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== 6.1#h(0.6em)Understand Percent

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

Translate “the ratio of #math.equation(block: false, alt: "33")[$33$] to #math.equation(block: false, alt: "5”")[$"5”"$] into an algebraic expression. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 33 over 5")[$frac(33, 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write #math.equation(block: false, alt: "the fraction 3 over 5")[$frac(3, 5)$] as a decimal. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "0.6")[$0.6$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write #math.equation(block: false, alt: "0.62")[$0.62$] as a fraction. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 31 over 50")[$frac(31, 50)$]
]
]


=== Use the Definition of Percent

How many cents are in one dollar? There are #math.equation(block: false, alt: "100")[$100$] cents in a dollar. How many years are in a century? There are #math.equation(block: false, alt: "100")[$100$] years in a century. Does this give you a clue about what the word “percent” means? It is really two words, “per cent,” and means per one hundred. A #strong[percent] is a ratio whose denominator is #math.equation(block: false, alt: "100 .")[$100 .$] We use the percent symbol #math.equation(block: false, alt: "%,")[$"%,"$] to show percent.

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

A percent is a ratio whose denominator is #math.equation(block: false, alt: "100 .")[$100 .$]
]

According to data from the American Association of Community Colleges #math.equation(block: false, alt: "open parenthesis 2015 close parenthesis ,")[$( 2015 ) ","$] about #math.equation(block: false, alt: "57%")[$"57%"$] of community college students are female. This means #math.equation(block: false, alt: "57")[$57$] out of every #math.equation(block: false, alt: "100")[$100$] community college students are female, as  shows. Out of the #math.equation(block: false, alt: "100")[$100$] squares on the grid, #math.equation(block: false, alt: "57")[$57$] are shaded, which we write as the ratio #math.equation(block: false, alt: "the fraction 57 over 100 .")[$frac(57, 100) .$]

#figure(figph[The figure shows a hundred flat with 57 units shaded.], alt: "The figure shows a hundred flat with 57 units shaded.", caption: [Among every #math.equation(block: false, alt: "100")[$100$] community college students, #math.equation(block: false, alt: "57")[$57$] are female.])

Similarly, #math.equation(block: false, alt: "25%")[$"25%"$] means a ratio of #math.equation(block: false, alt: "the fraction 25 over 100 , 3%")[$frac(25, 100) , "3%"$] means a ratio of #math.equation(block: false, alt: "the fraction 3 over 100")[$frac(3, 100)$] and #math.equation(block: false, alt: "100%")[$"100%"$] means a ratio of #math.equation(block: false, alt: "the fraction 100 over 100 .")[$frac(100, 100) .$] In words, "one hundred percent" means the total #math.equation(block: false, alt: "100%")[$"100%"$] is #math.equation(block: false, alt: "the fraction 100 over 100 ,")[$frac(100, 100) ,$] and since #math.equation(block: false, alt: "the fraction 100 over 100 equals 1 ,")[$frac(100, 100) = 1 ,$] we see that #math.equation(block: false, alt: "100%")[$"100%"$] means #math.equation(block: false, alt: "1")[$1$] whole.

#examplebox("Example 1")[][
According to the Public Policy Institute of California #math.equation(block: false, alt: "open parenthesis 2010 close parenthesis , 44%")[$( 2010 ) "," #h(0.2em) "44%"$] of parents of public school children would like their youngest child to earn a graduate degree. Write this percent as a ratio.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([The amount we want to convert is 44%.], [#math.equation(block: false, alt: "44%")[$44%$]]),
  [Write the percent as a ratio. Remember that #emph[percent] means per 100.], [#math.equation(block: false, alt: "the fraction 44 over 100")[$frac(44, 100)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the percent as a ratio.

According to a survey, #math.equation(block: false, alt: "89%")[$"89%"$] of college students have a smartphone.

#solutionbox[
#math.equation(block: true, alt: "the fraction 89 over 100")[$frac(89, 100)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the percent as a ratio.

A study found that #math.equation(block: false, alt: "72%")[$"72%"$] of U.S. teens send text messages regularly.

#solutionbox[
#math.equation(block: true, alt: "the fraction 72 over 100")[$frac(72, 100)$]
]
]

#examplebox("Example 2")[][
In #math.equation(block: false, alt: "2007 ,")[$2007 ,$] according to a U.S. Department of Education report, #math.equation(block: false, alt: "21")[$21$] out of every #math.equation(block: false, alt: "100")[$100$] first-time freshmen college students at #math.equation(block: false, alt: "4-year")[$"4-year"$] public institutions took at least one remedial course. Write this as a ratio and then as a percent.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([The amount we want to convert is #math.equation(block: false, alt: "21")[$21$] out of #math.equation(block: false, alt: "100")[$100$].], [#math.equation(block: false, alt: "21")[$21$] out of #math.equation(block: false, alt: "100")[$100$]]),
  [Write as a ratio.], [#math.equation(block: false, alt: "the fraction 21 over 100")[$frac(21, 100)$]],
  [Convert the 21 per 100 to percent.], [#math.equation(block: false, alt: "21%")[$21%$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a ratio and then as a percent: The American Association of Community Colleges reported that #math.equation(block: false, alt: "62")[$62$] out of #math.equation(block: false, alt: "100")[$100$] full-time community college students balance their studies with full-time or part time employment.

#solutionbox[
#math.equation(block: true, alt: "the fraction 62 over 100 , 62%")[$frac(62, 100) , "62%"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a ratio and then as a percent: In response to a student survey, #math.equation(block: false, alt: "41")[$41$] out of #math.equation(block: false, alt: "100")[$100$] Santa Ana College students expressed a goal of earning an Associate's degree or transferring to a four-year college.

#solutionbox[
#math.equation(block: true, alt: "the fraction 41 over 100 , 41%")[$frac(41, 100) , "41%"$]
]
]


=== Convert Percents to Fractions and Decimals

Since percents are ratios, they can easily be expressed as fractions. Remember that #strong[percent] means per #math.equation(block: false, alt: "100 ,")[$100 ,$] so the denominator of the fraction is #math.equation(block: false, alt: "100 .")[$100 .$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Convert a percent to a fraction.]

+ Write the percent as a ratio with the denominator #math.equation(block: false, alt: "100 .")[$100 .$]
+ Simplify the fraction if possible.
]

#examplebox("Example 3")[][
Convert each percent to a fraction:

+ ⓐ #math.equation(block: false, alt: "36%")[$#h(0.2em) #h(0.2em) "36%" #h(0.2em) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "125%")[$#h(0.2em) "125%"$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "36%")[$36%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 36 over 100")[$frac(36, 100)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 9 over 25")[$frac(9, 25)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "125%")[$125%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 125 over 100")[$frac(125, 100)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 5 over 4")[$frac(5, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each percent to a fraction:

+ ⓐ #math.equation(block: false, alt: "48%")[$#h(0.2em) "48%" #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "110%")[$#h(0.2em) "110%"$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 12 over 25")[$#h(0.2em) frac(12, 25) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 11 over 10")[$#h(0.2em) frac(11, 10)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each percent to a fraction:

+ ⓐ #math.equation(block: false, alt: "64%")[$#h(0.2em) "64%" #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "150%")[$#h(0.2em) "150%"$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 16 over 25")[$#h(0.2em) frac(16, 25) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 3 over 2")[$#h(0.2em) frac(3, 2)$]
]
]

The previous example shows that a #strong[percent] can be greater than #math.equation(block: false, alt: "1 .")[$1 .$] We saw that #math.equation(block: false, alt: "125%")[$"125%"$] means #math.equation(block: false, alt: "the fraction 125 over 100 ,")[$frac(125, 100) ,$] or #math.equation(block: false, alt: "the fraction 5 over 4 .")[$frac(5, 4) .$] These are improper fractions, and their values are greater than one.

#examplebox("Example 4")[][
Convert each percent to a fraction:

+ ⓐ #math.equation(block: false, alt: "24.5%")[$#h(0.2em) "24.5%" #h(0.2em) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "33 the fraction 1 over 3 %")[$#h(0.2em) 33 frac(1, 3) %$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "24.5%")[$24.5%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 24.5 over 100")[$frac(24.5, 100)$]],
  [Clear the decimal by multiplying numerator and denominator by 10.], [#math.equation(block: false, alt: "the fraction 24.5 open parenthesis 10 close parenthesis over 100 open parenthesis 10 close parenthesis")[$frac(24.5 ( 10 ), 100 ( 10 ))$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 245 over 1000")[$frac(245, 1000)$]],
  [Rewrite showing common factors.], [#math.equation(block: false, alt: "the fraction 5 times 49 over 5 times 200")[$frac(5 · 49, 5 · 200)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 49 over 200")[$frac(49, 200)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "33 the fraction 1 over 3 %")[$33 frac(1, 3) %$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 33 the fraction 1 over 3 over 100")[$frac(33 frac(1, 3), 100)$]],
  [Write the numerator as an improper fraction.], [#math.equation(block: false, alt: "the fraction the fraction 100 over 3 over 100")[$frac(#h(0.2em) #h(0.2em) #h(0.2em) frac(100, 3) #h(0.2em) #h(0.2em) #h(0.2em) #h(0.2em), 100)$]],
  [Rewrite as fraction division, replacing 100 with #math.equation(block: false, alt: "the fraction 100 over 1")[$frac(100, 1)$].], [#math.equation(block: false, alt: "the fraction 100 over 3 divided by the fraction 100 over 1")[$frac(100, 3) ÷ frac(100, 1)$]],
  [Multiply by the reciprocal.], [#math.equation(block: false, alt: "the fraction 100 over 3 times the fraction 1 over 100")[$frac(100, 3) ⋅ frac(1, 100)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each percent to a fraction:

+ ⓐ #math.equation(block: false, alt: "64.4%")[$#h(0.2em) "64.4%" #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "66 the fraction 2 over 3 %")[$#h(0.2em) 66 frac(2, 3) %$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 161 over 250")[$#h(0.2em) frac(161, 250) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 2 over 3")[$#h(0.2em) frac(2, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each percent to a fraction:

+ ⓐ #math.equation(block: false, alt: "42.5%")[$#h(0.2em) "42.5%" #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "8 the fraction 3 over 4 %")[$#h(0.2em) 8 frac(3, 4) %$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 17 over 40")[$#h(0.2em) frac(17, 40) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 7 over 80")[$#h(0.2em) frac(7, 80)$]
]
]

In Decimals, we learned how to convert fractions to decimals. To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Convert a percent to a decimal.]

+ Write the percent as a ratio with the denominator #math.equation(block: false, alt: "100 .")[$100 .$]
+ Convert the fraction to a decimal by dividing the numerator by the denominator.
]

#examplebox("Example 5")[][
Convert each percent to a decimal:

+ ⓐ #math.equation(block: false, alt: "6%")[$#h(0.4em) "6%" #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "78%")[$#h(0.2em) "78%"$]

#solutionbox[
Because we want to change to a decimal, we will leave the fractions with denominator #math.equation(block: false, alt: "100")[$100$] instead of removing common factors.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "6%")[$6%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 6 over 100")[$frac(6, 100)$]],
  [Change the fraction to a decimal by dividing the numerator by the denominator.], [#math.equation(block: false, alt: "0.06")[$0.06$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "78%")[$78%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 78 over 100")[$frac(78, 100)$]],
  [Change the fraction to a decimal by dividing the numerator by the denominator.], [#math.equation(block: false, alt: "0.78")[$0.78$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each percent to a decimal:

+ ⓐ #math.equation(block: false, alt: "9%")[$#h(0.3em) "9%" #h(0.3em)$]
+ ⓑ #math.equation(block: false, alt: "87%")[$#h(0.3em) "87%"$]

#solutionbox[
+ ⓐ 0.09
+ ⓑ 0.87
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each percent to a decimal:

+ ⓐ #math.equation(block: false, alt: "3%")[$#h(0.3em) "3%" #h(0.3em)$]
+ ⓑ #math.equation(block: false, alt: "91%")[$#h(0.3em) "91%"$]

#solutionbox[
+ ⓐ 0.03
+ ⓑ 0.91
]
]

#examplebox("Example 6")[][
Convert each percent to a decimal:

+ ⓐ #math.equation(block: false, alt: "135%")[$#h(0.3em) "135%" #h(0.3em)$]
+ ⓑ #math.equation(block: false, alt: "12.5%")[$#h(0.3em) "12.5%"$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "135%")[$135%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 135 over 100")[$frac(135, 100)$]],
  [Change the fraction to a decimal by dividing the numerator by the denominator.], [#math.equation(block: false, alt: "1.35")[$1.35$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "12.5%")[$12.5%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 12.5 over 100")[$frac(12.5, 100)$]],
  [Change the fraction to a decimal by dividing the numerator by the denominator.], [#math.equation(block: false, alt: "0.125")[$0.125$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each percent to a decimal:

+ ⓐ #math.equation(block: false, alt: "115%")[$#h(0.3em) "115%" #h(0.3em)$]
+ ⓑ #math.equation(block: false, alt: "23.5%")[$#h(0.3em) "23.5%"$]

#solutionbox[
+ ⓐ 1.15
+ ⓑ 0.235
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each percent to a decimal:

+ ⓐ #math.equation(block: false, alt: "123%")[$#h(0.3em) "123%" #h(0.3em)$]
+ ⓑ #math.equation(block: false, alt: "16.8%")[$#h(0.3em) "16.8%"$]

#solutionbox[
+ ⓐ 1.23
+ ⓑ 0.168
]
]

Let's summarize the results from the previous examples in , and look for a pattern we could use to quickly convert a percent number to a decimal number.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Percent], [Decimal]),
  [#math.equation(block: false, alt: "6%")[$"6%"$]], [#math.equation(block: false, alt: "0.06")[$0.06$]],
  [#math.equation(block: false, alt: "78%")[$"78%"$]], [#math.equation(block: false, alt: "0.78")[$0.78$]],
  [#math.equation(block: false, alt: "135%")[$"135%"$]], [#math.equation(block: false, alt: "1.35")[$1.35$]],
  [#math.equation(block: false, alt: "12.5%")[$"12.5%"$]], [#math.equation(block: false, alt: "0.125")[$0.125$]],
))

Do you see the pattern?

To convert a #strong[percent] number to a decimal number, we move the decimal point two places to the left and remove the #math.equation(block: false, alt: "%")[$%$] sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer #math.equation(block: false, alt: "6")[$6$] as #math.equation(block: false, alt: "6.0 ,")[$6.0 ,$] we can think of #math.equation(block: false, alt: "6%")[$"6%"$] as #math.equation(block: false, alt: "6.0% .")[$"6.0%" .$]) Notice that we may need to add zeros in front of the number when moving the decimal to the left.

uses the percents in  and shows visually how to convert them to decimals by moving the decimal point two places to the left.

#figure(figph[The figures shows two columns and five rows . The  first row is a header row and it labels each column “Percent” and “Decimal”. Under the “Percent” column are the values: 6%, 78%, 135%, 12.5%. Under the “Decimal” column are the values: 0.06, 0.78, 1.35, 0.125. There are two jumps for each percent to show how to convert it to a decimal.], alt: "The figures shows two columns and five rows . The first row is a header row and it labels each column “Percent” and “Decimal”. Under the “Percent” column are the values: 6%, 78%, 135%, 12.5%. Under the “Decimal” column are the values: 0.06, 0.78, 1.35, 0.125. There are two jumps for each percent to show how to convert it to a decimal.", caption: none)

#examplebox("Example 7")[][
Among a group of business leaders, #math.equation(block: false, alt: "77%")[$"77%"$] believe that poor math and science education in the U.S. will lead to higher unemployment rates.

Convert the percent to: ⓐ a fraction ⓑ a decimal

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "77%")[$77%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 77 over 100")[$frac(77, 100)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "the fraction 77 over 100")[$frac(77, 100)$]],
  [Change the fraction to a decimal by dividing the numerator by the denominator.], [#math.equation(block: false, alt: "0.77")[$0.77$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert the percent to: ⓐ a fraction and
ⓑ a decimal

Twitter's share of web traffic jumped #math.equation(block: false, alt: "24%")[$"24%"$] when one celebrity tweeted live on air.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 6 over 25")[$#h(0.2em) frac(6, 25) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "0.24")[$#h(0.2em) 0.24$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert the percent to:
ⓐ a fraction and
ⓑ a decimal

The U.S. Census estimated that in #math.equation(block: false, alt: "2013 , 44%")[$2013 , "44%"$] of the population of Boston age #math.equation(block: false, alt: "25")[$25$] or older have a bachelor's or higher degrees.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 11 over 25")[$#h(0.2em) frac(11, 25) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "0.44")[$#h(0.2em) 0.44$]
]
]

#examplebox("Example 8")[][
There are four suits of cards in a deck of cards—hearts, diamonds, clubs, and spades. The probability of randomly choosing a heart from a shuffled deck of cards is #math.equation(block: false, alt: "25% .")[$"25%" .$] Convert the percent to:

+ ⓐ a fraction
+ ⓑ a decimal

#figure(figph[The figure shows someone holding a deck of cards.], alt: "The figure shows someone holding a deck of cards.", caption: [(credit: Riles32807, Wikimedia Commons)])

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "25%")[$25%$]],
  [Write as a ratio with denominator 100.], [#math.equation(block: false, alt: "the fraction 25 over 100")[$frac(25, 100)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]]),
  [Change the fraction to a decimal by dividing the numerator by the denominator.], [#math.equation(block: false, alt: "0.25")[$0.25$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert the percent to:
ⓐ a fraction, and

ⓑ a decimal

The probability that it will rain Monday is #math.equation(block: false, alt: "30% .")[$"30%" .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 3 over 10")[$#h(0.2em) frac(3, 10) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "0.3")[$#h(0.2em) 0.3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert the percent to:
ⓐ a fraction, and
ⓑ a decimal

The probability of getting heads three times when tossing a coin three times is #math.equation(block: false, alt: "12.5% .")[$"12.5%" .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 1 over 8")[$#h(0.2em) frac(1, 8) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "0.125")[$#h(0.2em) 0.125$]
]
]


=== Convert Decimals and Fractions to Percents

To convert a decimal to a #strong[percent], remember that percent means per hundred. If we change the decimal to a fraction whose denominator is #math.equation(block: false, alt: "100 ,")[$100 ,$] it is easy to change that fraction to a percent.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Convert a decimal to a percent.]

+ Write the decimal as a fraction.
+ If the denominator of the fraction is not #math.equation(block: false, alt: "100 ,")[$100 ,$] rewrite it as an equivalent fraction with denominator #math.equation(block: false, alt: "100 .")[$100 .$]
+ Write this ratio as a percent.
]

#examplebox("Example 9")[][
Convert each decimal to a percent:
ⓐ #math.equation(block: false, alt: "0.05")[$#h(0.2em) 0.05 #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "0.83")[$#h(0.2em) 0.83$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "0.05")[$0.05$]],
  [Write as a fraction. The denominator is 100.], [#math.equation(block: false, alt: "the fraction 5 over 100")[$frac(5, 100)$]],
  [Write this ratio as a percent.], [#math.equation(block: false, alt: "5%")[$5%$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "0.83")[$0.83$]],
  [The denominator is 100.], [#math.equation(block: false, alt: "the fraction 83 over 100")[$frac(83, 100)$]],
  [Write this ratio as a percent.], [#math.equation(block: false, alt: "83%")[$83%$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each decimal to a percent:
ⓐ #math.equation(block: false, alt: "0.01")[$#h(0.2em) 0.01 #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "0.17 .")[$#h(0.2em) 0.17 .$]

#solutionbox[
+ ⓐ 1%
+ ⓑ 17%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each decimal to a percent:
ⓐ #math.equation(block: false, alt: "0.04")[$#h(0.2em) 0.04 #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "0.41")[$#h(0.2em) 0.41$]

#solutionbox[
+ ⓐ 4%
+ ⓑ 41%
]
]

To convert a mixed number to a percent, we first write it as an improper fraction.

#examplebox("Example 10")[][
Convert each decimal to a percent:
ⓐ #math.equation(block: false, alt: "1.05")[$#h(0.2em) 1.05 #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "0.075")[$#h(0.2em) 0.075$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "1.05")[$1.05$]],
  [Write as a fraction.], [#math.equation(block: false, alt: "1 the fraction 5 over 100")[$1 frac(5, 100)$]],
  [Write as an improper fraction. The denominator is 100.], [#math.equation(block: false, alt: "the fraction 105 over 100")[$frac(105, 100)$]],
  [Write this ratio as a percent.], [#math.equation(block: false, alt: "105%")[$105%$]],
))

Notice that since #math.equation(block: false, alt: "1.05 greater than 1 ,")[$1.05 > 1 ,$] the result is more than #math.equation(block: false, alt: "100%.")[$"100%."$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "0.075")[$0.075$]],
  [Write as a fraction. The denominator is 1,000.], [#math.equation(block: false, alt: "the fraction 75 over 1,000")[$frac(75, 1","000)$]],
  [Divide the numerator and denominator by 10, so that the denominator is 100.], [#math.equation(block: false, alt: "the fraction 7.5 over 100")[$frac(7.5, 100)$]],
  [Write this ratio as a percent.], [#math.equation(block: false, alt: "7.5%")[$7.5%$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each decimal to a percent:
ⓐ #math.equation(block: false, alt: "1.75")[$#h(0.2em) 1.75 #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "0.0825")[$#h(0.2em) 0.0825$]

#solutionbox[
+ ⓐ 175%
+ ⓑ 8.25%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each decimal to a percent:
ⓐ #math.equation(block: false, alt: "2.25")[$#h(0.2em) 2.25 #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "0.0925")[$#h(0.2em) 0.0925$]

#solutionbox[
+ ⓐ 225%
+ ⓑ 9.25%
]
]

Let's summarize the results from the previous examples in  so we can look for a pattern.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Decimal], [Percent]),
  [#math.equation(block: false, alt: "0.05")[$0.05$]], [#math.equation(block: false, alt: "5%")[$"5%"$]],
  [#math.equation(block: false, alt: "0.83")[$0.83$]], [#math.equation(block: false, alt: "83%")[$"83%"$]],
  [#math.equation(block: false, alt: "1.05")[$1.05$]], [#math.equation(block: false, alt: "105%")[$"105%"$]],
  [#math.equation(block: false, alt: "0.075")[$0.075$]], [#math.equation(block: false, alt: "7.5%")[$"7.5%"$]],
))

Do you see the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

uses the decimal numbers in  and shows visually to convert them to percents by moving the decimal point two places to the right and then writing the #math.equation(block: false, alt: "%")[$%$] sign.

#figure(figph[The figure shows two columns and five rows. The  first row is a header row and it labels each column “Decimal” and “Percent”. Under the “Decimal” column are the values: 0.05, 0.83, 1.05, 0.075, 0.3. Under the “Percent” column are the values: 5%, 83%, 105%, 7.5%, 30%. There are two jumps for each decimal to show how to convert it to a percent.], alt: "The figure shows two columns and five rows. The first row is a header row and it labels each column “Decimal” and “Percent”. Under the “Decimal” column are the values: 0.05, 0.83, 1.05, 0.075, 0.3. Under the “Percent” column are the values: 5%, 83%, 105%, 7.5%, 30%. There are two jumps for each decimal to show how to convert it to a percent.", caption: none)

In Decimals, we learned how to convert fractions to decimals. Now we also know how to change decimals to percents. So to convert a fraction to a percent, we first change it to a decimal and then convert that decimal to a percent.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Convert a fraction to a percent.]

+ Convert the fraction to a decimal.
+ Convert the decimal to a percent.
]

#examplebox("Example 11")[][
Convert each fraction or mixed number to a percent:
ⓐ #math.equation(block: false, alt: "the fraction 3 over 4")[$#h(0.2em) frac(3, 4) #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "the fraction 11 over 8")[$#h(0.2em) frac(11, 8) #h(0.2em)$]
ⓒ #math.equation(block: false, alt: "2 the fraction 1 over 5")[$#h(0.2em) 2 frac(1, 5)$]

#solutionbox[
To convert a fraction to a decimal, divide the numerator by the denominator.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [Change to a decimal.], [#math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$]],
  [Write as a percent by moving the decimal two places.], [#figure(figph[The decimal number 0.75 is shown, with two blue arrows below it, likely illustrating movement of the decimal point or place value shifts in a mathematical context.], alt: "The decimal number 0.75 is shown, with two blue arrows below it, likely illustrating movement of the decimal point or place value shifts in a mathematical context.", caption: none)],
  [], [#math.equation(block: false, alt: "75%")[$75%$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [Change to a decimal.], [#math.equation(block: false, alt: "the fraction 11 over 8")[$frac(11, 8)$]],
  [Write as a percent by moving the decimal two places.], [#figure(figph[The number 1.375 is shown with light blue curved arrows beneath the decimal point, highlighting the digits '3' and '75' to emphasize the decimal places.], alt: "The number 1.375 is shown with light blue curved arrows beneath the decimal point, highlighting the digits '3' and '75' to emphasize the decimal places.", caption: none)],
  [], [#math.equation(block: false, alt: "137.5%")[$137.5%$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [Write as an improper fraction.], [#math.equation(block: false, alt: "2 the fraction 1 over 5")[$2 frac(1, 5)$]],
  [Change to a decimal.], [#math.equation(block: false, alt: "the fraction 11 over 5")[$frac(11, 5)$]],
  [Write as a percent.], [#figure(figph[The number 2.20 is displayed with two blue arcs, visually representing the shift of the decimal point two places to the right, a common step in percentage conversions.], alt: "The number 2.20 is displayed with two blue arcs, visually representing the shift of the decimal point two places to the right, a common step in percentage conversions.", caption: none)],
  [], [#math.equation(block: false, alt: "220%")[$220%$]],
))

Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each fraction or mixed number to a percent:
ⓐ #math.equation(block: false, alt: "the fraction 5 over 8")[$frac(5, 8) #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "the fraction 11 over 4")[$frac(11, 4) #h(0.2em)$]
ⓒ #math.equation(block: false, alt: "3 the fraction 2 over 5")[$3 frac(2, 5)$]

#solutionbox[
+ ⓐ 62.5%
+ ⓑ 275%
+ ⓒ 340%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert each fraction or mixed number to a percent:
ⓐ #math.equation(block: false, alt: "the fraction 7 over 8")[$#h(0.2em) frac(7, 8) #h(0.2em)$]
ⓑ #math.equation(block: false, alt: "the fraction 9 over 4")[$#h(0.2em) frac(9, 4) #h(0.2em)$]
ⓒ #math.equation(block: false, alt: "1 the fraction 3 over 5")[$#h(0.2em) 1 frac(3, 5)$]

#solutionbox[
+ ⓐ 87.5%
+ ⓑ 225%
+ ⓒ 160%
]
]

Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we will #strong[round] off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place. For most other cases in this book we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.

#examplebox("Example 12")[][
Convert #math.equation(block: false, alt: "the fraction 5 over 7")[$frac(5, 7)$] to a percent.

#solutionbox[
To change a fraction to a decimal, we divide the numerator by the denominator.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 5 over 7")[$frac(5, 7)$]]),
  [Change to a decimal—rounding to the nearest thousandth.], [#math.equation(block: false, alt: "0.714")[$0.714$]],
  [Write as a percent.], [#math.equation(block: false, alt: "71.4%")[$71.4%$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert the fraction to a percent: #math.equation(block: false, alt: "the fraction 3 over 7")[$frac(3, 7)$]

#solutionbox[
42.9%
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert the fraction to a percent: #math.equation(block: false, alt: "the fraction 4 over 7")[$frac(4, 7)$]

#solutionbox[
57.1%
]
]

When we first looked at fractions and decimals, we saw that some fractions converted to a #strong[repeating decimal]. For example, when we converted the fraction #math.equation(block: false, alt: "the fraction 4 over 3")[$frac(4, 3)$] to a decimal, we wrote the answer as #math.equation(block: false, alt: "1 . 3 ¯ .")[$1 . overline(3) .$] We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

#examplebox("Example 13")[][
An article in a medical journal claimed that approximately #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] of American adults are obese. Convert the fraction #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] to a percent.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]]),
  [Change to a decimal.], [#figure(figph[Long division showing 1 divided by 3 equals the repeating decimal 0.333... This illustrates how a fraction translates to an infinite decimal.], alt: "Long division showing 1 divided by 3 equals the repeating decimal 0.333... This illustrates how a fraction translates to an infinite decimal.", caption: none)],
  [Write as a repeating decimal.], [#math.equation(block: false, alt: "0.333 and so on")[$0.333 …$]],
  [Write as a percent.], [#math.equation(block: false, alt: "33 the fraction 1 over 3 %")[$33 frac(1, 3) %$]],
))

We could also write the percent as #math.equation(block: false, alt: "33 . 3 _ %")[$33 . overline(3) %$].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert the fraction to a percent:

According to the U.S. Census Bureau, about #math.equation(block: false, alt: "the fraction 1 over 9")[$frac(1, 9)$] of United States housing units have just #math.equation(block: false, alt: "1")[$1$] bedroom.

#solutionbox[
#math.equation(block: true, alt: "11. 1 minus %, or 11 the fraction 1 over 9 %")[$11. limits(1)^(−) "%," #h(0.2em) "or" #h(0.2em) 11 frac(1, 9) %$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert the fraction to a percent:

According to the U.S. Census Bureau, about #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] of Colorado residents speak a language other than English at home.

#solutionbox[
#math.equation(block: true, alt: "16 . 6 minus %, or 16 the fraction 2 over 3 %")[$16 . limits(6)^(−) "%," #h(0.2em) "or" #h(0.2em) 16 frac(2, 3) %$]
]
]


=== Key Concepts

- #emph[Convert a percent to a fraction.]
+ Write the percent as a ratio with the denominator 100.
+ Simplify the fraction if possible.
- #emph[Convert a percent to a decimal.]
+ Write the percent as a ratio with the denominator 100.
+ Convert the fraction to a decimal by dividing the numerator by the denominator.
- #emph[Convert a decimal to a percent.]
+ Write the decimal as a fraction.
+ If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100.
+ Write this ratio as a percent.
- #emph[Convert a fraction to a percent.]
+ Convert the fraction to a decimal.
+ Convert the decimal to a percent.


==== Practice Makes Perfect

#strong[Use the Definition of Percents]

In the following exercises, write each percent as a ratio.

In #math.equation(block: false, alt: "2014 ,")[$2014 ,$] the unemployment rate for those with only a high school degree was #math.equation(block: false, alt: "6.0% .")[$"6.0%" .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 over 100")[$frac(6, 100)$]
]

In #math.equation(block: false, alt: "2015 ,")[$2015 ,$] among the unemployed, #math.equation(block: false, alt: "29%")[$"29%"$] were long-term unemployed.

The unemployment rate for those with Bachelor's degrees was #math.equation(block: false, alt: "3.2%")[$"3.2%"$] in #math.equation(block: false, alt: "2014 .")[$2014 .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 32 over 1000")[$frac(32, 1000)$]
]

The unemployment rate in Michigan in #math.equation(block: false, alt: "2014")[$2014$] was #math.equation(block: false, alt: "7.3% .")[$"7.3%" .$]

In the following exercises, write as

+ ⓐ a ratio and
+ ⓑ a percent

#math.equation(block: false, alt: "57")[$57$] out of #math.equation(block: false, alt: "100")[$100$] nursing candidates received their degree at a community college.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 57 over 100")[$#h(0.2em) frac(57, 100) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "57%")[$#h(0.2em) "57%"$]
]

#math.equation(block: false, alt: "80")[$80$] out of #math.equation(block: false, alt: "100")[$100$] firefighters and law enforcement officers were educated at a community college.

#math.equation(block: false, alt: "42")[$42$] out of #math.equation(block: false, alt: "100")[$100$] first-time freshmen students attend a community college.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 42 over 100")[$#h(0.2em) frac(42, 100) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "42%")[$#h(0.2em) "42%"$]
]

#math.equation(block: false, alt: "71")[$71$] out of #math.equation(block: false, alt: "100")[$100$] full-time community college faculty have a master's degree.

#strong[Convert Percents to Fractions and Decimals]

In the following exercises, convert each percent to a fraction and simplify all fractions.

#math.equation(block: true, alt: "4%")[$"4%"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 25")[$frac(1, 25)$]
]

#math.equation(block: true, alt: "8%")[$"8%"$]

#math.equation(block: true, alt: "17%")[$"17%"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 17 over 100")[$frac(17, 100)$]
]

#math.equation(block: true, alt: "19%")[$"19%"$]

#math.equation(block: true, alt: "52%")[$"52%"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over 25")[$frac(13, 25)$]
]

#math.equation(block: true, alt: "78%")[$"78%"$]

#math.equation(block: true, alt: "125%")[$"125%"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 4")[$frac(5, 4)$]
]

#math.equation(block: true, alt: "135%")[$"135%"$]

#math.equation(block: true, alt: "37.5%")[$"37.5%"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 8")[$frac(3, 8)$]
]

#math.equation(block: true, alt: "42.5%")[$"42.5%"$]

#math.equation(block: true, alt: "18.4%")[$"18.4%"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 23 over 125")[$frac(23, 125)$]
]

#math.equation(block: true, alt: "46.4%")[$"46.4%"$]

#math.equation(block: true, alt: "9 the fraction 1 over 2 %")[$9 frac(1, 2) %$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 19 over 200")[$frac(19, 200)$]
]

#math.equation(block: true, alt: "8 the fraction 1 over 2 %")[$8 frac(1, 2) %$]

#math.equation(block: true, alt: "5 the fraction 1 over 3 %")[$5 frac(1, 3) %$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 75")[$frac(4, 75)$]
]

#math.equation(block: true, alt: "6 the fraction 2 over 3 %")[$6 frac(2, 3) %$]

In the following exercises, convert each percent to a decimal.

#math.equation(block: true, alt: "5%")[$"5%"$]

#solutionbox[
0.05
]

#math.equation(block: true, alt: "9%")[$"9%"$]

#math.equation(block: true, alt: "1%")[$"1%"$]

#solutionbox[
0.01
]

#math.equation(block: true, alt: "2%")[$"2%"$]

#math.equation(block: true, alt: "63%")[$"63%"$]

#solutionbox[
0.63
]

#math.equation(block: true, alt: "71%")[$"71%"$]

#math.equation(block: true, alt: "40%")[$"40%"$]

#solutionbox[
0.4
]

#math.equation(block: true, alt: "50%")[$"50%"$]

#math.equation(block: true, alt: "115%")[$"115%"$]

#solutionbox[
1.15
]

#math.equation(block: true, alt: "125%")[$"125%"$]

#math.equation(block: true, alt: "150%")[$"150%"$]

#solutionbox[
1.5
]

#math.equation(block: true, alt: "250%")[$"250%"$]

#math.equation(block: true, alt: "21.4%")[$"21.4%"$]

#solutionbox[
0.214
]

#math.equation(block: true, alt: "39.3%")[$"39.3%"$]

#math.equation(block: true, alt: "7.8%")[$"7.8%"$]

#solutionbox[
0.078
]

#math.equation(block: true, alt: "6.4%")[$"6.4%"$]

In the following exercises, convert each percent to

+ ⓐ a simplified fraction and
+ ⓑ a decimal

In #math.equation(block: false, alt: "2010 , 1.5%")[$2010 , "1.5%"$] of home sales had owner financing. (#emph[Source:] Bloomberg Businessweek, 5/23–29/2011)

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 3 over 200")[$#h(0.2em) frac(3, 200) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "0.015")[$#h(0.2em) 0.015$]
]

In #math.equation(block: false, alt: "2000 , 4.2%")[$2000 , "4.2%"$] of the United States population was of Asian descent. (#emph[Source:] www.census.gov)

According to government data, in #math.equation(block: false, alt: "2013")[$2013$] the number of cell phones in India was #math.equation(block: false, alt: "70.23%")[$"70.23%"$] of the population.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 7023 over 10,000")[$#h(0.2em) frac(7023, 10","000) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "0.7023")[$#h(0.2em) 0.7023$]
]

According to the U.S. Census Bureau, among Americans age #math.equation(block: false, alt: "25")[$25$] or older who had doctorate degrees in #math.equation(block: false, alt: "2014 , 37.1%")[$2014 , "37.1%"$] are women.

A couple plans to have two children. The probability they will have two girls is #math.equation(block: false, alt: "25% .")[$"25%" .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 1 over 4")[$#h(0.2em) frac(1, 4) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "0.25")[$#h(0.2em) 0.25$]
]

Javier will choose one digit at random from #math.equation(block: false, alt: "0")[$0$] through #math.equation(block: false, alt: "9 .")[$9 .$] The probability he will choose #math.equation(block: false, alt: "3")[$3$] is #math.equation(block: false, alt: "10% .")[$"10%" .$]

According to the local weather report, the probability of thunderstorms in New York City on July #math.equation(block: false, alt: "15")[$15$] is #math.equation(block: false, alt: "60% .")[$"60%" .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 3 over 5")[$#h(0.2em) frac(3, 5) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "0.6")[$#h(0.2em) 0.6$]
]

A club sells #math.equation(block: false, alt: "50")[$50$] tickets to a raffle. Osbaldo bought one ticket. The probability he will win the raffle is #math.equation(block: false, alt: "2% .")[$"2%" .$]

#strong[Convert Decimals and Fractions to Percents]

In the following exercises, convert each decimal to a percent.

#math.equation(block: true, alt: "0.01")[$0.01$]

#solutionbox[
1%
]

#math.equation(block: true, alt: "0.03")[$0.03$]

#math.equation(block: true, alt: "0.18")[$0.18$]

#solutionbox[
18%
]

#math.equation(block: true, alt: "0.15")[$0.15$]

#math.equation(block: true, alt: "1.35")[$1.35$]

#solutionbox[
135%
]

#math.equation(block: true, alt: "1.56")[$1.56$]

#math.equation(block: true, alt: "3")[$3$]

#solutionbox[
300%
]

#math.equation(block: true, alt: "4")[$4$]

#math.equation(block: true, alt: "0.009")[$0.009$]

#solutionbox[
0.9%
]

#math.equation(block: true, alt: "0.008")[$0.008$]

#math.equation(block: true, alt: "0.0875")[$0.0875$]

#solutionbox[
8.75%
]

#math.equation(block: true, alt: "0.0625")[$0.0625$]

#math.equation(block: true, alt: "1.5")[$1.5$]

#solutionbox[
150%
]

#math.equation(block: true, alt: "2.2")[$2.2$]

#math.equation(block: true, alt: "2.254")[$2.254$]

#solutionbox[
225.4%
]

#math.equation(block: true, alt: "2.317")[$2.317$]

In the following exercises, convert each fraction to a percent.

#math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$]

#solutionbox[
25%
]

#math.equation(block: true, alt: "the fraction 1 over 5")[$frac(1, 5)$]

#math.equation(block: true, alt: "the fraction 3 over 8")[$frac(3, 8)$]

#solutionbox[
37.5%
]

#math.equation(block: true, alt: "the fraction 5 over 8")[$frac(5, 8)$]

#math.equation(block: true, alt: "the fraction 7 over 4")[$frac(7, 4)$]

#solutionbox[
175%
]

#math.equation(block: true, alt: "the fraction 9 over 8")[$frac(9, 8)$]

#math.equation(block: true, alt: "6 the fraction 4 over 5")[$6 frac(4, 5)$]

#solutionbox[
680%
]

#math.equation(block: true, alt: "5 the fraction 1 over 4")[$5 frac(1, 4)$]

#math.equation(block: true, alt: "the fraction 5 over 12")[$frac(5, 12)$]

#solutionbox[
#math.equation(block: true, alt: "41 the fraction 2 over 3 % o r 41. 6 minus %")[$41 frac(2, 3) "%" #h(0.2em) o r #h(0.2em) 41. accent(6, −) "%"$]
]

#math.equation(block: true, alt: "the fraction 11 over 12")[$frac(11, 12)$]

#math.equation(block: true, alt: "2 the fraction 2 over 3")[$2 frac(2, 3)$]

#solutionbox[
#math.equation(block: true, alt: "266 . 6 minus %")[$266 . limits(6)^(−) "%"$]
]

#math.equation(block: true, alt: "1 the fraction 2 over 3")[$1 frac(2, 3)$]

#math.equation(block: true, alt: "the fraction 3 over 7")[$frac(3, 7)$]

#solutionbox[
42.9%
]

#math.equation(block: true, alt: "the fraction 6 over 7")[$frac(6, 7)$]

#math.equation(block: true, alt: "the fraction 5 over 9")[$frac(5, 9)$]

#solutionbox[
#math.equation(block: true, alt: "55. 5 minus %")[$55. accent(5, −) "%"$]
]

#math.equation(block: true, alt: "the fraction 4 over 9")[$frac(4, 9)$]

In the following exercises, convert each fraction to a percent.

#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] of washing machines needed repair.

#solutionbox[
25%
]

#math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] of dishwashers needed repair.

In the following exercises, convert each fraction to a percent.

According to the National Center for Health Statistics, in #math.equation(block: false, alt: "2012 , the fraction 7 over 20")[$2012 , frac(7, 20)$] of American adults were obese.

#solutionbox[
35%
]

The U.S. Census Bureau estimated that in #math.equation(block: false, alt: "2013 , 85%")[$2013 , "85%"$] of Americans lived in the same house as they did #math.equation(block: false, alt: "1")[$1$] year before.

In the following exercises, complete the table.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Fraction], [Decimal], [Percent]),
  [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [], [],
  [], [#math.equation(block: false, alt: "0.45")[$0.45$]], [],
  [], [], [#math.equation(block: false, alt: "18 %")[$18 "%"$]],
  [#math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]], [], [],
  [], [#math.equation(block: false, alt: "0.008")[$0.008$]], [],
  [#math.equation(block: false, alt: "2")[$2$]], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Fraction], [Decimal], [Percent]),
  [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "50 %")[$50 "%"$]],
  [#math.equation(block: false, alt: "the fraction 9 over 20")[$frac(9, 20)$]], [#math.equation(block: false, alt: "0.45")[$0.45$]], [#math.equation(block: false, alt: "45 %")[$45 "%"$]],
  [#math.equation(block: false, alt: "the fraction 9 over 50")[$frac(9, 50)$]], [#math.equation(block: false, alt: "0.18")[$0.18$]], [#math.equation(block: false, alt: "18 %")[$18 "%"$]],
  [#math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]], [#math.equation(block: false, alt: "0.3")[$0.3$]], [#math.equation(block: false, alt: "33.3 %")[$33.3 "%"$]],
  [#math.equation(block: false, alt: "the fraction 1 over 125")[$frac(1, 125)$]], [#math.equation(block: false, alt: "0.008")[$0.008$]], [#math.equation(block: false, alt: "0.8 %")[$0.8 "%"$]],
  [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "2.0")[$2.0$]], [#math.equation(block: false, alt: "200 %")[$200 "%"$]],
))
]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Fraction], [Decimal], [Percent]),
  [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]], [], [],
  [], [#math.equation(block: false, alt: "0.65")[$0.65$]], [],
  [], [], [#math.equation(block: false, alt: "22 %")[$22 "%"$]],
  [#math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$]], [], [],
  [], [#math.equation(block: false, alt: "0.004")[$0.004$]], [],
  [#math.equation(block: false, alt: "3")[$3$]], [], [],
))


==== Everyday Math

#strong[Sales tax] Felipa says she has an easy way to estimate the sales tax when she makes a purchase. The sales tax in her city is #math.equation(block: false, alt: "9.05% .")[$"9.05%" .$] She knows this is a little less than #math.equation(block: false, alt: "10% .")[$"10%" .$]

+ ⓐ Convert #math.equation(block: false, alt: "10%")[$"10%"$] to a fraction.
+ ⓑ Use your answer from ⓐ to estimate the sales tax Felipa would pay on a #math.equation(block: false, alt: "$95")[$"$95"$] dress.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 1 over 10")[$#h(0.2em) frac(1, 10) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "approximately $9.50")[$#h(0.2em) "approximately $9.50"$]
]

#strong[Savings] Ryan has #math.equation(block: false, alt: "25%")[$"25%"$] of each paycheck automatically deposited in his savings account.

+ ⓐ Write #math.equation(block: false, alt: "25%")[$"25%"$] as a fraction.
+ ⓑ Use your answer from ⓐ to find the amount that goes to savings from Ryan's #math.equation(block: false, alt: "$2,400")[$"$2,400"$] paycheck.

Amelio is shopping for textbooks online. He found three sellers that are offering a book he needs for the same price, including shipping. To decide which seller to buy from he is comparing their customer satisfaction ratings. The ratings are given in the chart.

Use the chart to answer the following questions

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Seller], [Rating]),
  [#math.equation(block: false, alt: "A")[$"A"$]], [#math.equation(block: false, alt: "4/5")[$"4/5"$]],
  [#math.equation(block: false, alt: "B")[$"B"$]], [#math.equation(block: false, alt: "3.5/4")[$"3.5/4"$]],
  [#math.equation(block: false, alt: "C")[$"C"$]], [#math.equation(block: false, alt: "85%")[$"85%"$]],
))

Write seller #math.equation(block: false, alt: "C’s")[$"C’s"$] rating as a fraction and a decimal.

#solutionbox[
#math.equation(block: true, alt: "the fraction 17 over 20 ; 0.85")[$frac(17, 20) ; #h(0.2em) 0.85$]
]

Write seller #math.equation(block: false, alt: "B’s")[$"B’s"$] rating as a percent and a decimal.

Write seller #math.equation(block: false, alt: "A’s")[$"A’s"$] rating as a percent and a decimal.

#solutionbox[
80%; 0.8
]

Which seller should Amelio buy from and why?


==== Writing Exercises

Convert #math.equation(block: false, alt: "25% , 50% , 75% , and 100%")[$"25%" , "50%" , "75%" , "and" #h(0.2em) "100%"$] to fractions. Do you notice a pattern? Explain what the pattern is.

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 4 , the fraction 1 over 2 , the fraction 3 over 4 , 1 .")[$frac(1, 4) , frac(1, 2) , frac(3, 4) , 1 .$]
]

Convert #math.equation(block: false, alt: "the fraction 1 over 10 , the fraction 2 over 10 , the fraction 3 over 10 , the fraction 4 over 10 , the fraction 5 over 10 , the fraction 6 over 10 , the fraction 7 over 10 , the fraction 8 over 10 ,")[$frac(1, 10) , frac(2, 10) , frac(3, 10) , frac(4, 10) , frac(5, 10) , frac(6, 10) , frac(7, 10) , frac(8, 10) ,$] and #math.equation(block: false, alt: "the fraction 9 over 10")[$frac(9, 10)$] to percents. Do you notice a pattern? Explain what the pattern is.

When the Szetos sold their home, the selling price was #math.equation(block: false, alt: "500%")[$"500%"$] of what they had paid for the house #math.equation(block: false, alt: "30 years")[$"30 years"$] ago. Explain what #math.equation(block: false, alt: "500%")[$"500%"$] means in this context.

#solutionbox[
The Szetos sold their home for five times what they paid 30 years ago.
]

According to cnn.com, cell phone use in #math.equation(block: false, alt: "2008")[$2008$] was #math.equation(block: false, alt: "600%")[$"600%"$] of what it had been in #math.equation(block: false, alt: "2001 .")[$2001 .$] Explain what #math.equation(block: false, alt: "600%")[$"600%"$] means in this context.


==== Self Check

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

#figure(figph[A self-assessment chart for students to evaluate their understanding of percentages, including defining percent and converting between percents, fractions, and decimals.], alt: "A self-assessment chart for students to evaluate their understanding of percentages, including defining percent and converting between percents, fractions, and decimals.", caption: none)

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.