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== 6.2#h(0.6em)Solve General Applications of Percent

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

Translate and solve: #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "24 .")[$24 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "32")[$32$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 4.5 close parenthesis open parenthesis 2.38 close parenthesis .")[$( 4.5 ) ( 2.38 ) .$] #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: "5")[$5$]
]
]


=== Translate and Solve Basic Percent Equations

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. In the past, you may have solved percent problems by setting them up as proportions. That was the best method available when you did not have the tools of algebra. Now as a prealgebra student, you can translate word sentences into algebraic equations, and then solve the equations.

We'll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.

When Aolani and her friends ate dinner at a restaurant, the bill came to #math.equation(block: false, alt: "$80 .")[$"$80" .$] They wanted to leave a #math.equation(block: false, alt: "20%")[$"20%"$] tip. What amount would the tip be?

To solve this, we want to find what #emph[amount] is #math.equation(block: false, alt: "20%")[$"20%"$] of #math.equation(block: false, alt: "$80 .")[$"$80" .$] The #math.equation(block: false, alt: "$80")[$"$80"$] is called the #emph[base]. The amount of the tip would be #math.equation(block: false, alt: "0.20 open parenthesis 80 close parenthesis ,")[$0.20 ( 80 ) ,$] or #math.equation(block: false, alt: "$16")[$"$16"$] To find the amount of the tip, we multiplied the percent by the base.

#figure(figph[The figure shows a customer copy of a restaurant receipt with the amount of the bill, \$80, and the amount of the tip, \$16. There is a group of bills totaling \$16.], alt: "The figure shows a customer copy of a restaurant receipt with the amount of the bill, $80, and the amount of the tip, $16. There is a group of bills totaling $16.", caption: [A #math.equation(block: false, alt: "20%")[$"20%"$] tip for an #math.equation(block: false, alt: "$80")[$"$80"$] restaurant bill comes out to #math.equation(block: false, alt: "$16 .")[$"$16" .$]])

In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.

#examplebox("Example 1")[][
What number is #math.equation(block: false, alt: "35%")[$"35%"$] of #math.equation(block: false, alt: "90 ?")[$90 ?$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate into algebra. Let #math.equation(block: false, alt: "n equals")[$n = #h(0.2em)$]the number.], [#figure(figph[A visual representation of translating the word problem 'What number is 35% of 90?' into the algebraic equation 'n = 0.35 \* 90'.], alt: "A visual representation of translating the word problem 'What number is 35% of 90?' into the algebraic equation 'n = 0.35 * 90'.", caption: none)]),
  [Multiply.], [#figure(figph[A mathematical equation is displayed on a white background, showing the variable 'n' equals '31.5'.], alt: "A mathematical equation is displayed on a white background, showing the variable 'n' equals '31.5'.", 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"))[
What number is #math.equation(block: false, alt: "45%")[$"45%"$] of #math.equation(block: false, alt: "80 ?")[$80 ?$]

#solutionbox[
36
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
What number is #math.equation(block: false, alt: "55%")[$"55%"$] of #math.equation(block: false, alt: "60 ?")[$60 ?$]

#solutionbox[
33
]
]

#examplebox("Example 2")[][
#math.equation(block: false, alt: "125%")[$"125%"$] of #math.equation(block: false, alt: "28")[$28$] is what number?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate into algebra. Let #math.equation(block: false, alt: "a equals")[$a #h(0.2em) = #h(0.2em)$]the number.], [#figure(figph[An image illustrating the conversion of the word problem '125% of 28 is what number?' into the algebraic equation '1.25 \* 28 = a', with visual cues mapping words to their mathematical forms.], alt: "An image illustrating the conversion of the word problem '125% of 28 is what number?' into the algebraic equation '1.25 * 28 = a', with visual cues mapping words to their mathematical forms.", caption: none)]),
  [Multiply.], [#figure(figph[A simple mathematical equation shows '35 = a' with the number 35 equal to the variable 'a'.], alt: "A simple mathematical equation shows '35 = a' with the number 35 equal to the variable 'a'.", caption: none)],
  [], [#math.equation(block: false, alt: "125%")[$125%$] of #math.equation(block: false, alt: "28")[$28$] is #math.equation(block: false, alt: "35")[$35$].],
))

Remember that a percent over #math.equation(block: false, alt: "100")[$100$] is a number greater than #math.equation(block: false, alt: "1 .")[$1 .$] We found that #math.equation(block: false, alt: "125%")[$"125%"$] of #math.equation(block: false, alt: "28")[$28$] is #math.equation(block: false, alt: "35 ,")[$35 ,$] which is greater than #math.equation(block: false, alt: "28 .")[$28 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "150%")[$"150%"$] of #math.equation(block: false, alt: "78")[$78$] is what number?

#solutionbox[
117
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "175%")[$"175%"$] of #math.equation(block: false, alt: "72")[$72$] is what number?

#solutionbox[
126
]
]

In the next examples, we are asked to find the base.

#examplebox("Example 3")[][
Translate and solve: #math.equation(block: false, alt: "36")[$36$] is #math.equation(block: false, alt: "75%")[$"75%"$] of what number?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate. Let #math.equation(block: false, alt: "b equals")[$b =$] the number.], [#figure(figph[A diagram illustrates how to translate the word problem '36 is 75% of what number?' into the mathematical equation '36 = 0.75 . b' by breaking down each part of the sentence.], alt: "A diagram illustrates how to translate the word problem '36 is 75% of what number?' into the mathematical equation '36 = 0.75 . b' by breaking down each part of the sentence.", caption: none)]),
  [Divide both sides by 0.75.], [#figure(figph[A mathematical equation is displayed where both sides of an equality are divided by 0.75 to solve for the variable 'b', represented as '36 / 0.75 = 0.75b / 0.75'.], alt: "A mathematical equation is displayed where both sides of an equality are divided by 0.75 to solve for the variable 'b', represented as '36 / 0.75 = 0.75b / 0.75'.", caption: none)],
  [Simplify.], [#figure(figph[The image displays two lines of text: '48 = b' and '36 is 75% of 48', illustrating a mathematical relationship between numbers and a percentage.], alt: "The image displays two lines of text: '48 = b' and '36 is 75% of 48', illustrating a mathematical relationship between numbers and a percentage.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "17")[$17$] is #math.equation(block: false, alt: "25%")[$"25%"$] of what number?

#solutionbox[
68
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "40")[$40$] is #math.equation(block: false, alt: "62.5%")[$"62.5%"$] of what number?

#solutionbox[
64
]
]

#examplebox("Example 4")[][
#math.equation(block: false, alt: "6.5%")[$"6.5%"$] of what number is #math.equation(block: false, alt: "$1.17 ?")[$"$1.17" ?$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate. Let #math.equation(block: false, alt: "b equals")[$b =$] the number.], [#figure(figph[Translating the percentage word problem '6.5% of what number is \$1.17?' into the equation '0.065 \* b = 1.17', showing term-by-term correspondence.], alt: "Translating the percentage word problem '6.5% of what number is $1.17?' into the equation '0.065 * b = 1.17', showing term-by-term correspondence.", caption: none)]),
  [Divide both sides by 0.065.], [#figure(figph[A mathematical equation shows '0.065n over 0.065 equals 1.17 over 0.065' on a white background, demonstrating the step of dividing both sides of an equation by 0.065 to solve for 'n'.], alt: "A mathematical equation shows '0.065n over 0.065 equals 1.17 over 0.065' on a white background, demonstrating the step of dividing both sides of an equation by 0.065 to solve for 'n'.", caption: none)],
  [Simplify.], [#figure(figph[The image displays two lines of text: 'n = 18' at the top right, and '6.5% of \$18 is \$1.17.' at the bottom left, illustrating a percentage calculation.], alt: "The image displays two lines of text: 'n = 18' at the top right, and '6.5% of $18 is $1.17.' at the bottom left, illustrating a percentage calculation.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "7.5%")[$"7.5%"$] of what number is #math.equation(block: false, alt: "$1.95 ?")[$"$1.95" ?$]

#solutionbox[
\$26
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "8.5%")[$"8.5%"$] of what number is #math.equation(block: false, alt: "$3.06 ?")[$"$3.06" ?$]

#solutionbox[
\$36
]
]

In the next examples, we will solve for the percent.

#examplebox("Example 5")[][
What percent of #math.equation(block: false, alt: "36")[$36$] is #math.equation(block: false, alt: "9 ?")[$9 ?$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate into algebra. Let #math.equation(block: false, alt: "p equals")[$p =$] the percent.], [#figure(figph[A visual guide translating the phrase 'What percent of 36 is 9?' into the algebraic equation 'p . 36 = 9', showing the correspondence between words and mathematical symbols.], alt: "A visual guide translating the phrase 'What percent of 36 is 9?' into the algebraic equation 'p . 36 = 9', showing the correspondence between words and mathematical symbols.", caption: none)]),
  [Divide by 36.], [#figure(figph[A mathematical equation shows '36p over 36 equals 9 over 36'.], alt: "A mathematical equation shows '36p over 36 equals 9 over 36'.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation on a white background, stating 'p = 1/4'. The letter 'p' is shown in italics, followed by an equals sign, and then the fraction one over four.], alt: "The image displays a mathematical equation on a white background, stating 'p = 1/4'. The letter 'p' is shown in italics, followed by an equals sign, and then the fraction one over four.", caption: none)],
  [Convert to decimal form.], [#figure(figph[The image displays the equation 'p = 0.25' in a clear, sans-serif font against a plain white background, indicating a probability or a numerical value.], alt: "The image displays the equation 'p = 0.25' in a clear, sans-serif font against a plain white background, indicating a probability or a numerical value.", caption: none)],
  [Convert to percent.], [#figure(figph[The image displays the mathematical statement '25% of 36 is 9.' and also shows 'p = 25%,' illustrating a percentage calculation and its variable representation.], alt: "The image displays the mathematical statement '25% of 36 is 9.' and also shows 'p = 25%,' illustrating a percentage calculation and its variable representation.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
What percent of #math.equation(block: false, alt: "76")[$76$] is #math.equation(block: false, alt: "57 ?")[$57 ?$]

#solutionbox[
75%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
What percent of #math.equation(block: false, alt: "120")[$120$] is #math.equation(block: false, alt: "96 ?")[$96 ?$]

#solutionbox[
80%
]
]

#examplebox("Example 6")[][
#math.equation(block: false, alt: "144")[$144$] is what percent of #math.equation(block: false, alt: "96 ?")[$96 ?$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate into algebra. Let #math.equation(block: false, alt: "p equals")[$p =$] the percent.], [#figure(figph[Translating the phrase '144 is what percent of 96?' into an algebraic equation '144 = p . 96' to solve for the unknown percentage 'p'.], alt: "Translating the phrase '144 is what percent of 96?' into an algebraic equation '144 = p . 96' to solve for the unknown percentage 'p'.", caption: none)]),
  [Divide by 96.], [#figure(figph[A mathematical equation shows '144 divided by 96 equals 96p divided by 96' on a white background.], alt: "A mathematical equation shows '144 divided by 96 equals 96p divided by 96' on a white background.", caption: none)],
  [Simplify.], [#figure(figph[The equation 1.5 = p is displayed in black text on a white background, representing a simple algebraic statement or a given value for the variable 'p'.], alt: "The equation 1.5 = p is displayed in black text on a white background, representing a simple algebraic statement or a given value for the variable 'p'.", caption: none)],
  [Convert to percent.], [#figure(figph[Two lines of text display mathematical concepts: the first line shows '150% = p', and the second line states '144 is 150% of 96.'], alt: "Two lines of text display mathematical concepts: the first line shows '150% = p', and the second line states '144 is 150% of 96.'", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "110")[$110$] is what percent of #math.equation(block: false, alt: "88 ?")[$88 ?$]

#solutionbox[
125%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "126")[$126$] is what percent of #math.equation(block: false, alt: "72 ?")[$72 ?$]

#solutionbox[
175%
]
]


=== Solve Applications of Percent

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we'll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.

We will update the strategy we used in our earlier applications to include equations now. Notice that we will translate a sentence into an equation.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve an application]

+ Identify what you are asked to find and choose a variable to represent it.
+ Write a sentence that gives the information to find it.
+ Translate the sentence into an equation.
+ Solve the equation using good algebra techniques.
+ Check the answer in the problem and make sure it makes sense.
+ Write a complete sentence that answers the question.
]

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 we'll solve involve everyday situations, you can rely on your own experience.

#examplebox("Example 7")[][
Dezohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was #math.equation(block: false, alt: "$68.50 .")[$"$68.50" .$] They want to leave an #math.equation(block: false, alt: "18%")[$"18%"$] tip. If the tip will be #math.equation(block: false, alt: "18%")[$"18%"$] of the total bill, how much should the tip be?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([What are you asked to find?], [the amount of the tip]),
  [Choose a variable to represent it.], [Let #math.equation(block: false, alt: "t equals")[$t =$] amount of tip.],
  [Write a sentence that give the information to find it.], [The tip is 18% of the total bill.],
  [Translate the sentence into an equation.], [#figure(figph[Algebraic equation representing a calculation: The variable "t" equals zero point one eight times sixty eight dollars and fifty cents. This shows how to convert eighteen percent to a decimal and interpret "of" as multiplication.], alt: "Algebraic equation representing a calculation: The variable \"t\" equals zero point one eight times sixty eight dollars and fifty cents. This shows how to convert eighteen percent to a decimal and interpret \"of\" as multiplication.", caption: none)],
  [Multiply.], [#figure(figph[The image shows a mathematical expression], alt: "The image shows a mathematical expression", caption: none)],
  [Check. Is this answer reasonable?], [],
  [If we approximate the bill to \$70 and the percent to 20%, we would have a tip of \$14.  #linebreak()  So a tip of \$12.33 seems reasonable.], [],
  [Write a complete  sentence that answers the question.], [The couple should leave a tip of \$12.33.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Cierra and her sister enjoyed a special dinner in a restaurant, and the bill was #math.equation(block: false, alt: "$81.50 .")[$"$81.50" .$] If she wants to leave #math.equation(block: false, alt: "18%")[$"18%"$] 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 #math.equation(block: false, alt: "15%")[$"15%"$] of the total bill as her tip. If her bill was #math.equation(block: false, alt: "$14.40 ,")[$"$14.40" ,$] how much will she leave for the tip?

#solutionbox[
\$2.16
]
]

#examplebox("Example 8")[][
The label on Masao's breakfast cereal said that one serving of cereal provides #math.equation(block: false, alt: "85")[$85$] milligrams (mg) of potassium, which is #math.equation(block: false, alt: "2%")[$"2%"$] of the recommended daily amount. What is the total recommended daily amount of potassium?

#figure(figph[The figures shows the nutrition facts for cereal.], alt: "The figures shows the nutrition facts for cereal.", caption: none)

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([What are you asked to find?], [the total amount of potassium recommended]),
  [Choose a variable to represent it.], [Let #math.equation(block: false, alt: "a equals")[$a =$] total amount of potassium.],
  [Write a sentence that gives the information to find it.], [85 mg is 2% of the total amount.],
  [Translate the sentence into an equation.], [#figure(figph[Translating the word problem '85 mg is 2% of a?' into the algebraic equation '85 = 0.02 \* a'.], alt: "Translating the word problem '85 mg is 2% of a?' into the algebraic equation '85 = 0.02 * a'.", caption: none)],
  [Divide both sides by 0.02.], [#figure(figph[A mathematical equation shows both sides being divided by 0.02: 85/0.02 = 0.02a/0.02.], alt: "A mathematical equation shows both sides being divided by 0.02: 85/0.02 = 0.02a/0.02.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation: 4,250 equals the variable 'a'.], alt: "The image displays a mathematical equation: 4,250 equals the variable 'a'.", caption: none)],
  [Check: Is this answer reasonable?], [],
  [Yes. 2% is a small percent and 85 is a small part of 4,250.], [],
  [Write a complete  sentence that answers the question.], [The amount of potassium that is recommended is 4250 mg.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One serving of wheat square cereal has #math.equation(block: false, alt: "7")[$7$] grams of fiber, which is #math.equation(block: false, alt: "29%")[$"29%"$] of the recommended daily amount. What is the total recommended daily amount of fiber?

#solutionbox[
24.1 grams
]
]

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

#solutionbox[
2,375 mg
]
]

#examplebox("Example 9")[][
Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was #math.equation(block: false, alt: "480")[$480$] calories, and had #math.equation(block: false, alt: "240")[$240$] calories of fat. What percent of the total calories in each brownie comes from fat?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([What are you asked to find?], [the percent of the total calories from fat]),
  [Choose a variable to represent it.], [Let #math.equation(block: false, alt: "p equals")[$p =$] percent from fat.],
  [Write a sentence that gives the information to find it.], [What percent of 480 is 240?],
  [Translate the sentence into an equation.], [#figure(figph[Visual representation of translating 'What percent of 480 is 240?' into the algebraic equation 'p \* 480 = 240', showing how words map to mathematical symbols.], alt: "Visual representation of translating 'What percent of 480 is 240?' into the algebraic equation 'p * 480 = 240', showing how words map to mathematical symbols.", caption: none)],
  [Divide both sides by 480.], [#figure(figph[A mathematical equation shows p multiplied by 480 and divided by 480, which is equal to 240 divided by 480.], alt: "A mathematical equation shows p multiplied by 480 and divided by 480, which is equal to 240 divided by 480.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression p = 0.5 is displayed, suggesting a probability or a specific value for the variable p.], alt: "The mathematical expression p = 0.5 is displayed, suggesting a probability or a specific value for the variable p.", caption: none)],
  [Convert to percent form.], [#figure(figph[The image displays the mathematical expression 'p = 50%' in black text against a plain white background, indicating a probability or percentage value.], alt: "The image displays the mathematical expression 'p = 50%' in black text against a plain white background, indicating a probability or percentage value.", caption: none)],
  [Check. Is this answer reasonable?], [],
  [Yes. 240 is half of 480, so 50% makes sense.], [],
  [Write a complete  sentence that answers the question.], [Of the total calories in each brownie, 50% is fat.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Veronica is planning to make muffins from a mix. The package says each muffin will be #math.equation(block: false, alt: "230")[$230$] calories and #math.equation(block: false, alt: "60")[$60$] calories will be from fat. What percent of the total calories is from fat? (Round to the nearest whole percent.)

#solutionbox[
26%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The brownie mix Ricardo plans to use says that each brownie will be #math.equation(block: false, alt: "190")[$190$] calories, and #math.equation(block: false, alt: "70")[$70$] calories are from fat. What percent of the total calories are from fat?

#solutionbox[
37%
]
]


=== 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 #strong[percent].

To find the #strong[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.

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

Step 1. Find the amount of increase.

- #math.equation(block: false, alt: "increase equals new amount minus original amount")[$"increase" = "new amount" − "original amount"$]

Step 2. Find the percent increase as a percent of the original amount.
]

#examplebox("Example 10")[][
In #math.equation(block: false, alt: "2011 ,")[$2011 ,$] the California governor proposed raising community college fees from #math.equation(block: false, alt: "$26")[$"$26"$] per unit to #math.equation(block: false, alt: "$36")[$"$36"$] per unit. Find the percent increase. (Round to the nearest tenth of a percent.)

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([What are you asked to find?], [the percent increase]),
  [Choose a variable to represent it.], [Let #math.equation(block: false, alt: "p equals")[$p =$] percent.],
  [Find the amount of increase.], [#figure(figph[A mathematical equation illustrating the calculation of an increase: 36 (new amount) - 26 (original amount) = 10 (increase).], alt: "A mathematical equation illustrating the calculation of an increase: 36 (new amount) - 26 (original amount) = 10 (increase).", caption: none)],
  [Find the percent increase.], [The increase is what percent of the original amount?],
  [Translate to an equation.], [#figure(figph[A mathematical problem asks '10 is what percent of 26?' above its algebraic translation: '10 = p . 26'. Each part of the word problem is shown with a bracket above its corresponding mathematical symbol below.], alt: "A mathematical problem asks '10 is what percent of 26?' above its algebraic translation: '10 = p . 26'. Each part of the word problem is shown with a bracket above its corresponding mathematical symbol below.", caption: none)],
  [Divide both sides by 26.], [#figure(figph[An algebraic equation showing 10 divided by 26 equals 26p divided by 26, presented in fraction form on a white background.], alt: "An algebraic equation showing 10 divided by 26 equals 26p divided by 26, presented in fraction form on a white background.", caption: none)],
  [Round to the nearest thousandth.], [#figure(figph[The equation 0.385 = p is displayed in black text on a white background.], alt: "The equation 0.385 = p is displayed in black text on a white background.", caption: none)],
  [Convert to percent form.], [#figure(figph[A mathematical equation displays '38.5% = p' in a simple, clear font on a white background. This equation assigns the percentage value of 38.5 to the variable 'p', indicating a direct equivalency.], alt: "A mathematical equation displays '38.5% = p' in a simple, clear font on a white background. This equation assigns the percentage value of 38.5 to the variable 'p', indicating a direct equivalency.", caption: none)],
  [Write a complete  sentence.], [The new fees represent a 38.5% increase over the old fees.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
In #math.equation(block: false, alt: "2011 ,")[$2011 ,$] the IRS increased the deductible mileage cost to #math.equation(block: false, alt: "55.5")[$55.5$] cents from #math.equation(block: false, alt: "51")[$51$] cents. Find the percent increase. (Round to the nearest tenth of a percent.)

#solutionbox[
8.8%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
In #math.equation(block: false, alt: "1995 ,")[$1995 ,$] the standard bus fare in Chicago was #math.equation(block: false, alt: "$1.50 .")[$"$1.50" .$] In #math.equation(block: false, alt: "2008 ,")[$2008 ,$] the standard bus fare was #math.equation(block: false, alt: "$2.25 .")[$"$2.25" .$] Find the percent increase. (Round to the nearest tenth of a percent.)

#solutionbox[
50%
]
]

Finding the #strong[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.

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

+ Find the amount of decrease.- #math.equation(block: false, alt: "decrease equals original amount minus new amount")[$"decrease" = "original amount" − "new amount"$]
+ Find the percent decrease as a percent of the original amount.
]

#examplebox("Example 11")[][
The average price of a gallon of gas in one city in June #math.equation(block: false, alt: "2014")[$2014$] was #math.equation(block: false, alt: "$3.71 .")[$"$3.71" .$] The average price in that city in July was #math.equation(block: false, alt: "$3.64 .")[$"$3.64" .$] Find the percent decrease.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([What are you asked to find?], [the percent decrease]),
  [Choose a variable to represent it.], [Let #math.equation(block: false, alt: "p equals")[$p =$] percent.],
  [Find the amount of decrease.], [#figure(figph[An image displays the subtraction 3.71 - 3.64 = 0.07. The number 3.71 is labeled 'original amount', 3.64 is labeled 'new amount', and 0.07 is labeled 'decrease'.], alt: "An image displays the subtraction 3.71 - 3.64 = 0.07. The number 3.71 is labeled 'original amount', 3.64 is labeled 'new amount', and 0.07 is labeled 'decrease'.", caption: none)],
  [Find the percent of decrease.], [The decrease is what percent of the original amount?],
  [Translate to an equation.], [#figure(figph[A visual representation of translating the English phrase '0.07 is what percent of 3.71?' into the algebraic equation '0.07 = p \* 3.71' for solving percentages.], alt: "A visual representation of translating the English phrase '0.07 is what percent of 3.71?' into the algebraic equation '0.07 = p * 3.71' for solving percentages.", caption: none)],
  [Divide both sides by 3.71.], [#figure(figph[A mathematical equation shows both sides being divided by 3.71: 0.07 / 3.71 = 3.71p / 3.71.], alt: "A mathematical equation shows both sides being divided by 3.71: 0.07 / 3.71 = 3.71p / 3.71.", caption: none)],
  [Round to the nearest thousandth.], [#figure(figph[A mathematical equation displays '0.019 = p' in black text against a plain white background.], alt: "A mathematical equation displays '0.019 = p' in black text against a plain white background.", caption: none)],
  [Convert to percent form.], [#figure(figph[The image displays a mathematical equation: 1.9% = p. The equation shows the percentage 1.9% equated to the variable p, suggesting a conversion or a given value for p.], alt: "The image displays a mathematical equation: 1.9% = p. The equation shows the percentage 1.9% equated to the variable p, suggesting a conversion or a given value for p.", caption: none)],
  [Write a complete sentence.], [The price of gas decreased 1.9%.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The population of one city was about #math.equation(block: false, alt: "672,000")[$672,000$] in #math.equation(block: false, alt: "2010 .")[$2010 .$] The population of the city is projected to be about #math.equation(block: false, alt: "630,000")[$630,000$] in #math.equation(block: false, alt: "2020 .")[$2020 .$] Find the percent decrease. (Round to the nearest tenth of a percent.)

#solutionbox[
6.3%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Last year Sheila's salary was #math.equation(block: false, alt: "$42,000 .")[$"$42,000" .$] Because of furlough days, this year her salary was #math.equation(block: false, alt: "$37,800 .")[$"$37,800" .$] Find the percent decrease. (Round to the nearest tenth of a percent.)

#solutionbox[
10%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://www.openstax.org/l/24percentincdec")[Percent Increase and Percent Decrease Visualization]
]


=== Key Concepts

- #emph[Solve an application.]
+ Identify what you are asked to find and choose a variable to represent it.
+ Write a sentence that gives the information to find it.
+ Translate the sentence into an equation.
+ Solve the equation using good algebra techniques.
+ Write a complete sentence that answers the question.
+ Check the answer in the problem and make sure it makes sense.
- #emph[Find percent increase.]
+ Find the amount of increase:   #linebreak() 
#math.equation(block: false, alt: "increase equals new amount minus original amount")[$"increase" = "new amount" − "original amount"$]
+ Find the percent increase as a percent of the original amount.
- #emph[Find percent decrease.]
+ Find the amount of decrease. #linebreak() 
#math.equation(block: false, alt: "decrease equals original amount minus new amount")[$"decrease" = "original amount" − "new amount"$]
+ Find the percent decrease as a percent of the original amount.


==== Practice Makes Perfect

#strong[Translate and Solve Basic Percent Equations]

In the following exercises, translate and solve.

What number is #math.equation(block: false, alt: "45%")[$"45%"$] of #math.equation(block: false, alt: "120 ?")[$120 ?$]

#solutionbox[
54
]

What number is #math.equation(block: false, alt: "65%")[$"65%"$] of #math.equation(block: false, alt: "100 ?")[$100 ?$]

What number is #math.equation(block: false, alt: "24%")[$"24%"$] of #math.equation(block: false, alt: "112 ?")[$112 ?$]

#solutionbox[
26.88
]

What number is #math.equation(block: false, alt: "36%")[$"36%"$] of #math.equation(block: false, alt: "124 ?")[$124 ?$]

#math.equation(block: false, alt: "250%")[$"250%"$] of #math.equation(block: false, alt: "65")[$65$] is what number?

#solutionbox[
162.5
]

#math.equation(block: false, alt: "150%")[$"150%"$] of #math.equation(block: false, alt: "90")[$90$] is what number?

#math.equation(block: false, alt: "800%")[$"800%"$] of #math.equation(block: false, alt: "2,250")[$2,250$] is what number?

#solutionbox[
18,000
]

#math.equation(block: false, alt: "600%")[$"600%"$] of #math.equation(block: false, alt: "1,740")[$1,740$] is what number?

#math.equation(block: false, alt: "28")[$28$] is #math.equation(block: false, alt: "25%")[$"25%"$] of what number?

#solutionbox[
112
]

#math.equation(block: false, alt: "36")[$36$] is #math.equation(block: false, alt: "25%")[$"25%"$] of what number?

#math.equation(block: false, alt: "81")[$81$] is #math.equation(block: false, alt: "75%")[$"75%"$] of what number?

#solutionbox[
108
]

#math.equation(block: false, alt: "93")[$93$] is #math.equation(block: false, alt: "75%")[$"75%"$] of what number?

#math.equation(block: false, alt: "8.2%")[$"8.2%"$] of what number is #math.equation(block: false, alt: "$2.87 ?")[$"$2.87" ?$]

#solutionbox[
\$35
]

#math.equation(block: false, alt: "6.4%")[$"6.4%"$] of what number is #math.equation(block: false, alt: "$2.88 ?")[$"$2.88" ?$]

#math.equation(block: false, alt: "11.5%")[$"11.5%"$] of what number is #math.equation(block: false, alt: "$108.10 ?")[$"$108.10" ?$]

#solutionbox[
\$940
]

#math.equation(block: false, alt: "12.3%")[$"12.3%"$] of what number is #math.equation(block: false, alt: "$92.25 ?")[$"$92.25" ?$]

What percent of #math.equation(block: false, alt: "260")[$260$] is #math.equation(block: false, alt: "78 ?")[$78 ?$]

#solutionbox[
30%
]

What percent of #math.equation(block: false, alt: "215")[$215$] is #math.equation(block: false, alt: "86 ?")[$86 ?$]

What percent of #math.equation(block: false, alt: "1,500")[$1,500$] is #math.equation(block: false, alt: "540 ?")[$540 ?$]

#solutionbox[
36%
]

What percent of #math.equation(block: false, alt: "1,800")[$1,800$] is #math.equation(block: false, alt: "846 ?")[$846 ?$]

#math.equation(block: false, alt: "30")[$30$] is what percent of #math.equation(block: false, alt: "20 ?")[$20 ?$]

#solutionbox[
150%
]

#math.equation(block: false, alt: "50")[$50$] is what percent of #math.equation(block: false, alt: "40 ?")[$40 ?$]

#math.equation(block: false, alt: "840")[$840$] is what percent of #math.equation(block: false, alt: "480 ?")[$480 ?$]

#solutionbox[
175%
]

#math.equation(block: false, alt: "790")[$790$] is what percent of #math.equation(block: false, alt: "395 ?")[$395 ?$]

#strong[Solve Applications of Percents]

In the following exercises, solve the applications of percents.

Geneva treated her parents to dinner at their favorite restaurant. The bill was #math.equation(block: false, alt: "$74.25 .")[$"$74.25" .$] She wants to leave #math.equation(block: false, alt: "16%")[$"16%"$] 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 the bill was #math.equation(block: false, alt: "$90.50 .")[$"$90.50" .$] They want to leave #math.equation(block: false, alt: "18%")[$"18%"$] of the total bill as a tip. How much should the tip be?

Trong has #math.equation(block: false, alt: "12%")[$"12%"$] of each paycheck automatically deposited to his savings account. His last paycheck was #math.equation(block: false, alt: "$2,165 .")[$"$2,165" .$] How much money was deposited to Trong's savings account?

#solutionbox[
\$259.80
]

Cherise deposits #math.equation(block: false, alt: "8%")[$"8%"$] of each paycheck into her retirement account. Her last paycheck was #math.equation(block: false, alt: "$1,485 .")[$"$1,485" .$] How much did Cherise deposit into her retirement account?

One serving of oatmeal has #math.equation(block: false, alt: "8")[$8$] grams of fiber, which is #math.equation(block: false, alt: "33%")[$"33%"$] of the recommended daily amount. What is the total recommended daily amount of fiber?

#solutionbox[
24.2 grams
]

One serving of trail mix has #math.equation(block: false, alt: "67")[$67$] grams of carbohydrates, which is #math.equation(block: false, alt: "22%")[$"22%"$] of the recommended daily amount. What is the total recommended daily amount of carbohydrates?

A bacon cheeseburger at a popular fast food restaurant contains #math.equation(block: false, alt: "2,070")[$2,070$] milligrams (mg) of sodium, which is #math.equation(block: false, alt: "86%")[$"86%"$] of the recommended daily amount. What is the total recommended daily amount of sodium?

#solutionbox[
2,407 mg
]

A grilled chicken salad at a popular fast food restaurant contains #math.equation(block: false, alt: "650")[$650$] milligrams (mg) of sodium, which is #math.equation(block: false, alt: "27%")[$"27%"$] of the recommended daily amount. What is the total recommended daily amount of sodium?

The nutrition fact sheet at a fast food restaurant says the fish sandwich has #math.equation(block: false, alt: "380")[$380$] calories, and #math.equation(block: false, alt: "171")[$171$] 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 #math.equation(block: false, alt: "190")[$190$] calories, and #math.equation(block: false, alt: "114")[$114$] calories are from fat. What percent of the total calories is from fat?

Emma gets paid #math.equation(block: false, alt: "$3,000")[$"$3,000"$] per month. She pays #math.equation(block: false, alt: "$750")[$"$750"$] a month for rent. What percent of her monthly pay goes to rent?

#solutionbox[
25%
]

Dimple gets paid #math.equation(block: false, alt: "$3,200")[$"$3,200"$] per month. She pays #math.equation(block: false, alt: "$960")[$"$960"$] a month for rent. What percent of her monthly pay goes to rent?

#strong[Find Percent Increase and Percent Decrease]

In the following exercises, find the percent increase or percent decrease.

Tamanika got a raise in her hourly pay, from #math.equation(block: false, alt: "$15.50")[$"$15.50"$] to #math.equation(block: false, alt: "$17.55 .")[$"$17.55" .$] Find the percent increase.

#solutionbox[
13.2%
]

Ayodele got a raise in her hourly pay, from #math.equation(block: false, alt: "$24.50")[$"$24.50"$] to #math.equation(block: false, alt: "$25.48 .")[$"$25.48" .$] Find the percent increase.

Annual student fees at the University of California rose from about #math.equation(block: false, alt: "$4,000")[$"$4,000"$] in #math.equation(block: false, alt: "2000")[$2000$] to about #math.equation(block: false, alt: "$9,000")[$"$9,000"$] in #math.equation(block: false, alt: "2014 .")[$2014 .$] Find the percent increase.

#solutionbox[
125%
]

The price of a share of one stock rose from #math.equation(block: false, alt: "$12.50")[$"$12.50"$] to #math.equation(block: false, alt: "$50 .")[$"$50" .$] Find the percent increase.

According to Time magazine #math.equation(block: false, alt: "open parenthesis 7/19/2011 close parenthesis")[$( "7/19/2011" )$] annual global seafood consumption rose from #math.equation(block: false, alt: "22")[$22$] pounds per person in #math.equation(block: false, alt: "1960")[$1960$] to #math.equation(block: false, alt: "38")[$38$] pounds per person today. 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 #math.equation(block: false, alt: "$225,400")[$"$225,400"$] to #math.equation(block: false, alt: "$241,500 .")[$"$241,500" .$] Find the percent increase. (Round to the nearest tenth of a percent.)

A grocery store reduced the price of a loaf of bread from #math.equation(block: false, alt: "$2.80")[$"$2.80"$] to #math.equation(block: false, alt: "$2.73 .")[$"$2.73" .$] Find the percent decrease.

#solutionbox[
2.5%
]

The price of a share of one stock fell from #math.equation(block: false, alt: "$8.75")[$"$8.75"$] to #math.equation(block: false, alt: "$8.54 .")[$"$8.54" .$] Find the percent decrease.

Hernando's salary was #math.equation(block: false, alt: "$49,500")[$"$49,500"$] last year. This year his salary was cut to #math.equation(block: false, alt: "$44,055 .")[$"$44,055" .$] Find the percent decrease.

#solutionbox[
11%
]

From #math.equation(block: false, alt: "2000")[$2000$] to #math.equation(block: false, alt: "2010 ,")[$2010 ,$] the population of Detroit fell from about #math.equation(block: false, alt: "951,000")[$951,000$] to about #math.equation(block: false, alt: "714,000 .")[$714,000 .$] Find the percent decrease. (Round to the nearest tenth of a percent.)

In one month, the median home price in the West fell from #math.equation(block: false, alt: "$203,400")[$"$203,400"$] to #math.equation(block: false, alt: "$192,300 .")[$"$192,300" .$] Find the percent decrease. (Round to the nearest tenth of a percent.)

#solutionbox[
5.5%
]

Sales of video games and consoles fell from #math.equation(block: false, alt: "$1,150")[$"$1,150"$] million to #math.equation(block: false, alt: "$1,030")[$"$1,030"$] million in one year. Find the percent decrease. (Round to the nearest tenth of a percent.)


==== Everyday Math

#strong[Tipping] At the campus coffee cart, a medium coffee costs #math.equation(block: false, alt: "$1.65 .")[$"$1.65" .$] MaryAnne brings #math.equation(block: false, alt: "$2.00")[$"$2.00"$] 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[Late Fees] Alison was late paying her credit card bill of #math.equation(block: false, alt: "$249 .")[$"$249" .$] She was charged a #math.equation(block: false, alt: "5%")[$"5%"$] late fee. What was the amount of the late fee?


==== Writing Exercises

Without solving the problem #math.equation(block: false, alt: "“ 44")[$“ 44$] is #math.equation(block: false, alt: "80%")[$"80%"$] of what number”, think about what the solution might be. Should it be a number that is greater than #math.equation(block: false, alt: "44")[$44$] or less than #math.equation(block: false, alt: "44 ?")[$44 ?$] Explain your reasoning.

#solutionbox[
The original number should be greater than 44.80% is less than 100%, so when 80% is converted to a decimal and multiplied to the base in the percent equation, the resulting amount of 44 is less. 44 is only the larger number in cases where the percent is greater than 100%.
]

Without solving the problem “What is #math.equation(block: false, alt: "20%")[$"20%"$] of #math.equation(block: false, alt: "300 ?”")[$300 ?”$] think about what the solution might be. Should it be a number that is greater than #math.equation(block: false, alt: "300")[$300$] or less than #math.equation(block: false, alt: "300 ?")[$300 ?$] Explain your reasoning.

After returning from vacation, Alex said he should have packed #math.equation(block: false, alt: "50%")[$"50%"$] fewer shorts and #math.equation(block: false, alt: "200%")[$"200%"$] more shirts. Explain what Alex meant.

#solutionbox[
Alex should have packed half as many shorts and twice as many shirts.
]

Because of road construction in one city, commuters were advised to plan their Monday morning commute to take #math.equation(block: false, alt: "150%")[$"150%"$] 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[A self-assessment table titled 'I can...' asks users to rate their ability to 'translate and solve basic percent equations,' 'solve applications of percent,' and 'find percent increase and percent decrease' with options 'Confidently,' 'With some help,' or 'No - I don't get it!'], alt: "A self-assessment table titled 'I can...' asks users to rate their ability to 'translate and solve basic percent equations,' 'solve applications of percent,' and 'find percent increase and percent decrease' with options 'Confidently,' 'With some help,' or 'No - I don't get it!'", caption: none)

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