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== 5.1#h(0.6em)Decimals

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

Name the number #math.equation(block: false, alt: "4,926,015")[$4,926,015$] in words. #linebreak() 
If you missed this problem, review .

#solutionbox[
Four million, nine hundred twenty-six thousand, fifteen
]
]

#notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round #math.equation(block: false, alt: "748")[$748$] to the nearest ten. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "750")[$750$]
]
]

#notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Locate #math.equation(block: false, alt: "the fraction 3 over 10")[$frac(3, 10)$] on a number line. #linebreak() 
If you missed this problem, review .

#solutionbox[
#figure(figph[This image shows a number line from negative 5 to 5. A point is plotted at three tenths on the number line.], alt: "This image shows a number line from negative 5 to 5. A point is plotted at three tenths on the number line.", caption: none)
]
]


=== Name Decimals

You probably already know quite a bit about decimals based on your experience with money. Suppose you buy a sandwich and a bottle of water for lunch. If the sandwich costs #math.equation(block: false, alt: "$3.45")[$"$3.45"$], the bottle of water costs #math.equation(block: false, alt: "$1.25")[$"$1.25"$], and the total sales tax is #math.equation(block: false, alt: "$0.33")[$"$0.33"$], what is the total cost of your lunch?

#figure(figph[A vertical addition problem is shown. The top line shows \$3.45 for a sandwich, the next line shows \$1.25 for water, and the last line shows \$0.33 for tax. The total is shown to be \$5.03.], alt: "A vertical addition problem is shown. The top line shows $3.45 for a sandwich, the next line shows $1.25 for water, and the last line shows $0.33 for tax. The total is shown to be $5.03.", caption: none)

The total is #math.equation(block: false, alt: "$5.03.")[$"$5.03" .$] Suppose you pay with a #math.equation(block: false, alt: "$5")[$"$5"$] bill and #math.equation(block: false, alt: "3")[$3$] pennies. Should you wait for change? No, #math.equation(block: false, alt: "$5")[$"$5"$] and #math.equation(block: false, alt: "3")[$3$] pennies is the same as #math.equation(block: false, alt: "$5.03.")[$"$5.03" .$]

Because #math.equation(block: false, alt: "100 pennies equals $1 ,")[$"100 pennies" = "$1" ,$] each penny is worth #math.equation(block: false, alt: "the fraction 1 over 100")[$frac(1, 100)$] of a dollar. We write the value of one penny as #math.equation(block: false, alt: "$0.01 ,")[$\$0.01 ,$] since #math.equation(block: false, alt: "0.01 equals the fraction 1 over 100.")[$0.01 = frac(1, 100) .$]

Writing a number with a decimal is known as decimal notation. It is a way of showing parts of a whole when the whole is a power of ten. In other words, decimals are another way of writing fractions whose denominators are powers of ten. Just as the counting numbers are based on powers of ten, decimals are based on powers of ten.  shows the counting numbers.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Counting number], [Name]),
  [#math.equation(block: false, alt: "1")[$1$]], [One],
  [#math.equation(block: false, alt: "10 equals 10")[$10 = 10$]], [Ten],
  [#math.equation(block: false, alt: "10 times 10 equals 100")[$10 · 10 = 100$]], [One hundred],
  [#math.equation(block: false, alt: "10 times 10 times 10 equals 1000")[$10 · 10 · 10 = 1000$]], [One thousand],
  [#math.equation(block: false, alt: "10 times 10 times 10 times 10 equals 10,000")[$10 · 10 · 10 · 10 = 10,000$]], [Ten thousand],
))

How are decimals related to fractions?  shows the relation.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Decimal], [Fraction], [Name]),
  [#math.equation(block: false, alt: "0.1")[$0.1$]], [#math.equation(block: false, alt: "the fraction 1 over 10")[$frac(1, 10)$]], [One tenth],
  [#math.equation(block: false, alt: "0.01")[$0.01$]], [#math.equation(block: false, alt: "the fraction 1 over 100")[$frac(1, 100)$]], [One hundredth],
  [#math.equation(block: false, alt: "0.001")[$0.001$]], [#math.equation(block: false, alt: "the fraction 1 over 1,000")[$frac(1, 1","000)$]], [One thousandth],
  [#math.equation(block: false, alt: "0.0001")[$0.0001$]], [#math.equation(block: false, alt: "the fraction 1 over 10,000")[$frac(1, 10","000)$]], [One ten-thousandth],
))

When we name a whole number, the name corresponds to the place value based on the powers of ten. In Whole Numbers, we learned to read #math.equation(block: false, alt: "10,000")[$10,000$] as #emph[ten thousand]. Likewise, the names of the decimal places correspond to their #strong[fraction] values. Notice how the place value names in  relate to the names of the fractions from .

#figure(figph[A chart is shown labeled “Place Value”. There are 12 columns. The columns are labeled, from left to right, Hundred thousands, Ten thousands, Thousands, Hundreds, Tens, Ones, Decimal Point, Tenths, Hundredths, Thousandths, Ten-thousandths, Hundred-thousandths.], alt: "A chart is shown labeled “Place Value”. There are 12 columns. The columns are labeled, from left to right, Hundred thousands, Ten thousands, Thousands, Hundreds, Tens, Ones, Decimal Point, Tenths, Hundredths, Thousandths, Ten-thousandths, Hundred-thousandths.", caption: [This chart illustrates place values to the left and right of the decimal point.])

Notice two important facts shown.

- The “th” at the end of the name means the number is a fraction. “One thousand” is a number larger than one, but “one thousandth” is a number smaller than one.
- The tenths place is the first place to the right of the decimal, but the tens place is two places to the left of the decimal.

Remember that #math.equation(block: false, alt: "$5.03")[$"$5" . 0 3$] lunch? We read #math.equation(block: false, alt: "$5.03")[$"$5.03"$] as #emph[five dollars and three cents]. Naming decimals (those that don’t represent money) is done in a similar way. We read the number #math.equation(block: false, alt: "5.03")[$5.03$] as #emph[five and three hundredths].

We sometimes need to translate a number written in #strong[decimal notation] into words. As shown in , we write the amount on a check in both words and numbers.

#figure(figph[An image of a check is shown. The check is made out to Jane Doe. It shows the number \$152.65 and says in words, “One hundred fifty two and 65 over 100 dollars.”], alt: "An image of a check is shown. The check is made out to Jane Doe. It shows the number $152.65 and says in words, “One hundred fifty two and 65 over 100 dollars.”", caption: [When we write a check, we write the amount as a decimal number as well as in words. The bank looks at the check to make sure both numbers match. This helps prevent errors.])

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Let’s try naming a decimal, such as 15.68.], []),
  [We start by naming the number to the left of the decimal.], [fifteen\_\_\_\_\_\_],
  [We use the word “and” to indicate the decimal point.], [fifteen and\_\_\_\_\_],
  [Then we name the number to the right of the decimal point as if it were a whole number.], [fifteen and sixty-eight\_\_\_\_\_],
  [Last, name the decimal place of the last digit.], [fifteen and sixty-eight hundredths],
))

The number #math.equation(block: false, alt: "15.68")[$15.68$] is read #emph[fifteen and sixty-eight hundredths].

#notebox("How To", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Name a decimal number.]

- Name the number to the left of the decimal point.
- Write “and” for the decimal point.
- Name the “number” part to the right of the decimal point as if it were a whole number.
- Name the decimal place of the last digit.
]

#examplebox("Example 1")[][
Name each decimal: ⓐ #math.equation(block: false, alt: "4.3")[$#h(0.2em) 4.3$] ⓑ #math.equation(block: false, alt: "2.45")[$#h(0.2em) 2.45$] ⓒ #math.equation(block: false, alt: "0.009")[$#h(0.2em) 0.009$] ⓓ #math.equation(block: false, alt: "−15.571.")[$#h(0.2em) −15.571 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [4.3],
  [Name the number to the left of the decimal point.], [four\_\_\_\_\_],
  [Write "and" for the decimal point.], [four and\_\_\_\_\_],
  [Name the number to the right of the decimal point as if it were a whole number.], [four and three\_\_\_\_\_],
  [Name the decimal place of the last digit.], [four and three tenths],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [2.45],
  [Name the number to the left of the decimal point.], [two\_\_\_\_\_],
  [Write "and" for the decimal point.], [two and\_\_\_\_\_],
  [Name the number to the right of the decimal point as if it were a whole number.], [two and forty-five\_\_\_\_\_],
  [Name the decimal place of the last digit.], [two and forty-five hundredths],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [0.009],
  [Name the number to the left of the decimal point.], [Zero is the number to the left of the decimal; it is not included in the name.],
  [Name the number to the right of the decimal point as if it were a whole number.], [nine\_\_\_\_\_],
  [Name the decimal place of the last digit.], [nine thousandths],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓓ], []),
  [], [#math.equation(block: false, alt: "minus 15.571")[$− 15.571$]],
  [Name the number to the left of the decimal point.], [negative fifteen],
  [Write "and" for the decimal point.], [negative fifteen and\_\_\_\_\_],
  [Name the number to the right of the decimal point as if it were a whole number.], [negative fifteen and five hundred seventy-one\_\_\_\_\_],
  [Name the decimal place of the last digit.], [negative fifteen and five hundred seventy-one thousandths],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Name each decimal:

ⓐ #math.equation(block: false, alt: "6.7")[$#h(0.2em) 6.7$] ⓑ #math.equation(block: false, alt: "19.58")[$#h(0.2em) 19.58$] ⓒ #math.equation(block: false, alt: "0.018")[$#h(0.2em) 0.018$] ⓓ #math.equation(block: false, alt: "−2.053")[$#h(0.2em) #h(0.2em) −2.053$]

#solutionbox[
+ ⓐ six and seven tenths
+ ⓑ nineteen and fifty-eight hundredths
+ ⓒ eighteen thousandths
+ ⓓ negative two and fifty-three thousandths
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Name each decimal:

ⓐ #math.equation(block: false, alt: "5.8")[$#h(0.2em) 5.8$] ⓑ #math.equation(block: false, alt: "3.57")[$#h(0.2em) 3.57$] ⓒ #math.equation(block: false, alt: "0.005")[$#h(0.2em) 0.005$] ⓓ #math.equation(block: false, alt: "−13.461")[$#h(0.2em) −13.461$]

#solutionbox[
+ ⓐ five and eight tenths
+ ⓑ three and fifty-seven hundredths
+ ⓒ five thousandths
+ ⓓ negative thirteen and four hundred sixty-one thousandths
]
]


=== Write Decimals

Now we will translate the name of a decimal number into decimal notation. We will reverse the procedure we just used.

Let’s start by writing the number six and seventeen hundredths:

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [six and seventeen hundredths]),
  [The word #emph[and] tells us to place a decimal point.], [\_\_\_.\_\_\_],
  [The word before #emph[and] is the whole number; write it to the left of the decimal point.], [6.\_\_\_\_\_],
  [The decimal part is seventeen hundredths.  #linebreak()  Mark two places to the right of the decimal point for hundredths.], [6.\_ \_],
  [Write the numerals for seventeen in the places marked.], [6.17],
))

#examplebox("Example 2")[][
Write fourteen and thirty-seven hundredths as a decimal.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [fourteen and thirty-seven hundredths]),
  [Place a decimal point under the word ‘and’.], [\_\_\_\_\_\_. \_\_\_\_\_\_\_\_\_],
  [Translate the words before ‘and’ into the whole number and place it to the left of the decimal point.], [14. \_\_\_\_\_\_\_\_\_],
  [Mark two places to the right of the decimal point for “hundredths”.], [14.\_\_ \_\_],
  [Translate the words after “and” and write the number to the right of the decimal point.], [14.37],
  [], [Fourteen and thirty-seven hundredths is written 14.37.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a decimal: thirteen and sixty-eight hundredths.

#solutionbox[
13.68
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a decimal: five and eight hundred ninety-four thousandths.

#solutionbox[
5.894
]
]

#notebox("How To", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Write a decimal number from its name.]

+ Look for the word “and”—it locates the decimal point.
+ Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.

- Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point.
- If there is no “and,” write a “0” with a decimal point to its right.
+ Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
+ Fill in zeros for place holders as needed.
]

The second bullet in Step 2 is needed for decimals that have no whole number part, like ‘nine thousandths’. We recognize them by the words that indicate the place value after the decimal – such as ‘tenths’ or ‘hundredths.’  Since there is no whole number, there is no ‘and.’ We start by placing a zero to the left of the decimal and continue by filling in the numbers to the right, as we did above.

#examplebox("Example 3")[][
Write twenty-four thousandths as a decimal.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [twenty-four thousandths]),
  [Look for the word "and".], [There is no "and" so start with 0  #linebreak()  0.],
  [To the right of the decimal point, put three decimal places for thousandths.], [#figure(figph[A math problem template showing '0.' followed by three blank spaces, labeled respectively 'tenths', 'hundredths', and 'thousandths', illustrating decimal place values.], alt: "A math problem template showing '0.' followed by three blank spaces, labeled respectively 'tenths', 'hundredths', and 'thousandths', illustrating decimal place values.", caption: none)],
  [Write the number 24 with the 4 in the thousandths place.], [#figure(figph[A decimal number is shown as 0. \_ (blank for tenths), 2 (hundredths), and 4 (thousandths). The place values 'tenths', 'hundredths', and 'thousandths' are explicitly labeled below their respective positions.], alt: "A decimal number is shown as 0. _ (blank for tenths), 2 (hundredths), and 4 (thousandths). The place values 'tenths', 'hundredths', and 'thousandths' are explicitly labeled below their respective positions.", caption: none)],
  [Put zeros as placeholders in the remaining decimal places.], [0.024],
  [], [So, twenty-four thousandths is written 0.024],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a decimal: fifty-eight thousandths.

#solutionbox[
0.058
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a decimal: sixty-seven thousandths.

#solutionbox[
0.067
]
]

Before we move on to our next objective, think about money again. We know that #math.equation(block: false, alt: "$1")[$"$1"$] is the same as #math.equation(block: false, alt: "$1.00.")[$"$1.00" .$] The way we write #math.equation(block: false, alt: "$1 open parenthesis or $1.00 close parenthesis")[$"$1" #h(0.2em) ( "or" #h(0.2em) "$1.00" )$] depends on the context. In the same way, integers can be written as decimals with as many zeros as needed to the right of the decimal.

#math.equation(block: true, alt: "5 equals 5.0, −2 equals −2.0; 5 equals 5.00, −2 equals −2.00; 5 equals 5.000, −2 equals −2.000")[$5 = 5.0 &  &  & −2 = −2.0 \ 5 = 5.00 &  &  & −2 = −2.00 \ 5 = 5.000 &  &  & −2 = −2.000$]#math.equation(block: true, alt: "and so on…")[$"and so on…"$]
=== Convert Decimals to Fractions or Mixed Numbers

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that #math.equation(block: false, alt: "$5.03")[$"$5.03"$] means #math.equation(block: false, alt: "5")[$5$] dollars and #math.equation(block: false, alt: "3")[$3$] cents. Since there are #math.equation(block: false, alt: "100")[$100$] cents in one dollar, #math.equation(block: false, alt: "3")[$3$] cents means #math.equation(block: false, alt: "the fraction 3 over 100")[$frac(3, 100)$] of a dollar, so #math.equation(block: false, alt: "0.03 equals the fraction 3 over 100.")[$0.03 = frac(3, 100) .$]

We convert decimals to fractions by identifying the place value of the farthest right digit. In the decimal #math.equation(block: false, alt: "0.03 ,")[$0.03 ,$] the #math.equation(block: false, alt: "3")[$3$] is in the hundredths place, so #math.equation(block: false, alt: "100")[$100$] is the denominator of the fraction equivalent to #math.equation(block: false, alt: "0.03.")[$0.03 .$]

#math.equation(block: true, alt: "0.03 equals the fraction 3 over 100")[$0.03 = frac(3, 100)$]For our #math.equation(block: false, alt: "$5.03")[$"$5.03"$] lunch, we can write the decimal #math.equation(block: false, alt: "5.03")[$5.03$] as a mixed number.

#math.equation(block: true, alt: "5.03 equals 5 the fraction 3 over 100")[$5.03 = 5 frac(3, 100)$]Notice that when the number to the left of the decimal is zero, we get a proper fraction. When the number to the left of the decimal is not zero, we get a mixed number.

#notebox("How To", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Convert a decimal number to a fraction or mixed number.]

+ Look at the number to the left of the decimal.
   - If it is zero, the decimal converts to a proper fraction.
- If it is not zero, the decimal converts to a mixed number.
   		- Write the whole number.
+ Determine the place value of the final digit.
+ Write the fraction.
	- numerator—the ‘numbers’ to the right of the decimal point
- denominator—the place value corresponding to the final digit
+ Simplify the fraction, if possible.
]

#examplebox("Example 4")[][
Write each of the following decimal numbers as a fraction or a mixed number:

ⓐ #math.equation(block: false, alt: "4.09")[$#h(0.2em) 4.09$] ⓑ #math.equation(block: false, alt: "3.7")[$#h(0.2em) 3.7$] ⓒ #math.equation(block: false, alt: "−0.286")[$#h(0.2em) −0.286$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [4.09],
  [There is a 4 to the left of the decimal point.  #linebreak()  Write "4" as the whole number part of the mixed number.], [#figure(figph[The number 4 is followed by a fraction with an empty square numerator and an empty square denominator, suggesting an incomplete mathematical expression or a prompt to fill in the blanks.], alt: "The number 4 is followed by a fraction with an empty square numerator and an empty square denominator, suggesting an incomplete mathematical expression or a prompt to fill in the blanks.", caption: none)],
  [Determine the place value of the final digit.], [#figure(figph[A partially obscured image displays a decimal number, with '4.' visible, followed by '0' and '9'. Below '0' is the word 'tenths' in light blue, and below '9' is 'hundredths' in light blue, indicating place values.], alt: "A partially obscured image displays a decimal number, with '4.' visible, followed by '0' and '9'. Below '0' is the word 'tenths' in light blue, and below '9' is 'hundredths' in light blue, indicating place values.", caption: none)],
  [Write the fraction.  #linebreak()  Write 9 in the numerator as it is the number to the right of the decimal point.], [#figure(figph[A mathematical expression showing a mixed number where the whole number is 4, the numerator of the fraction is 9, and the denominator is represented by an empty rectangle or box, indicating a missing value.], alt: "A mathematical expression showing a mixed number where the whole number is 4, the numerator of the fraction is 9, and the denominator is represented by an empty rectangle or box, indicating a missing value.", caption: none)],
  [Write 100 in the denominator as the place value of the final digit, 9, is hundredth.], [#figure(figph[A mixed number is displayed, consisting of the whole number 4, followed by the fraction 9/100, where 9 is the numerator and 100 is the denominator. The expression represents four and nine hundredths.], alt: "A mixed number is displayed, consisting of the whole number 4, followed by the fraction 9/100, where 9 is the numerator and 100 is the denominator. The expression represents four and nine hundredths.", caption: none)],
  [The fraction is in simplest form.], [#figure(figph[The image shows the conversion of the decimal number 4.09 into a mixed fraction, expressed as 'So, 4.09 = 4 9/100' on a white background.], alt: "The image shows the conversion of the decimal number 4.09 into a mixed fraction, expressed as 'So, 4.09 = 4 9/100' on a white background.", caption: none)],
))

Did you notice that the number of zeros in the denominator is the same as the number of #strong[decimal] places?

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [3.7],
  [There is a 3 to the left of the decimal point.  #linebreak()  Write "3" as the whole number part of the mixed number.], [#figure(figph[The number 3 is shown next to a horizontal fraction line, with an empty square box above the line and another empty square box below the line, against a white background.], alt: "The number 3 is shown next to a horizontal fraction line, with an empty square box above the line and another empty square box below the line, against a white background.", caption: none)],
  [Determine the place value of the final digit.], [#figure(figph[The number '3. 7' is displayed above the word 'tenths', indicating the value 3 and 7 tenths.], alt: "The number '3. 7' is displayed above the word 'tenths', indicating the value 3 and 7 tenths.", caption: none)],
  [Write the fraction.  #linebreak()  Write 7 in the numerator as it is the number to the right of the decimal point.], [#figure(figph[A mathematical expression showing the number 3 next to a fraction bar with the number 7 above it and an empty square below it, representing an incomplete mixed number or fraction.], alt: "A mathematical expression showing the number 3 next to a fraction bar with the number 7 above it and an empty square below it, representing an incomplete mixed number or fraction.", caption: none)],
  [Write 10 in the denominator as the place value of the final digit, 7, is tenths.], [#figure(figph[The mixed number 3 and 7/10 is displayed in a dark blue font against a white background.], alt: "The mixed number 3 and 7/10 is displayed in a dark blue font against a white background.", caption: none)],
  [The fraction is in simplest form.], [#figure(figph[A mathematical equation showing the conversion of a decimal to a mixed number, specifically 'So, 3.7 = 3 7/10'.], alt: "A mathematical equation showing the conversion of a decimal to a mixed number, specifically 'So, 3.7 = 3 7/10'.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [−0.286],
  [There is a 0 to the left of the decimal point.  #linebreak()  Write a negative sign before the fraction.], [#figure(figph[A mathematical expression featuring a negative sign followed by a fraction, where both the numerator and denominator are depicted as empty square placeholders.], alt: "A mathematical expression featuring a negative sign followed by a fraction, where both the numerator and denominator are depicted as empty square placeholders.", caption: none)],
  [Determine the place value of the final digit and write it in the denominator.], [#figure(figph[A decimal number -0.286 is shown with its place values labeled: 2 is in the tenths place, 8 is in the hundredths place, and 6 is in the thousandths place.], alt: "A decimal number -0.286 is shown with its place values labeled: 2 is in the tenths place, 8 is in the hundredths place, and 6 is in the thousandths place.", caption: none)],
  [Write the fraction.  #linebreak()  Write 286 in the numerator as it is the number to the right of the decimal point. #linebreak() Write 1,000 in the denominator as the place value of the final digit, 6, is thousandths.], [#figure(figph[A negative fraction is displayed, with the number 286 in the numerator and 1000 in the denominator.], alt: "A negative fraction is displayed, with the number 286 in the numerator and 1000 in the denominator.", caption: none)],
  [We remove a common factor of 2 to simplify the fraction.], [#figure(figph[The image displays the negative fraction -143/500, a mathematical expression showing a division of 143 by 500 with a negative sign preceding it. It is presented on a plain white background.], alt: "The image displays the negative fraction -143/500, a mathematical expression showing a division of 143 by 500 with a negative sign preceding it. It is presented on a plain white background.", caption: none)],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a fraction or mixed number. Simplify the answer if possible.

ⓐ #math.equation(block: false, alt: "5.3")[$#h(0.2em) 5.3$] ⓑ #math.equation(block: false, alt: "6.07")[$#h(0.2em) 6.07$] ⓒ #math.equation(block: false, alt: "−0.234")[$#h(0.2em) −0.234$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "5 the fraction 3 over 10")[$#h(0.2em) 5 frac(3, 10)$]
+ ⓑ #math.equation(block: false, alt: "6 the fraction 7 over 100")[$#h(0.2em) 6 frac(7, 100)$]
+ ⓒ #math.equation(block: false, alt: "minus the fraction 117 over 500")[$#h(0.2em) − frac(117, 500)$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a fraction or mixed number. Simplify the answer if possible.

ⓐ #math.equation(block: false, alt: "8.7")[$#h(0.2em) 8.7$] ⓑ #math.equation(block: false, alt: "1.03")[$#h(0.2em) 1.03$] ⓒ #math.equation(block: false, alt: "−0.024")[$#h(0.2em) −0.024$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "8 the fraction 7 over 10")[$#h(0.2em) 8 frac(7, 10)$]
+ ⓑ #math.equation(block: false, alt: "1 the fraction 3 over 100")[$#h(0.2em) 1 frac(3, 100)$]
+ ⓒ #math.equation(block: false, alt: "minus the fraction 3 over 125")[$#h(0.2em) − frac(3, 125)$]
]
]


=== Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

#examplebox("Example 5")[][
Locate #math.equation(block: false, alt: "0.4")[$0.4$] on a number line.

#solutionbox[
The decimal #math.equation(block: false, alt: "0.4")[$0.4$] is equivalent to #math.equation(block: false, alt: "the fraction 4 over 10 ,")[$frac(4, 10) ,$] so #math.equation(block: false, alt: "0.4")[$0.4$] is located between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "1.")[$1 .$] On a number line, divide the interval between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "1")[$1$] into #math.equation(block: false, alt: "10")[$10$] equal parts and place marks to separate the parts.

Label the marks #math.equation(block: false, alt: "0.1 , 0.2 , 0.3 , 0.4 , 0.5 , 0.6 , 0.7 , 0.8 , 0.9 , 1.0.")[$0.1 , 0.2 , 0.3 , 0.4 , 0.5 , 0.6 , 0.7 , 0.8 , 0.9 , 1.0 .$] We write #math.equation(block: false, alt: "0")[$0$] as #math.equation(block: false, alt: "0.0")[$0.0$] and #math.equation(block: false, alt: "1")[$1$] as #math.equation(block: false, alt: "1.0 ,")[$1.0 ,$] so that the numbers are consistently in tenths. Finally, mark #math.equation(block: false, alt: "0.4")[$0.4$] on the number line. #linebreak() 
#figure(figph[A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a red dot at 0.4.], alt: "A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a red dot at 0.4.", caption: none)
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Locate #math.equation(block: false, alt: "0.6")[$0.6$] on a number line.

#solutionbox[
#figure(figph[This image shows a number line from 0.0 to 1.0 and segmented into tenths.  A point is plotted at 0.6 on the number line.], alt: "This image shows a number line from 0.0 to 1.0 and segmented into tenths. A point is plotted at 0.6 on the number line.", caption: none)
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Locate #math.equation(block: false, alt: "0.9")[$0.9$] on a number line.

#solutionbox[
#figure(figph[This image shows a number line from 0.0 to 1.0 and segmented into tenths.  A point is plotted at 0.9 on the number line.], alt: "This image shows a number line from 0.0 to 1.0 and segmented into tenths. A point is plotted at 0.9 on the number line.", caption: none)
]
]

#examplebox("Example 6")[][
Locate #math.equation(block: false, alt: "−0.74")[$−0.74$] on a number line.

#solutionbox[
The decimal #math.equation(block: false, alt: "−0.74")[$−0.74$] is equivalent to #math.equation(block: false, alt: "minus the fraction 74 over 100 ,")[$− frac(74, 100) ,$] so it is located between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "−1.")[$−1 .$] On a number line, mark off and label the multiples of #math.equation(block: false, alt: "-0.10")[$-0.10$] in the interval between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "−1")[$−1$] (#math.equation(block: false, alt: "−0.10")[$−0.10$], #math.equation(block: false, alt: "−0.20")[$−0.20$], etc.) and mark  #math.equation(block: false, alt: "−0.74")[$−0.74$] between #math.equation(block: false, alt: "−0.70")[$−0.70$] and #math.equation(block: false, alt: "−0.80,")[$−0.80,$] a little closer to #math.equation(block: false, alt: "−0.70")[$−0.70$].

 #linebreak() 
#figure(figph[A number line is shown with negative 1.00, negative 0.90, negative 0.80, negative 0.70, negative 0.60, negative 0.50, negative 0.40, negative 0.30, negative 0.20, negative 0.10, and 0.00 labeled. There is a red dot between negative 0.80 and negative 0.70 labeled as negative 0.74.], alt: "A number line is shown with negative 1.00, negative 0.90, negative 0.80, negative 0.70, negative 0.60, negative 0.50, negative 0.40, negative 0.30, negative 0.20, negative 0.10, and 0.00 labeled. There is a red dot between negative 0.80 and negative 0.70 labeled as negative 0.74.", caption: none)
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Locate #math.equation(block: false, alt: "−0.63")[$−0.63$] on a number line.

#solutionbox[
#figure(figph[This image shows a number line from -1.00 to 0.00 . A point is plotted at negative 0.63 on the number line.], alt: "This image shows a number line from -1.00 to 0.00 . A point is plotted at negative 0.63 on the number line.", caption: none)
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Locate #math.equation(block: false, alt: "−0.25")[$−0.25$] on a number line.

#solutionbox[
#figure(figph[This image shows a number line from -1.00 to 0.00 . A point is plotted at negative 0.25 on the number line.], alt: "This image shows a number line from -1.00 to 0.00 . A point is plotted at negative 0.25 on the number line.", caption: none)
]
]


=== Order Decimals

Which is larger, #math.equation(block: false, alt: "0.04")[$0.04$] or #math.equation(block: false, alt: "0.40 ?")[$0.40 ?$]

If you think of this as money, you know that #math.equation(block: false, alt: "$0.40")[$"$0.40"$] (forty cents) is greater than #math.equation(block: false, alt: "$0.04")[$"$0.04"$] (four cents). So,

#math.equation(block: true, alt: "0.40 greater than 0.04")[$0.40 > 0.04$]In previous chapters, we used the number line to order numbers.

#math.equation(block: true, alt: "a less than b ‘ a is less than b ’ when a is to the left of b on the number line; a greater than b ‘ a is greater than b ’ when a is to the right of b on the number line")[$a < b #h(0.5em) ‘ a #h(0.2em) "is less than" #h(0.2em) b ’ #h(0.2em) "when" #h(0.2em) a #h(0.2em) "is to the left of" #h(0.2em) b #h(0.2em) "on the number line" \ a > b #h(0.5em) ‘ a #h(0.2em) "is greater than" #h(0.2em) b ’ #h(0.2em) "when" #h(0.2em) a #h(0.2em) "is to the right of" #h(0.2em) b #h(0.2em) "on the number line"$]Where are #math.equation(block: false, alt: "0.04")[$0.04$] and #math.equation(block: false, alt: "0.40")[$0.40$] located on the number line?

#figure(figph[A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a red dot between 0.0 and 0.1 labeled as 0.04. There is another red dot at 0.4.], alt: "A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a red dot between 0.0 and 0.1 labeled as 0.04. There is another red dot at 0.4.", caption: none)

We see that #math.equation(block: false, alt: "0.40")[$0.40$] is to the right of #math.equation(block: false, alt: "0.04.")[$0.04 .$] So we know #math.equation(block: false, alt: "0.40 greater than 0.04.")[$0.40 > 0.04 .$]

How does #math.equation(block: false, alt: "0.31")[$0.31$] compare to #math.equation(block: false, alt: "0.308 ?")[$0.308 ?$] This doesn’t translate into money to make the comparison easy. But if we convert #math.equation(block: false, alt: "0.31")[$0.31$] and #math.equation(block: false, alt: "0.308")[$0.308$] to fractions, we can tell which is larger.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "0.31")[$0.31$]], [#math.equation(block: false, alt: "0.308")[$0.308$]]),
  [Convert to fractions.], [#math.equation(block: false, alt: "the fraction 31 over 100")[$frac(31, 100)$]], [#math.equation(block: false, alt: "the fraction 308 over 1000")[$frac(308, 1000)$]],
  [We need a common denominator to compare them.], [#figure(figph[A fraction with a numerator of 31 multiplied by 10 and a denominator of 100 multiplied by 10. The number 10 is highlighted in red in both the numerator and denominator.], alt: "A fraction with a numerator of 31 multiplied by 10 and a denominator of 100 multiplied by 10. The number 10 is highlighted in red in both the numerator and denominator.", caption: none)], [#math.equation(block: false, alt: "the fraction 308 over 1000")[$frac(308, 1000)$]],
  [], [#math.equation(block: false, alt: "the fraction 310 over 1000")[$frac(310, 1000)$]], [#math.equation(block: false, alt: "the fraction 308 over 1000")[$frac(308, 1000)$]],
))

Because #math.equation(block: false, alt: "310 greater than 308 ,")[$310 > 308 ,$] we know that #math.equation(block: false, alt: "the fraction 310 over 1000 greater than the fraction 308 over 1000.")[$frac(310, 1000) > frac(308, 1000) .$] Therefore, #math.equation(block: false, alt: "0.31 greater than 0.308.")[$0.31 > 0.308 .$]

Notice what we did in converting #math.equation(block: false, alt: "0.31")[$0.31$] to a fraction—we started with the fraction #math.equation(block: false, alt: "the fraction 31 over 100")[$frac(31, 100)$] and ended with the equivalent fraction #math.equation(block: false, alt: "the fraction 310 over 1000.")[$frac(310, 1000) .$] Converting #math.equation(block: false, alt: "the fraction 310 over 1000")[$frac(310, 1000)$] back to a decimal gives #math.equation(block: false, alt: "0.310.")[$0.310 .$] So #math.equation(block: false, alt: "0.31")[$0.31$] is equivalent to #math.equation(block: false, alt: "0.310.")[$0.310 .$] Writing zeros at the end of a decimal does not change its value.

#math.equation(block: true, alt: "the fraction 31 over 100 equals the fraction 310 over 1000 and 0.31 equals 0.310")[$frac(31, 100) = frac(310, 1000) #h(0.4em) "and" #h(0.2em) 0.31 = 0.310$]If two decimals have the same value, they are said to be equivalent decimals.

#math.equation(block: true, alt: "0.31 equals 0.310")[$0.31 = 0.310$]We say #math.equation(block: false, alt: "0.31")[$0.31$] and #math.equation(block: false, alt: "0.310")[$0.310$] are equivalent decimals.

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

Two decimals are #strong[equivalent decimals] if they convert to equivalent fractions.
]

Remember, writing zeros at the end of a decimal does not change its value.

#notebox("How To", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Order decimals.]

+ Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match.
+ Compare the numbers to the right of the decimal point as if they were whole numbers.
+ Order the numbers using the appropriate inequality sign.
]

#examplebox("Example 7")[][
Order the following decimals using #math.equation(block: false, alt: "less than or >:")[$< #h(0.2em) "or" #h(0.2em) ">:"$]

+ ⓐ #math.equation(block: false, alt: "0.64 __ 0.6")[$#h(0.2em) 0.64 #h(0.2em) \_\_ 0.6$]
+ ⓑ #math.equation(block: false, alt: "0.83 __ 0.803")[$#h(0.2em) 0.83 #h(0.2em) \_\_ 0.803$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "0.64 __ 0.6")[$#h(0.2em) 0.64 #h(0.2em) \_\_ 0.6$]],
  [Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.6.], [#math.equation(block: false, alt: "0.64 __ 0.60")[$#h(0.2em) 0.64 #h(0.2em) \_\_ 0.60$]],
  [Compare the numbers to the right of the decimal point as if they were whole numbers.], [#math.equation(block: false, alt: "64 greater than 60")[$64 > 60$]],
  [Order the numbers using the appropriate inequality sign.], [#math.equation(block: false, alt: "0.64 greater than 0.60")[$0.64 > 0.60$] #linebreak()  #linebreak() #math.equation(block: false, alt: "0.64 greater than 0.6")[$0.64 > 0.6$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "0.83 __ 0.803")[$#h(0.2em) 0.83 #h(0.2em) \_\_ 0.803$]],
  [Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.83.], [#math.equation(block: false, alt: "0.830 __ 0.803")[$#h(0.2em) 0.830 #h(0.2em) \_\_ 0.803$]],
  [Compare the numbers to the right of the decimal point as if they were whole numbers.], [#math.equation(block: false, alt: "830 greater than 803")[$830 > 803$]],
  [Order the numbers using the appropriate inequality sign.], [#math.equation(block: false, alt: "0.830 greater than 0.803")[$0.830 > 0.803$] #linebreak()  #linebreak() #math.equation(block: false, alt: "0.83 greater than 0.803")[$0.83 > 0.803$]],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Order each of the following pairs of numbers, using #math.equation(block: false, alt: "less than or >:")[$< #h(0.2em) "or" #h(0.2em) ">:"$]

ⓐ #math.equation(block: false, alt: "0.42 __ 0.4")[$#h(0.2em) 0.42 \_\_ 0.4$] ⓑ #math.equation(block: false, alt: "0.76 __ 0.706")[$#h(0.2em) 0.76 \_\_ 0.706$]

#solutionbox[
+ ⓐ \>
+ ⓑ \>
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Order each of the following pairs of numbers, using #math.equation(block: false, alt: "less than or >:")[$< #h(0.2em) "or" #h(0.2em) ">:"$]

ⓐ #math.equation(block: false, alt: "0.1 __ 0.18")[$#h(0.2em) 0.1 \_\_ 0.18$] ⓑ #math.equation(block: false, alt: "0.305 __ 0.35")[$#h(0.2em) 0.305 \_\_ 0.35$]

#solutionbox[
+ ⓐ \<
+ ⓑ \<
]
]

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because #math.equation(block: false, alt: "−2")[$−2$] lies to the right of #math.equation(block: false, alt: "−3")[$−3$] on the number line, we know that #math.equation(block: false, alt: "−2 greater than −3.")[$−2 > −3 .$] Similarly, smaller numbers lie to the left on the number line. For example, because #math.equation(block: false, alt: "−9")[$−9$] lies to the left of #math.equation(block: false, alt: "−6")[$−6$] on the number line, we know that #math.equation(block: false, alt: "−9 less than −6.")[$−9 < −6 .$]

#figure(figph[A number line is shown with integers from negative 10 to 0. Blue dots are placed on negative nine and negative six. Red dots are placed at negative two and negative three.], alt: "A number line is shown with integers from negative 10 to 0. Blue dots are placed on negative nine and negative six. Red dots are placed at negative two and negative three.", caption: none)

If we zoomed in on the interval between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "−1 ,")[$−1 ,$] we would see in the same way that #math.equation(block: false, alt: "−0.2 greater than −0.3 and −0.9 less than −0.6.")[$−0.2 > −0.3 #h(0.2em) "and" #h(0.2em) −0.9 < −0.6 .$]

#examplebox("Example 8")[][
Use #math.equation(block: false, alt: "less than or greater than")[$< #h(0.2em) "or" #h(0.2em) >$] to order. #math.equation(block: false, alt: "−0.1 __ −0.8.")[$−0.1 \_\_ −0.8 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−0.1 __ −0.8")[$#h(0.2em) −0.1 #h(0.2em) \_\_ −0.8$]]),
  [Write the numbers one under the other, lining up the decimal points.], [#math.equation(block: false, alt: "−0.1")[$−0.1$] #linebreak()  #linebreak() #math.equation(block: false, alt: "−0.8")[$−0.8$]],
  [They have the same number of digits.], [],
  [Since #math.equation(block: false, alt: "−1 greater than −8 , −1")[$−1 > −8 , −1$] tenth is greater than #math.equation(block: false, alt: "−8")[$−8$] tenths.], [#math.equation(block: false, alt: "−0.1 greater than −0.8")[$−0.1 > −0.8$]],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Order each of the following pairs of numbers, using #math.equation(block: false, alt: "less than or >:")[$< #h(0.2em) "or" #h(0.2em) ">:"$]

#math.equation(block: true, alt: "−0.3 ___ −0.5")[$−0.3 \_\_\_ −0.5$]

#solutionbox[
\>
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Order each of the following pairs of numbers, using #math.equation(block: false, alt: "less than or >:")[$< #h(0.2em) "or" #h(0.2em) ">:"$]

#math.equation(block: true, alt: "−0.6 ___ −0.7")[$−0.6 \_\_\_ −0.7$]

#solutionbox[
\>
]
]


=== Round Decimals

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at #math.equation(block: false, alt: "$3.279")[$"$3.279"$] per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay #math.equation(block: false, alt: "$3.28")[$"$3.28"$], because the final price would be rounded to the nearest cent. In Whole Numbers, we saw that we round numbers to get an approximate value when the exact value is not needed. Suppose we wanted to round #math.equation(block: false, alt: "$2.72")[$"$2.72"$] to the nearest dollar. Is it closer to #math.equation(block: false, alt: "$2")[$"$2"$] or to #math.equation(block: false, alt: "$3 ?")[$"$3" ?$] What if we wanted to round #math.equation(block: false, alt: "$2.72")[$"$2.72"$] to the nearest ten cents; is it closer to #math.equation(block: false, alt: "$2.70")[$"$2.70"$] or to #math.equation(block: false, alt: "$2.80 ?")[$"$2.80" ?$] The number lines in  can help us answer those questions.

#figure(figph[In part a, a number line is shown with 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 3. There is a dot between 2.7 and 2.8 labeled as 2.72.  In part b, a number line is shown with 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, and 2.80. There is a dot at 2.72. ], alt: "In part a, a number line is shown with 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 3. There is a dot between 2.7 and 2.8 labeled as 2.72. In part b, a number line is shown with 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, and 2.80. There is a dot at 2.72.", caption: [ⓐ We see that #math.equation(block: false, alt: "2.72")[$2.72$] is closer to #math.equation(block: false, alt: "3")[$3$] than to #math.equation(block: false, alt: "2.")[$2 .$] So, #math.equation(block: false, alt: "2.72")[$2.72$] rounded to the nearest whole number is #math.equation(block: false, alt: "3.")[$3 .$]  #linebreak() ⓑ We see that #math.equation(block: false, alt: "2.72")[$2.72$] is closer to #math.equation(block: false, alt: "2.70")[$2.70$] than #math.equation(block: false, alt: "2.80.")[$2.80 .$] So we say that #math.equation(block: false, alt: "2.72")[$2.72$] rounded to the nearest tenth is #math.equation(block: false, alt: "2.7.")[$2.7 .$]])

Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.

#notebox("How To", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Round a decimal.]

+ Locate the given place value and mark it with an arrow.
+ Underline the digit to the right of the given place value.
+ Is this digit greater than or equal to #math.equation(block: false, alt: "5 ?")[$5 ?$]
- Yes - add #math.equation(block: false, alt: "1")[$1$] to the digit in the given place value.
- No - do not change the digit in the given place value
+ Rewrite the number, removing all digits to the right of the given place value.
]

#examplebox("Example 9")[][
Round #math.equation(block: false, alt: "18.379")[$18.379$] to the nearest hundredth.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The number 18.379 is prominently displayed on a white background.], alt: "The number 18.379 is prominently displayed on a white background.", caption: none)]),
  [Locate the hundredths place and mark it with an arrow.], [#figure(figph[The digit 7 occupies the hundredths place in the decimal number 18.379, as indicated by the arrow pointing from 'hundredths place' to the 7.], alt: "The digit 7 occupies the hundredths place in the decimal number 18.379, as indicated by the arrow pointing from 'hundredths place' to the 7.", caption: none)],
  [Underline the digit to the right of the 7.], [#figure(figph[An arrow points from the text 'hundredths place' to the digit '7' in the number 18.379, indicating its position.], alt: "An arrow points from the text 'hundredths place' to the digit '7' in the number 18.379, indicating its position.", caption: none)],
  [Because 9 is greater than or equal to 5, add 1 to the 7.], [#figure(figph[An image illustrating a numerical operation, where the number 18.379 is shown with instructions to 'delete' the '.379' part and 'add 1' to the integer portion, effectively rounding up to 19.], alt: "An image illustrating a numerical operation, where the number 18.379 is shown with instructions to 'delete' the '.379' part and 'add 1' to the integer portion, effectively rounding up to 19.", caption: none)],
  [Rewrite the number, deleting all digits to the right of the hundredths place.], [#figure(figph[The numbers 18.38 are displayed in a dark teal font against a white background.], alt: "The numbers 18.38 are displayed in a dark teal font against a white background.", caption: none)],
  [], [18.38 is 18.379 rounded to the nearest hundredth.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to the nearest hundredth: #math.equation(block: false, alt: "1.047.")[$1.047 .$]

#solutionbox[
1.05
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to the nearest hundredth: #math.equation(block: false, alt: "9.173.")[$9.173 .$]

#solutionbox[
9.17
]
]

#examplebox("Example 10")[][
Round #math.equation(block: false, alt: "18.379")[$18.379$] to the nearest ⓐ tenth ⓑ whole number.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ Round 18.379 to the nearest tenth.], []),
  [], [#figure(figph[The number 18.379 is displayed in a dark teal font against a white background.], alt: "The number 18.379 is displayed in a dark teal font against a white background.", caption: none)],
  [Locate the tenths place and mark it with an arrow.], [#figure(figph[An arrow points from the text 'tenths place' to the digit '3' in the number '18.379', indicating the tenths place value in a decimal.], alt: "An arrow points from the text 'tenths place' to the digit '3' in the number '18.379', indicating the tenths place value in a decimal.", caption: none)],
  [Underline the digit to the right of the tenths digit.], [#figure(figph[An arrow points from 'tenths place' to the number 18.379, with the '3' underlined, illustrating the tenths place value in a decimal.], alt: "An arrow points from 'tenths place' to the number 18.379, with the '3' underlined, illustrating the tenths place value in a decimal.", caption: none)],
  [Because 7 is greater than or equal to 5, add 1 to the 3.], [#figure(figph[Illustration of rounding 18.379 to the nearest whole number. Because the tenths digit '3' (indicated by 'add 1') is less than 5, the decimal part is 'delete'd, resulting in 18.], alt: "Illustration of rounding 18.379 to the nearest whole number. Because the tenths digit '3' (indicated by 'add 1') is less than 5, the decimal part is 'delete'd, resulting in 18.", caption: none)],
  [Rewrite the number, deleting all digits to the right of the tenths place.], [#figure(figph[The number 18.4 is displayed in a teal or bluish-green color against a clean white background.], alt: "The number 18.4 is displayed in a teal or bluish-green color against a clean white background.", caption: none)],
  [], [So, 18.379 rounded to the nearest tenth is 18.4.],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ Round 18.379 to the nearest whole number.], []),
  [], [#figure(figph[A numerical value, '18.379', is displayed in a dark teal font against a plain white background.], alt: "A numerical value, '18.379', is displayed in a dark teal font against a plain white background.", caption: none)],
  [Locate the ones place and mark it with an arrow.], [#figure(figph[An arrow points from the text 'ones place' to the number '18.379', illustrating the concept of place value for the digit 8 in the ones place.], alt: "An arrow points from the text 'ones place' to the number '18.379', illustrating the concept of place value for the digit 8 in the ones place.", caption: none)],
  [Underline the digit to the right of the ones place.], [#figure(figph[An illustration of place value, with an arrow pointing from 'ones place' to the digit '8' in 18.379, demonstrating its position. The digit '3' is also underlined.], alt: "An illustration of place value, with an arrow pointing from 'ones place' to the digit '8' in 18.379, demonstrating its position. The digit '3' is also underlined.", caption: none)],
  [Since 3 is not greater than or equal to 5, do not add 1 to the 8.], [#figure(figph[An image illustrating a numerical operation, showing the number 18.379 with instructions to 'delete' the .379 portion and 'do not add 1' (implying no rounding up) for the remaining 18.], alt: "An image illustrating a numerical operation, showing the number 18.379 with instructions to 'delete' the .379 portion and 'do not add 1' (implying no rounding up) for the remaining 18.", caption: none)],
  [Rewrite the number, deleting all digits to the right of the ones place.], [#figure(figph[The number 18 is displayed in a dark blue-grey font on a clean white background, standing out with its simple yet clear presentation.], alt: "The number 18 is displayed in a dark blue-grey font on a clean white background, standing out with its simple yet clear presentation.", caption: none)],
  [], [So 18.379 rounded to the nearest whole number is 18.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round #math.equation(block: false, alt: "6.582")[$6.582$] to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

#solutionbox[
+ ⓐ 6.58
+ ⓑ 6.6
+ ⓒ 7
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round #math.equation(block: false, alt: "15.2175")[$15.2175$] to the nearest ⓐ thousandth ⓑ hundredth ⓒ tenth.

#solutionbox[
+ ⓐ 15.218
+ ⓑ 15.22
+ ⓒ 15.2
]
]

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

- #link("https://www.openstax.org/l/24decmlnotat")[Introduction to Decimal Notation]
- #link("https://www.openstax.org/l/24word2dcmlnot")[Write a Number in Decimal Notation from Words]
- #link("https://www.openstax.org/l/24decmlnumline")[Identify Decimals on the Number Line]
- #link("https://www.openstax.org/l/24rounddecml")[Rounding Decimals]
- #link("https://www.openstax.org/l/24decmlsimpfrac")[Writing a Decimal as a Simplified Fraction]
]


=== Key Concepts

- #strong[Name a decimal number.]
+ Name the number to the left of the decimal point.
+ Write “and” for the decimal point.
+ Name the “number” part to the right of the decimal point as if it were a whole number.
+ Name the decimal place of the last digit.
- #strong[Write a decimal number from its name.]
+ Look for the word “and”—it locates the decimal point. #linebreak() 
Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point. #linebreak() 
If there is no “and,” write a “0” with a decimal point to its right.
+ Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
+ Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
+ Fill in zeros for place holders as needed.
- #strong[Convert a decimal number to a fraction or mixed number.]
+ Look at the number to the left of the decimal. #linebreak() 
If it is zero, the decimal converts to a proper fraction. #linebreak() 
If it is not zero, the decimal converts to a mixed number. #linebreak() 
Write the whole number.
+ Determine the place value of the final digit.
+ Write the fraction.
numerator—the ‘numbers’ to the right of the decimal point
denominator—the place value corresponding to the final digit
+ Simplify the fraction, if possible.
- #strong[Order decimals.]
+ Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match.
+ Compare the numbers to the right of the decimal point as if they were whole numbers.
+ Order the numbers using the appropriate inequality sign.
- #strong[Round a decimal.]
+ Locate the given place value and mark it with an arrow.
+ Underline the digit to the right of the given place value.
+ Is this digit greater than or equal to 5? #linebreak() 
Yes - add 1 to the digit in the given place value. #linebreak() 
No - do not change the digit in the given place value
+ Rewrite the number, removing all digits to the right of the given place value.


=== Practice Makes Perfect

#strong[Name Decimals]

In the following exercises, name each decimal.

#math.equation(block: true, alt: "5.5")[$5.5$]

#solutionbox[
five and five tenths
]

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

#math.equation(block: true, alt: "5.01")[$5.01$]

#solutionbox[
five and one hundredth
]

#math.equation(block: true, alt: "14.02")[$14.02$]

#math.equation(block: true, alt: "8.71")[$8.71$]

#solutionbox[
eight and seventy-one hundredths
]

#math.equation(block: true, alt: "2.64")[$2.64$]

#math.equation(block: true, alt: "0.002")[$0.002$]

#solutionbox[
two thousandths
]

#math.equation(block: true, alt: "0.005")[$0.005$]

#math.equation(block: true, alt: "0.381")[$0.381$]

#solutionbox[
three hundred eighty-one thousandths
]

#math.equation(block: true, alt: "0.479")[$0.479$]

#math.equation(block: true, alt: "−17.9")[$−17.9$]

#solutionbox[
negative seventeen and nine tenths
]

#math.equation(block: true, alt: "−31.4")[$−31.4$]

#strong[Write Decimals]

In the following exercises, translate the name into a decimal number.

Eight and three hundredths

#solutionbox[
8.03
]

Nine and seven hundredths

Twenty-nine and eighty-one hundredths

#solutionbox[
29.81
]

Sixty-one and seventy-four hundredths

Seven tenths

#solutionbox[
0.7
]

Six tenths

One thousandth

#solutionbox[
0.001
]

Nine thousandths

Twenty-nine thousandths

#solutionbox[
0.029
]

Thirty-five thousandths

Negative eleven and nine ten-thousandths

#solutionbox[
−11.0009
]

Negative fifty-nine and two ten-thousandths

Thirteen and three hundred ninety-five ten thousandths

#solutionbox[
13.0395
]

Thirty and two hundred seventy-nine thousandths

#strong[Convert Decimals to Fractions or Mixed Numbers]

In the following exercises, convert each decimal to a fraction or mixed number.

#math.equation(block: true, alt: "1.99")[$1.99$]

#solutionbox[
#math.equation(block: true, alt: "1 the fraction 99 over 100")[$1 frac(99, 100)$]
]

#math.equation(block: true, alt: "5.83")[$5.83$]

#math.equation(block: true, alt: "15.7")[$15.7$]

#solutionbox[
#math.equation(block: true, alt: "15 the fraction 7 over 10")[$15 frac(7, 10)$]
]

#math.equation(block: true, alt: "18.1")[$18.1$]

#math.equation(block: true, alt: "0.239")[$0.239$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 239 over 1000")[$frac(239, 1000)$]
]

#math.equation(block: true, alt: "0.373")[$0.373$]

#math.equation(block: true, alt: "0.13")[$0.13$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over 100")[$frac(13, 100)$]
]

#math.equation(block: true, alt: "0.19")[$0.19$]

#math.equation(block: true, alt: "0.011")[$0.011$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 1000")[$frac(11, 1000)$]
]

#math.equation(block: true, alt: "0.049")[$0.049$]

#math.equation(block: true, alt: "−0.00007")[$−0.00007$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 7 over 100000")[$− frac(7, 100000)$]
]

#math.equation(block: true, alt: "−0.00003")[$−0.00003$]

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

#solutionbox[
#math.equation(block: true, alt: "6 the fraction 2 over 5")[$6 frac(2, 5)$]
]

#math.equation(block: true, alt: "5.2")[$5.2$]

#math.equation(block: true, alt: "7.05")[$7.05$]

#solutionbox[
#math.equation(block: true, alt: "7 the fraction 1 over 20")[$7 frac(1, 20)$]
]

#math.equation(block: true, alt: "9.04")[$9.04$]

#math.equation(block: true, alt: "4.006")[$4.006$]

#solutionbox[
#math.equation(block: true, alt: "4 the fraction 3 over 500")[$4 frac(3, 500)$]
]

#math.equation(block: true, alt: "2.008")[$2.008$]

#math.equation(block: true, alt: "10.25")[$10.25$]

#solutionbox[
#math.equation(block: true, alt: "10 the fraction 1 over 4")[$10 frac(1, 4)$]
]

#math.equation(block: true, alt: "12.75")[$12.75$]

#math.equation(block: true, alt: "1.324")[$1.324$]

#solutionbox[
#math.equation(block: true, alt: "1 the fraction 81 over 250")[$1 frac(81, 250)$]
]

#math.equation(block: true, alt: "2.482")[$2.482$]

#math.equation(block: true, alt: "14.125")[$14.125$]

#solutionbox[
#math.equation(block: true, alt: "14 the fraction 1 over 8")[$14 frac(1, 8)$]
]

#math.equation(block: true, alt: "20.375")[$20.375$]

#strong[Locate Decimals on the Number Line]

In the following exercises, locate each number on a number line.

#math.equation(block: true, alt: "0.8")[$0.8$]

#solutionbox[
#figure(figph[There is a number line shown with integers from negative 4 to 4. There is a red dot between 0 and 1 labeled 0.8.], alt: "There is a number line shown with integers from negative 4 to 4. There is a red dot between 0 and 1 labeled 0.8.", caption: none)
]

#math.equation(block: true, alt: "0.3")[$0.3$]

#math.equation(block: true, alt: "−0.2")[$−0.2$]

#solutionbox[
#figure(figph[There is a number line shown with integers from negative 4 to 4. There is a red dot between negative 1 and  0 labeled negative 0.2.], alt: "There is a number line shown with integers from negative 4 to 4. There is a red dot between negative 1 and 0 labeled negative 0.2.", caption: none)
]

#math.equation(block: true, alt: "−0.9")[$−0.9$]

#math.equation(block: true, alt: "3.1")[$3.1$]

#solutionbox[
#figure(figph[This is an image of a number line. It spans from negative 5 on the left to 5 on the right. To the right of 0 are tick marks with the numbers 1, 2, 3, 4, 5 on the number line. To the left of the zero are tick marks with the numbers negative 1, negative 2, negative 3, negative 4, and negative 5. A point is plotted at 3.1.], alt: "This is an image of a number line. It spans from negative 5 on the left to 5 on the right. To the right of 0 are tick marks with the numbers 1, 2, 3, 4, 5 on the number line. To the left of the zero are tick marks with the numbers negative 1, negative 2, negative 3, negative 4, and negative 5. A point is plotted at 3.1.", caption: none)
]

#math.equation(block: true, alt: "2.7")[$2.7$]

#math.equation(block: true, alt: "−2.5")[$−2.5$]

#solutionbox[
#figure(figph[There is a number line shown with integers from negative 4 to 4. There is a red dot between negative 3 and negative 2 labeled negative 2.5.], alt: "There is a number line shown with integers from negative 4 to 4. There is a red dot between negative 3 and negative 2 labeled negative 2.5.", caption: none)
]

#math.equation(block: true, alt: "−1.6")[$−1.6$]

#strong[Order Decimals]

In the following exercises, order each of the following pairs of numbers, using #math.equation(block: false, alt: "less than or >.")[$< #h(0.2em) "or" #h(0.2em) >.$]

#math.equation(block: true, alt: "0.9 __ 0.6")[$0.9 \_\_ 0.6$]

#solutionbox[
\>
]

#math.equation(block: true, alt: "0.7 __ 0.8")[$0.7 \_\_ 0.8$]

#math.equation(block: true, alt: "0.37 __ 0.63")[$0.37 \_\_ 0.63$]

#solutionbox[
\<
]

#math.equation(block: true, alt: "0.86 __ 0.69")[$0.86 \_\_ 0.69$]

#math.equation(block: true, alt: "0.6 __ 0.59")[$0.6 \_\_ 0.59$]

#solutionbox[
\>
]

#math.equation(block: true, alt: "0.27 __ 0.3")[$0.27 \_\_ 0.3$]

#math.equation(block: true, alt: "0.91 __ 0.901")[$0.91 \_\_ 0.901$]

#solutionbox[
\>
]

#math.equation(block: true, alt: "0.415 __ 0.41")[$0.415 \_\_ 0.41$]

#math.equation(block: true, alt: "−0.5 __ −0.3")[$−0.5 \_\_ −0.3$]

#solutionbox[
\<
]

#math.equation(block: true, alt: "−0.1 _ −0.4")[$−0.1 \_ −0.4$]

#math.equation(block: true, alt: "−0.62 _ −0.619")[$−0.62 \_ −0.619$]

#solutionbox[
\<
]

#math.equation(block: true, alt: "−7.31 _ −7.3")[$−7.31 \_ −7.3$]

#strong[Round Decimals]

In the following exercises, round each number to the nearest tenth.

#math.equation(block: true, alt: "0.67")[$0.67$]

#solutionbox[
0.7
]

#math.equation(block: true, alt: "0.49")[$0.49$]

#math.equation(block: true, alt: "2.84")[$2.84$]

#solutionbox[
2.8
]

#math.equation(block: true, alt: "4.63")[$4.63$]

In the following exercises, round each number to the nearest hundredth.

#math.equation(block: true, alt: "0.845")[$0.845$]

#solutionbox[
0.85
]

#math.equation(block: true, alt: "0.761")[$0.761$]

#math.equation(block: true, alt: "5.7932")[$5.7932$]

#solutionbox[
5.79
]

#math.equation(block: true, alt: "3.6284")[$3.6284$]

#math.equation(block: true, alt: "0.299")[$0.299$]

#solutionbox[
0.30
]

#math.equation(block: true, alt: "0.697")[$0.697$]

#math.equation(block: true, alt: "4.098")[$4.098$]

#solutionbox[
4.10
]

#math.equation(block: true, alt: "7.096")[$7.096$]

In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

#math.equation(block: true, alt: "5.781")[$5.781$]

#solutionbox[
+ ⓐ 5.78
+ ⓑ 5.8
+ ⓒ 6
]

#math.equation(block: true, alt: "1.638")[$1.638$]

#math.equation(block: true, alt: "63.479")[$63.479$]

#solutionbox[
+ ⓐ 63.48
+ ⓑ 63.5
+ ⓒ 63
]

#math.equation(block: true, alt: "84.281")[$84.281$]


=== Everyday Math

#strong[Salary Increase] Danny got a raise and now makes #math.equation(block: false, alt: "$58,965.95")[$"$58,965.95"$] a year. Round this number to the nearest:

ⓐ dollar

ⓑ thousand dollars

ⓒ ten thousand dollars.

#solutionbox[
+ ⓐ \$58,966
+ ⓑ \$59,000
+ ⓒ \$60,000
]

#strong[New Car Purchase] Selena’s new car cost #math.equation(block: false, alt: "$23,795.95.")[$"$23,795.95" .$] Round this number to the nearest:

ⓐ dollar

ⓑ thousand dollars

ⓒ ten thousand dollars.

#strong[Sales Tax] Hyo Jin lives in San Diego. She bought a refrigerator for #math.equation(block: false, alt: "$1624.99")[$"$1624.99"$] and when the clerk calculated the sales tax it came out to exactly #math.equation(block: false, alt: "$142.186625.")[$"$142.186625" .$] Round the sales tax to the nearest ⓐ penny ⓑ dollar.

#solutionbox[
+ ⓐ \$142.19
+ ⓑ \$142
]

#strong[Sales Tax] Jennifer bought a #math.equation(block: false, alt: "$1,038.99")[$"$1,038.99"$] dining room set for her home in Cincinnati. She calculated the sales tax to be exactly #math.equation(block: false, alt: "$67.53435.")[$"$67.53435" .$] Round the sales tax to the nearest ⓐ penny ⓑ dollar.


=== Writing Exercises

How does your knowledge of money help you learn about decimals?

#solutionbox[
Answers will vary.
]

Explain how you write “three and nine hundredths” as a decimal.

Jim ran a #math.equation(block: false, alt: "100-meter")[$100-"meter"$] race in #math.equation(block: false, alt: "12.32 seconds .")[$"12.32 seconds" .$] Tim ran the same race in #math.equation(block: false, alt: "12.3 seconds .")[$"12.3 seconds" .$] Who had the faster time, Jim or Tim? How do you know?

#solutionbox[
Tim had the faster time. 12.3 is less than 12.32, so Tim had the faster time.
]

Gerry saw a sign advertising postcards marked for sale at #math.equation(block: false, alt: "“ 10 for 0.99 ¢.”")[$“ 10 #h(0.2em) "for" #h(0.2em) 0.99 "¢.”"$] What is wrong with the advertised price?


=== 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 gauge their understanding of decimals, with categories: Confidently, With some help, and No-I don't get it! Tasks include naming, writing, converting, locating, ordering, and rounding decimals.], alt: "A self-assessment chart for students to gauge their understanding of decimals, with categories: Confidently, With some help, and No-I don't get it! Tasks include naming, writing, converting, locating, ordering, and rounding 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.