goes to rent?

#strong[Find Percent Increase and Percent Decrease]

In the following exercises, solve.

Tamanika got a raise in her hourly pay, from \$15.50 to \$17.36. Find the percent increase.

#solutionbox[
12%
]

Ayodele got a raise in her hourly pay, from \$24.50 to \$25.48. Find the percent increase.

Annual student fees at the University of California rose from about \$4,000 in 2000 to about \$12,000 in 2010. Find the percent increase.

#solutionbox[
200%
]

The price of a share of one stock rose from \$12.50 to \$50. Find the percent increase.

According to #emph[Time] magazine annual global seafood consumption rose from 22 pounds per person in the 1960s to 38 pounds per person in 2011. Find the percent increase. (Round to the nearest tenth of a percent.)

#solutionbox[
72.7%
]

In one month, the median home price in the Northeast rose from \$225,400 to \$241,500. Find the percent increase. (Round to the nearest tenth of a percent.)

A grocery store reduced the price of a loaf of bread from \$2.80 to \$2.73. Find the percent decrease.

#solutionbox[
2.5%
]

The price of a share of one stock fell from \$8.75 to \$8.54. Find the percent decrease.

Hernando’s salary was \$49,500 last year. This year his salary was cut to \$44,055. Find the percent decrease.

#solutionbox[
11%
]

In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.

In 1 month, the median home price in the West fell from \$203,400 to \$192,300. Find the percent decrease. (Round to the nearest tenth of a percent.)

#solutionbox[
5.5%
]

Sales of video games and consoles fell from \$1,150 million to \$1,030 million in 1 year. Find the percent decrease. (Round to the nearest tenth of a percent.)

#strong[Solve Simple Interest Applications]

In the following exercises, solve.

Casey deposited \$1,450 in a bank account with interest rate 4%. How much interest was earned in two years?

#solutionbox[
\$116
]

Terrence deposited \$5,720 in a bank account with interest rate 6%. How much interest was earned in 4 years?

Robin deposited \$31,000 in a bank account with interest rate 5.2%. How much interest was earned in 3 years?

#solutionbox[
\$4,836
]

Carleen deposited \$16,400 in a bank account with interest rate 3.9%. How much interest was earned in 8 years?

Hilaria borrowed \$8,000 from her grandfather to pay for college. Five years later, she paid him back the \$8,000, plus \$1,200 interest. What was the rate of interest?

#solutionbox[
3%
]

Kenneth loaned his niece \$1,200 to buy a computer. Two years later, she paid him back the \$1,200, plus \$96 interest. What was the rate of interest?

Lebron loaned his daughter \$20,000 to help her buy a condominium. When she sold the condominium four years later, she paid him the \$20,000, plus \$3,000 interest. What was the rate of interest?

#solutionbox[
3.75%
]

Pablo borrowed \$50,000 to start a business. Three years later, he repaid the \$50,000, plus \$9,375 interest. What was the rate of interest?

In 10 years, a bank account that paid 5.25% earned \$18,375 interest. What was the principal of the account?

#solutionbox[
\$35,000
]

In 25 years, a bond that paid 4.75% earned \$2,375 interest. What was the principal of the bond?

Joshua’s computer loan statement said he would pay \$1,244.34 in interest for a 3-year loan at 12.4%. How much did Joshua borrow to buy the computer?

#solutionbox[
\$3,345
]

Margaret’s car loan statement said she would pay \$7,683.20 in interest for a 5-year loan at 9.8%. How much did Margaret borrow to buy the car?

#strong[Solve Applications with Discount or Mark-up]

In the following exercises, find the sale price.

Perla bought a cell phone that was on sale for \$50 off. The original price of the cell phone was \$189.

#solutionbox[
\$139
]

Sophie saw a dress she liked on sale for \$15 off. The original price of the dress was \$96.

Rick wants to buy a tool set with original price \$165. Next week the tool set will be on sale for \$40 off.

#solutionbox[
\$125
]

Angelo’s store is having a sale on televisions. One television, with original price \$859, is selling for \$125 off.

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

Janelle bought a beach chair on sale at 60% off. The original price was \$44.95.

#solutionbox[
ⓐ \$26.97 ⓑ \$17.98
]

Errol bought a skateboard helmet on sale at 40% off. The original price was \$49.95.

Kathy wants to buy a camera that lists for \$389. The camera is on sale with a 33% discount.

#solutionbox[
ⓐ \$128.37 ⓑ \$260.63
]

Colleen bought a suit that was discounted 25% from an original price of \$245.

Erys bought a treadmill on sale at 35% off. The original price was \$949.95 (round to the nearest cent.)

#solutionbox[
ⓐ \$332.48 ⓑ \$617.47
]

Jay bought a guitar on sale at 45% off. The original price was \$514.75 (round to the nearest cent.)

In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)

Larry and Donna bought a sofa at the sale price of \$1,344. The original price of the sofa was \$1,920.

#solutionbox[
ⓐ \$576 ⓑ 30%
]

Hiroshi bought a lawnmower at the sale price of \$240. The original price of the lawnmower is \$300.

Patty bought a baby stroller on sale for \$301.75. The original price of the stroller was \$355.

#solutionbox[
ⓐ \$53.25 ⓑ 15%
]

Bill found a book he wanted on sale for \$20.80. The original price of the book was \$32.

Nikki bought a patio set on sale for \$480. The original price was \$850. To the nearest tenth of a percent, what was the rate of discount?

#solutionbox[
ⓐ \$370 ⓑ 43.5%
]

Stella bought a dinette set on sale for \$725. The original price was \$1,299. To the nearest tenth of a percent, what was the rate of discount?

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Daria bought a bracelet at original cost \$16 to sell in her handicraft store. She marked the price up 45%.

#solutionbox[
ⓐ \$7.20 ⓑ \$23.20
]

Regina bought a handmade quilt at original cost \$120 to sell in her quilt store. She marked the price up 55%.

Tom paid \$0.60 a pound for tomatoes to sell at his produce store. He added a 33% mark-up.

#solutionbox[
ⓐ \$0.20 ⓑ \$0.80
]

Flora paid her supplier \$0.74 a stem for roses to sell at her flower shop. She added an 85% mark-up.

Alan bought a used bicycle for \$115. After re-conditioning it, he added 225% mark-up and then advertised it for sale.

#solutionbox[
ⓐ \$258.75 ⓑ \$373.75
]

Michael bought a classic car for \$8,500. He restored it, then added 150% mark-up before advertising it for sale.


==== Everyday Math

#strong[Leaving a Tip] At the campus coffee cart, a medium coffee costs \$1.65. MaryAnne brings \$2.00 with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?

#solutionbox[
21.2%
]

#strong[Splitting a Bill] Four friends went out to lunch and the bill came to \$53.75. They decided to add enough tip to make a total of \$64, so that they could easily split the bill evenly among themselves. What percent tip did they leave?


==== Writing Exercises

Without solving the problem “44 is 80% of what number” think about what the solution might be. Should it be a number that is greater than 44 or less than 44? Explain your reasoning.

#solutionbox[
The number should be greater than 44. Since 80% equals 0.8 in decimal form, 0.8 is less than one, and we must multiply the number by 0.8 to get 44, the number must be greater than 44.
]

Without solving the problem “What is 20% of 300?” think about what the solution might be. Should it be a number that is greater than 300 or less than 300? Explain your reasoning.

After returning from vacation, Alex said he should have packed 50% fewer shorts and 200% more shirts. Explain what Alex meant.

#solutionbox[
He meant that he should have packed half the shorts and twice the shirts.
]

Because of road construction in one city, commuters were advised to plan that their Monday morning commute would take 150% of their usual commuting time. Explain what this means.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This is a table that has two rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “translate and solve basic percent equations,” “solve percent applications,” “find percent increase and percent decrease,” “solve simple interest applications,” and “solve applications with discount or mark-up.” The rest of the cells are blank.], alt: "This is a table that has two rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “translate and solve basic percent equations,” “solve percent applications,” “find percent increase and percent decrease,” “solve simple interest applications,” and “solve applications with discount or mark-up.” The rest of the cells are blank.", caption: none)

ⓑ After reviewing this checklist, what will you do to become confident for all goals?