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== 5.6#h(0.6em)Ratios and Rate

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

Simplify: #math.equation(block: false, alt: "the fraction 16 over 24.")[$frac(16, 24) .$] #linebreak() 
If you missed this problem, review .

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

#notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "2.76 divided by 11.5.")[$2.76 ÷ 11.5 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "0.24")[$0.24$]
]
]

#notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 1 the fraction 1 over 2 over 2 the fraction 3 over 4.")[$frac(1 frac(1, 2), 2 frac(3, 4)) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 over 11")[$frac(6, 11)$]
]
]


=== Write a Ratio as a Fraction

When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. A #strong[ratio] compares two quantities that are measured with the same unit. If we compare #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$], the ratio is written as #math.equation(block: false, alt: "a to b , the fraction a over b , or a : b .")[$a #h(0.2em) "to" #h(0.2em) b , #h(0.2em) frac(a, b) , #h(0.2em) "or" #h(0.2em) "a" ":" "b" "."$]

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

A #strong[ratio] compares two numbers or two quantities that are measured with the same unit. The ratio of #math.equation(block: false, alt: "a")[$a$] to #math.equation(block: false, alt: "b")[$b$] is written #math.equation(block: false, alt: "a to b , the fraction a over b , or a : b .")[$a #h(0.2em) "to" #h(0.2em) b , #h(0.2em) frac(a, b) , #h(0.2em) "or" #h(0.2em) "a" ":" "b" "."$]
]

In this section, we will use the fraction notation. When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as #math.equation(block: false, alt: "the fraction 4 over 1")[$frac(4, 1)$] instead of simplifying it to #math.equation(block: false, alt: "4")[$4$] so that we can see the two parts of the ratio.

#examplebox("Example 1")[][
Write each ratio as a fraction: ⓐ #math.equation(block: false, alt: "15 to 27")[$#h(0.2em) 15 #h(0.2em) "to" #h(0.2em) 27 #h(0.2em)$]ⓑ #math.equation(block: false, alt: "45 to 18.")[$#h(0.2em) 45 #h(0.2em) "to" #h(0.2em) 18 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "15 to 27")[$"15 to 27"$]],
  [Write as a fraction with the first number in the numerator and the second in the denominator.], [#math.equation(block: false, alt: "the fraction 15 over 27")[$frac(15, 27)$]],
  [Simplify the fraction.], [#math.equation(block: false, alt: "the fraction 5 over 9")[$frac(5, 9)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "45 to 18")[$"45 to 18"$]],
  [Write as a fraction with the first number in the numerator and the second in the denominator.], [#math.equation(block: false, alt: "the fraction 45 over 18")[$frac(45, 18)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 5 over 2")[$frac(5, 2)$]],
))

We leave the ratio in ⓑ as an improper fraction.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each ratio as a fraction: ⓐ #math.equation(block: false, alt: "21 to 56")[$#h(0.2em) 21 #h(0.2em) "to" #h(0.2em) 56 #h(0.2em)$]ⓑ #math.equation(block: false, alt: "48 to 32.")[$#h(0.2em) 48 #h(0.2em) "to" #h(0.2em) 32 .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 3 over 8")[$#h(0.2em) frac(3, 8)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 3 over 2")[$#h(0.2em) frac(3, 2)$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each ratio as a fraction: ⓐ #math.equation(block: false, alt: "27 to 72")[$#h(0.2em) 27 #h(0.2em) "to" #h(0.2em) 72 #h(0.2em)$]ⓑ #math.equation(block: false, alt: "51 to 34.")[$#h(0.2em) 51 #h(0.2em) "to" #h(0.2em) 34 .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 3 over 8")[$#h(0.2em) frac(3, 8)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 3 over 2")[$#h(0.2em) frac(3, 2)$]
]
]


==== Ratios Involving Decimals

We will often work with ratios of decimals, especially when we have ratios involving money. In these cases, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator.

For example, consider the ratio #math.equation(block: false, alt: "0.8 to 0.05.")[$0.8 #h(0.2em) "to" #h(0.2em) 0.05 .$] We can write it as a fraction with decimals and then multiply the numerator and denominator by #math.equation(block: false, alt: "100")[$100$] to eliminate the decimals.

#figure(figph[A fraction is shown with 0.8 in the numerator and 0.05 in the denominator. Below it is the same fraction with both the numerator and denominator multiplied by 100. Below that is a fraction with 80 in the numerator and 5 in the denominator.], alt: "A fraction is shown with 0.8 in the numerator and 0.05 in the denominator. Below it is the same fraction with both the numerator and denominator multiplied by 100. Below that is a fraction with 80 in the numerator and 5 in the denominator.", caption: none)

Do you see a shortcut to find the equivalent fraction? Notice that #math.equation(block: false, alt: "0.8 equals the fraction 8 over 10")[$0.8 = frac(8, 10)$] and #math.equation(block: false, alt: "0.05 equals the fraction 5 over 100.")[$0.05 = frac(5, 100) .$] The least common denominator of #math.equation(block: false, alt: "the fraction 8 over 10")[$frac(8, 10)$] and #math.equation(block: false, alt: "the fraction 5 over 100")[$frac(5, 100)$] is #math.equation(block: false, alt: "100.")[$100 .$] By multiplying the numerator and denominator of #math.equation(block: false, alt: "the fraction 0.8 over 0.05")[$frac(0.8, 0.05)$] by #math.equation(block: false, alt: "100 ,")[$100 ,$] we ‘moved’ the decimal two places to the right to get the equivalent fraction with no decimals. Now that we understand the math behind the process, we can find the fraction with no decimals like this:

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The top line says 0.80 over 0.05. There are blue arrows moving the decimal points over 2 places to the right.], alt: "The top line says 0.80 over 0.05. There are blue arrows moving the decimal points over 2 places to the right.", caption: none)]),
  ["Move" the decimal 2 places.], [#math.equation(block: false, alt: "the fraction 80 over 5")[$frac(80, 5)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 16 over 1")[$frac(16, 1)$]],
))

You do not have to write out every step when you multiply the numerator and denominator by powers of ten. As long as you move both decimal places the same number of places, the ratio will remain the same.

#examplebox("Example 2")[][
Write each ratio as a fraction of whole numbers:

+ ⓐ #math.equation(block: false, alt: "4.8 to 11.2")[$#h(0.2em) 4.8 #h(0.2em) "to" #h(0.2em) 11.2$]
+ ⓑ #math.equation(block: false, alt: "2.7 to 0.54")[$#h(0.2em) 2.7 #h(0.2em) "to" #h(0.2em) 0.54$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ #math.equation(block: false, alt: "4.8 to 11.2")[$#h(0.2em) "4.8 to 11.2"$]], []),
  [Write as a fraction.], [#math.equation(block: false, alt: "the fraction 4.8 over 11.2")[$frac(4.8, 11.2)$]],
  [Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right.], [#math.equation(block: false, alt: "the fraction 48 over 112")[$frac(48, 112)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 over 7")[$frac(3, 7)$]],
))

So #math.equation(block: false, alt: "4.8 to 11.2")[$4.8 #h(0.2em) "to" #h(0.2em) 11.2$] is equivalent to #math.equation(block: false, alt: "the fraction 3 over 7.")[$frac(3, 7) .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ  #linebreak() 
The numerator has one decimal place and the denominator has #math.equation(block: false, alt: "2.")[$2 .$] To clear both decimals we need to move the decimal #math.equation(block: false, alt: "2")[$2$] places to the right. #linebreak() #math.equation(block: false, alt: "2.7 to 0.54")[$2.7 #h(0.2em) "to" #h(0.2em) 0.54$]], []),
  [Write as a fraction.], [#math.equation(block: false, alt: "the fraction 2.7 over 0.54")[$frac(2.7, 0.54)$]],
  [Move both decimals right two places.], [#math.equation(block: false, alt: "the fraction 270 over 54")[$frac(270, 54)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 5 over 1")[$frac(5, 1)$]],
))

So #math.equation(block: false, alt: "2.7 to 0.54")[$2.7 #h(0.2em) "to" #h(0.2em) 0.54$] is equivalent to #math.equation(block: false, alt: "the fraction 5 over 1.")[$frac(5, 1) .$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each ratio as a fraction: ⓐ #math.equation(block: false, alt: "4.6 to 11.5")[$#h(0.2em) 4.6 #h(0.2em) "to" #h(0.2em) 11.5 #h(0.2em)$]ⓑ #math.equation(block: false, alt: "2.3 to 0.69.")[$#h(0.2em) 2.3 #h(0.2em) "to" #h(0.2em) 0.69 .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 2 over 5")[$#h(0.2em) frac(2, 5) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 10 over 3")[$#h(0.2em) frac(10, 3)$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each ratio as a fraction: ⓐ #math.equation(block: false, alt: "3.4 to 15.3")[$#h(0.2em) 3.4 #h(0.2em) "to" #h(0.2em) 15.3 #h(0.2em)$]ⓑ #math.equation(block: false, alt: "3.4 to 0.68.")[$#h(0.2em) 3.4 #h(0.2em) "to" #h(0.2em) 0.68 .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 2 over 9")[$#h(0.2em) frac(2, 9)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 5 over 1")[$#h(0.2em) frac(5, 1)$]
]
]

Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.

#examplebox("Example 3")[][
Write the ratio of #math.equation(block: false, alt: "1 the fraction 1 over 4 to 2 the fraction 3 over 8")[$1 frac(1, 4) #h(0.2em) "to" #h(0.2em) 2 frac(3, 8)$] as a fraction.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "1 the fraction 1 over 4 to 2 the fraction 3 over 8")[$1 frac(1, 4) #h(0.2em) "to" #h(0.2em) 2 frac(3, 8)$]]),
  [Write as a fraction.], [#math.equation(block: false, alt: "the fraction 1 the fraction 1 over 4 over 2 the fraction 3 over 8")[$frac(1 frac(1, 4), 2 frac(3, 8))$]],
  [Convert the numerator and denominator to improper fractions.], [#math.equation(block: false, alt: "the fraction the fraction 5 over 4 over the fraction 19 over 8")[$frac(frac(5, 4), frac(19, 8))$]],
  [Rewrite as a division of fractions.], [#math.equation(block: false, alt: "the fraction 5 over 4 divided by the fraction 19 over 8")[$frac(5, 4) ÷ frac(19, 8)$]],
  [Invert the divisor and multiply.], [#math.equation(block: false, alt: "the fraction 5 over 4 times the fraction 8 over 19")[$frac(5, 4) · frac(8, 19)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 10 over 19")[$frac(10, 19)$]],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each ratio as a fraction: #math.equation(block: false, alt: "1 the fraction 3 over 4 to 2 the fraction 5 over 8.")[$1 frac(3, 4) #h(0.2em) "to" #h(0.2em) 2 frac(5, 8) .$]

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

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each ratio as a fraction: #math.equation(block: false, alt: "1 the fraction 1 over 8 to 2 the fraction 3 over 4.")[$1 frac(1, 8) #h(0.2em) "to" #h(0.2em) 2 frac(3, 4) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 9 over 22")[$frac(9, 22)$]
]
]


==== Applications of Ratios

One real-world application of ratios that affects many people involves measuring cholesterol in blood. The ratio of total cholesterol to HDL cholesterol is one way doctors assess a person's overall health. A ratio of less than #math.equation(block: false, alt: "5")[$5$] to #math.equation(block: false, alt: "1")[$1$] is considered good.

#examplebox("Example 4")[][
Hector's total cholesterol is #math.equation(block: false, alt: "249")[$249$] mg/dl and his HDL cholesterol is #math.equation(block: false, alt: "39")[$39$] mg/dl. ⓐ Find the ratio of his total cholesterol to his HDL cholesterol. ⓑ Assuming that a ratio less than #math.equation(block: false, alt: "5")[$5$] to #math.equation(block: false, alt: "1")[$1$] is considered good, what would you suggest to Hector?

#solutionbox[
ⓐ First, write the words that express the ratio. We want to know the ratio of Hector's total cholesterol to his HDL cholesterol.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Write as a fraction.], [#math.equation(block: false, alt: "the fraction total cholesterol over HDL cholesterol")[$frac("total cholesterol", "HDL cholesterol")$]]),
  [Substitute the values.], [#math.equation(block: false, alt: "the fraction 249 over 39")[$frac(249, 39)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 83 over 13")[$frac(83, 13)$]],
))

ⓑ Is Hector's cholesterol ratio ok? If we divide #math.equation(block: false, alt: "83")[$83$] by #math.equation(block: false, alt: "13")[$13$] we obtain approximately #math.equation(block: false, alt: "6.4 ,")[$6.4 ,$] so #math.equation(block: false, alt: "the fraction 83 over 13 approximately equals the fraction 6.4 over 1.")[$frac(83, 13) ≈ frac(6.4, 1) .$] Hector's cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the patient's ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is #math.equation(block: false, alt: "185")[$185$] mg/dL and HDL cholesterol is #math.equation(block: false, alt: "40")[$40$] mg/dL.

#solutionbox[
#math.equation(block: true, alt: "the fraction 37 over 8")[$frac(37, 8)$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is #math.equation(block: false, alt: "204")[$204$] mg/dL and HDL cholesterol is #math.equation(block: false, alt: "38")[$38$] mg/dL.

#solutionbox[
#math.equation(block: true, alt: "the fraction 102 over 19")[$frac(102, 19)$]
]
]

Ratios of Two Measurements in Different UnitsTo find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units.

We know that to simplify a fraction, we divide out common factors. Similarly in a ratio of measurements, we divide out the common unit.

#examplebox("Example 5")[][
The Americans with Disabilities Act (ADA) Guidelines for wheel chair ramps require a maximum vertical rise of #math.equation(block: false, alt: "1")[$1$] inch for every #math.equation(block: false, alt: "1")[$1$] foot of horizontal run. What is the ratio of the rise to the run?

#solutionbox[
In a ratio, the measurements must be in the same units. We can change feet to inches, or inches to feet. It is usually easier to convert to the smaller unit, since this avoids introducing more fractions into the problem.

Write the words that express the ratio.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [Ratio of the rise to the run]),
  [Write the ratio as a fraction.], [#math.equation(block: false, alt: "the fraction rise over run")[$frac("rise", "run")$]],
  [Substitute in the given values.], [#math.equation(block: false, alt: "the fraction 1 inch over 1 foot")[$frac("1 inch", "1 foot")$]],
  [Convert 1 foot to inches.], [#math.equation(block: false, alt: "the fraction 1 inch over 12 inches")[$frac("1 inch", "12 inches")$]],
  [Simplify, dividing out common factors and units.], [#math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$]],
))

So the ratio of rise to run is #math.equation(block: false, alt: "1")[$1$] to #math.equation(block: false, alt: "12.")[$12 .$] This means that the ramp should rise #math.equation(block: false, alt: "1")[$1$] inch for every #math.equation(block: false, alt: "12")[$12$] inches of horizontal run to comply with the guidelines.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the ratio of the first length to the second length: #math.equation(block: false, alt: "32")[$32$] inches to #math.equation(block: false, alt: "1")[$1$] foot.

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

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the ratio of the first length to the second length: #math.equation(block: false, alt: "1")[$1$] foot to #math.equation(block: false, alt: "54")[$54$] inches.

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


=== Write a Rate as a Fraction

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a #strong[rate]. Examples of rates are #math.equation(block: false, alt: "120")[$120$] miles in #math.equation(block: false, alt: "2")[$2$] hours, #math.equation(block: false, alt: "160")[$160$] words in #math.equation(block: false, alt: "4")[$4$] minutes, and #math.equation(block: false, alt: "$5")[$"$5"$] dollars per #math.equation(block: false, alt: "64")[$64$] ounces.

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

A #strong[rate] compares two quantities of different units. A rate is usually written as a fraction.
]

When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.

#examplebox("Example 6")[][
Bob drove his car #math.equation(block: false, alt: "525")[$525$] miles in #math.equation(block: false, alt: "9")[$9$] hours. Write this rate as a fraction.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "525 miles in 9 hours")[$"525 miles in 9 hours"$]]),
  [Write as a fraction, with 525 miles in the numerator and 9 hours in the denominator.], [#math.equation(block: false, alt: "the fraction 525 miles over 9 hours")[$frac("525 miles", "9 hours")$]],
  [], [#math.equation(block: false, alt: "the fraction 175 miles over 3 hours")[$frac("175 miles", "3 hours")$]],
))

So #math.equation(block: false, alt: "525")[$525$] miles in #math.equation(block: false, alt: "9")[$9$] hours is equivalent to #math.equation(block: false, alt: "the fraction 175 miles over 3 hours .")[$frac("175 miles", "3 hours") .$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the rate as a fraction: #math.equation(block: false, alt: "492")[$492$] miles in #math.equation(block: false, alt: "8")[$8$] hours.

#solutionbox[
#math.equation(block: true, alt: "the fraction 123 miles over 2 hours")[$frac("123 miles", "2 hours")$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the rate as a fraction: #math.equation(block: false, alt: "242")[$242$] miles in #math.equation(block: false, alt: "6")[$6$] hours.

#solutionbox[
#math.equation(block: true, alt: "the fraction 121 miles over 3 hours")[$frac("121 miles", "3 hours")$]
]
]


=== Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of #math.equation(block: false, alt: "the fraction 175 miles over 3 hours .")[$frac("175 miles", "3 hours") .$] This tells us that every three hours, Bob will travel #math.equation(block: false, alt: "175")[$175$] miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of #math.equation(block: false, alt: "1")[$1$] unit is referred to as a #strong[unit rate].

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

A #strong[unit rate] is a rate with denominator of #math.equation(block: false, alt: "1")[$1$] unit.
]

Unit rates are very common in our lives. For example, when we say that we are driving at a speed of #math.equation(block: false, alt: "68")[$68$] miles per hour we mean that we travel #math.equation(block: false, alt: "68")[$68$] miles in #math.equation(block: false, alt: "1")[$1$] hour. We would write this rate as #math.equation(block: false, alt: "68")[$68$] miles/hour (read #math.equation(block: false, alt: "68")[$68$] miles per hour). The common abbreviation for this is #math.equation(block: false, alt: "68")[$68$] mph. Note that when no number is written before a unit, it is assumed to be #math.equation(block: false, alt: "1.")[$1 .$]

So #math.equation(block: false, alt: "68")[$68$] miles/hour really means #math.equation(block: false, alt: "68 miles/1 hour.")[$"68 miles/1 hour."$]

Two rates we often use when driving can be written in different forms, as shown:

#figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Example], [Rate], [Write], [Abbreviate], [Read]),
  [#math.equation(block: false, alt: "68")[$68$] miles in #math.equation(block: false, alt: "1")[$1$] hour], [#math.equation(block: false, alt: "the fraction 68 miles over 1 hour")[$frac("68 miles", "1 hour")$]], [#math.equation(block: false, alt: "68")[$68$] miles/hour], [#math.equation(block: false, alt: "68")[$68$] mph], [#math.equation(block: false, alt: "68 miles per hour")[$"68 miles per hour"$]],
  [#math.equation(block: false, alt: "36")[$36$] miles to #math.equation(block: false, alt: "1")[$1$] gallon], [#math.equation(block: false, alt: "the fraction 36 miles over 1 gallon")[$frac("36 miles", "1 gallon")$]], [#math.equation(block: false, alt: "36")[$36$] miles/gallon], [#math.equation(block: false, alt: "36")[$36$] mpg], [#math.equation(block: false, alt: "36 miles per gallon")[$"36 miles per gallon"$]],
))

Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid #math.equation(block: false, alt: "$12.50")[$"$12.50"$] for each hour you work, you could write that your hourly (unit) pay rate is #math.equation(block: false, alt: "$12.50/hour")[$"$12.50/hour"$] (read #math.equation(block: false, alt: "$12.50")[$"$12.50"$] per hour.)

To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of #math.equation(block: false, alt: "1.")[$1 .$]

#examplebox("Example 7")[][
Anita was paid #math.equation(block: false, alt: "$384")[$"$384"$] last week for working #math.equation(block: false, alt: "32 hours .")[$"32 hours" .$] What is Anita’s hourly pay rate?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Start with a rate of dollars to hours. Then divide.], [#math.equation(block: false, alt: "$384 last week for 32 hours")[$"$384 last week for 32 hours"$]]),
  [Write as a rate.], [#math.equation(block: false, alt: "the fraction $384 over 32 hours")[$frac(\$384, "32 hours")$]],
  [Divide the numerator by the denominator.], [#math.equation(block: false, alt: "the fraction $12 over 1 hour")[$frac(\$12, "1 hour")$]],
  [Rewrite as a rate.], [#math.equation(block: false, alt: "$12 / hour")[$\$12 / "hour"$]],
))

Anita’s hourly pay rate is #math.equation(block: false, alt: "$12")[$"$12"$] per hour.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the unit rate: #math.equation(block: false, alt: "$630")[$"$630"$] for #math.equation(block: false, alt: "35")[$35$] hours.

#solutionbox[
\$18.00/hour
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the unit rate: #math.equation(block: false, alt: "$684")[$"$684"$] for #math.equation(block: false, alt: "36")[$36$] hours.

#solutionbox[
\$19.00/hour
]
]

#examplebox("Example 8")[][
Sven drives his car #math.equation(block: false, alt: "455")[$455$] miles, using #math.equation(block: false, alt: "14")[$14$] gallons of gasoline. How many miles per gallon does his car get?

#solutionbox[
Start with a rate of miles to gallons. Then divide.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "455 miles to 14 gallons of gas")[$"455 miles to 14 gallons of gas"$]]),
  [Write as a rate.], [#math.equation(block: false, alt: "the fraction 455 miles over 14 gallons")[$frac("455 miles", "14 gallons")$]],
  [Divide 455 by 14 to get the unit rate.], [#math.equation(block: false, alt: "the fraction 32.5 miles over 1 gallon")[$frac("32.5 miles", "1 gallon")$]],
))

Sven’s car gets #math.equation(block: false, alt: "32.5")[$32.5$] miles/gallon, or #math.equation(block: false, alt: "32.5")[$32.5$] mpg.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the unit rate: #math.equation(block: false, alt: "423")[$423$] miles to #math.equation(block: false, alt: "18")[$18$] gallons of gas.

#solutionbox[
23.5 mpg
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the unit rate: #math.equation(block: false, alt: "406")[$406$] miles to #math.equation(block: false, alt: "14.5")[$14.5$] gallons of gas.

#solutionbox[
28 mpg
]
]


=== Find Unit Price

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A #strong[unit price] is a unit rate for one item.

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

A #strong[unit price] is a unit rate that gives the price of one item.
]

#examplebox("Example 9")[][
The grocery store charges #math.equation(block: false, alt: "$3.99")[$"$3.99"$] for a case of #math.equation(block: false, alt: "24")[$24$] bottles of water. What is the unit price?

#solutionbox[
What are we asked to find? We are asked to find the unit price, which is the price per bottle.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Write as a rate.], [#math.equation(block: false, alt: "the fraction $3.99 over 24 bottles")[$frac(\$3.99, "24 bottles")$]]),
  [Divide to find the unit price.], [#math.equation(block: false, alt: "the fraction $0.16625 over 1 bottle")[$frac(\$0.16625, "1 bottle")$]],
  [Round the result to the nearest penny.], [#math.equation(block: false, alt: "the fraction $0.17 over 1 bottle")[$frac(\$0.17, "1 bottle")$]],
))

The unit price is approximately #math.equation(block: false, alt: "$0.17")[$"$0.17"$] per bottle. Each bottle costs about #math.equation(block: false, alt: "$0.17.")[$"$0.17" .$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the unit price. Round your answer to the nearest cent if necessary.

#math.equation(block: false, alt: "24-pack")[$"24-pack"$] of juice boxes for #math.equation(block: false, alt: "$6.99")[$"$6.99"$]

#solutionbox[
\$0.29/box
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the unit price. Round your answer to the nearest cent if necessary.

#math.equation(block: false, alt: "24-pack")[$"24-pack"$] of bottles of ice tea for #math.equation(block: false, alt: "$12.72")[$"$12.72"$]

#solutionbox[
\$0.53/bottle
]
]

Unit prices are very useful if you comparison shop. The #emph[better buy] is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.

#examplebox("Example 10")[][
Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at #math.equation(block: false, alt: "$14.99")[$"$14.99"$] for #math.equation(block: false, alt: "64")[$64$] loads of laundry and the same brand of powder detergent is priced at #math.equation(block: false, alt: "$15.99")[$"$15.99"$] for #math.equation(block: false, alt: "80")[$80$] loads.

Which detergent has the lowest cost per load?

#solutionbox[
To compare the prices, we first find the unit price for each type of detergent. #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [Liquid], [Powder]),
  [Write as a rate.], [#math.equation(block: false, alt: "the fraction $14.99 over 64 loads")[$frac("$14.99", "64 loads")$]], [#math.equation(block: false, alt: "the fraction $15.99 over 80 loads")[$frac("$15.99", "80 loads")$]],
  [Find the unit price.], [#math.equation(block: false, alt: "the fraction $0.234… over 1 load")[$frac("$0.234…", "1 load")$]], [#math.equation(block: false, alt: "the fraction $0.199… over 1 load")[$frac("$0.199…", "1 load")$]],
  [Round to the nearest cent.], [#math.equation(block: false, alt: "$0.23/load; (23 cents per load.)")[$"$0.23/load" \ "(23 cents per load.)"$]], [#math.equation(block: false, alt: "$0.20/load; (20 cents per load)")[$"$0.20/load" \ "(20 cents per load)"$]],
))

Now we compare the unit prices. The unit price of the liquid detergent is about #math.equation(block: false, alt: "$0.23")[$"$0.23"$] per load and the unit price of the powder detergent is about #math.equation(block: false, alt: "$0.20")[$"$0.20"$] per load. The powder is the better buy.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand A Storage Bags, #math.equation(block: false, alt: "$4.59")[$"$4.59"$] for #math.equation(block: false, alt: "40")[$40$] count, or Brand B Storage Bags, #math.equation(block: false, alt: "$3.99")[$"$3.99"$] for #math.equation(block: false, alt: "30")[$30$] count

#solutionbox[
Brand A costs \$0.11 per bag. Brand B costs \$0.13 per bag. Brand A is the better buy.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand C Chicken Noodle Soup, #math.equation(block: false, alt: "$1.89")[$"$1.89"$] for #math.equation(block: false, alt: "26")[$26$] ounces, or Brand D Chicken Noodle Soup, #math.equation(block: false, alt: "$0.95")[$"$0.95"$] for #math.equation(block: false, alt: "10.75")[$10.75$] ounces

#solutionbox[
Brand C costs \$0.07 per ounce. Brand D costs \$0.09 per ounce. Brand C is the better buy.
]
]

Notice in  that we rounded the unit price to the nearest cent. Sometimes we may need to carry the division to one more place to see the difference between the unit prices.


=== Translate Phrases to Expressions with Fractions

Have you noticed that the examples in this section used the comparison words #emph[ratio of, to, per, in, for, on], and #emph[from]? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction.

#examplebox("Example 11")[][
Translate the word phrase into an algebraic expression:

+ ⓐ #math.equation(block: false, alt: "427")[$#h(0.2em) 427$] miles per #math.equation(block: false, alt: "h")[$h$] hours
+ ⓑ #math.equation(block: false, alt: "x")[$#h(0.2em) x$] students to #math.equation(block: false, alt: "3")[$3$] teachers
+ ⓒ #math.equation(block: false, alt: "y")[$#h(0.2em) y$] dollars for #math.equation(block: false, alt: "18")[$18$] hours

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "427 miles per h hours")[$"427 miles per" #h(0.2em) h #h(0.2em) "hours"$]],
  [Write as a rate.], [#math.equation(block: false, alt: "the fraction 427 miles over h hours")[$frac("427 miles", h #h(0.2em) "hours")$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "x students to 3 teachers")[$x #h(0.2em) "students to 3 teachers"$]],
  [Write as a rate.], [#math.equation(block: false, alt: "the fraction x students over 3 teachers")[$frac(x #h(0.2em) "students", "3 teachers")$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [#math.equation(block: false, alt: "y dollars for 18 hours")[$y #h(0.2em) "dollars for 18 hours"$]],
  [Write as a rate.], [#math.equation(block: false, alt: "the fraction $ y over 18 hours")[$frac(\$ y, "18 hours")$]],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the word phrase into an algebraic expression.

ⓐ #math.equation(block: false, alt: "689")[$#h(0.2em) 689$] miles per #math.equation(block: false, alt: "h")[$h$] hours ⓑ #math.equation(block: false, alt: "y")[$y$] parents to #math.equation(block: false, alt: "22")[$22$] students ⓒ #math.equation(block: false, alt: "d")[$d$] dollars for #math.equation(block: false, alt: "9")[$9$] minutes

#solutionbox[
+ ⓐ 689 mi/#emph[h] hours
+ ⓑ #emph[y] parents/22 students
+ ⓒ \$#emph[d]/9 min
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the word phrase into an algebraic expression.

ⓐ #math.equation(block: false, alt: "m")[$m$] miles per #math.equation(block: false, alt: "9")[$9$] hours ⓑ #math.equation(block: false, alt: "x")[$x$] students to #math.equation(block: false, alt: "8")[$8$] buses ⓒ #math.equation(block: false, alt: "y")[$y$] dollars for #math.equation(block: false, alt: "40")[$40$] hours

#solutionbox[
+ ⓐ #emph[m] mi/9 h
+ ⓑ #emph[x] students/8 buses
+ ⓒ \$#emph[y]/40 h
]
]

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

- #link("https://www.openstax.org/l/24ratios")[Ratios]
- #link("https://www.openstax.org/l/24ratiosimpfrac")[Write Ratios as a Simplified Fractions Involving Decimals and Fractions]
- #link("https://www.openstax.org/l/24ratiosimp")[Write a Ratio as a Simplified Fraction]
- #link("https://www.openstax.org/l/24rates")[Rates and Unit Rates]
- #link("https://www.openstax.org/l/24unitrate")[Unit Rate for Cell Phone Plan]
]


=== Practice Makes Perfect

#strong[Write a Ratio as a Fraction]

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

#math.equation(block: false, alt: "20")[$20$] to #math.equation(block: false, alt: "36")[$36$]

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

#math.equation(block: false, alt: "20")[$20$] to #math.equation(block: false, alt: "32")[$32$]

#math.equation(block: false, alt: "42")[$42$] to #math.equation(block: false, alt: "48")[$48$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 8")[$frac(7, 8)$]
]

#math.equation(block: false, alt: "45")[$45$] to #math.equation(block: false, alt: "54")[$54$]

#math.equation(block: false, alt: "49")[$49$] to #math.equation(block: false, alt: "21")[$21$]

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

#math.equation(block: false, alt: "56")[$56$] to #math.equation(block: false, alt: "16")[$16$]

#math.equation(block: false, alt: "84")[$84$] to #math.equation(block: false, alt: "36")[$36$]

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

#math.equation(block: false, alt: "6.4")[$6.4$] to #math.equation(block: false, alt: "0.8")[$0.8$]

#math.equation(block: false, alt: "0.56")[$0.56$] to #math.equation(block: false, alt: "2.8")[$2.8$]

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

#math.equation(block: false, alt: "1.26")[$1.26$] to #math.equation(block: false, alt: "4.2")[$4.2$]

#math.equation(block: false, alt: "1 the fraction 2 over 3")[$1 frac(2, 3)$] to #math.equation(block: false, alt: "2 the fraction 5 over 6")[$2 frac(5, 6)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 10 over 17")[$frac(10, 17)$]
]

#math.equation(block: false, alt: "1 the fraction 3 over 4")[$1 frac(3, 4)$] to #math.equation(block: false, alt: "2 the fraction 5 over 8")[$2 frac(5, 8)$]

#math.equation(block: false, alt: "4 the fraction 1 over 6")[$4 frac(1, 6)$] to #math.equation(block: false, alt: "3 the fraction 1 over 3")[$3 frac(1, 3)$]

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

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

#math.equation(block: false, alt: "$18")[$"$18"$] to #math.equation(block: false, alt: "$63")[$"$63"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 7")[$frac(2, 7)$]
]

#math.equation(block: false, alt: "$16")[$"$16"$] to #math.equation(block: false, alt: "$72")[$"$72"$]

#math.equation(block: false, alt: "$1.21")[$"$1.21"$] to #math.equation(block: false, alt: "$0.44")[$"$0.44"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 4")[$frac(11, 4)$]
]

#math.equation(block: false, alt: "$1.38")[$"$1.38"$] to #math.equation(block: false, alt: "$0.69")[$"$0.69"$]

#math.equation(block: false, alt: "28")[$28$] ounces to #math.equation(block: false, alt: "84")[$84$] ounces

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

#math.equation(block: false, alt: "32")[$32$] ounces to #math.equation(block: false, alt: "128")[$128$] ounces

#math.equation(block: false, alt: "12")[$12$] feet to #math.equation(block: false, alt: "46")[$46$] feet

#solutionbox[
#math.equation(block: true, alt: "the fraction 6 over 23")[$frac(6, 23)$]
]

#math.equation(block: false, alt: "15")[$15$] feet to #math.equation(block: false, alt: "57")[$57$] feet

#math.equation(block: false, alt: "246")[$246$] milligrams to #math.equation(block: false, alt: "45")[$45$] milligrams

#solutionbox[
#math.equation(block: true, alt: "the fraction 82 over 15")[$frac(82, 15)$]
]

#math.equation(block: false, alt: "304")[$304$] milligrams to #math.equation(block: false, alt: "48")[$48$] milligrams

total cholesterol of #math.equation(block: false, alt: "175")[$175$] to HDL cholesterol of #math.equation(block: false, alt: "45")[$45$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 35 over 9")[$frac(35, 9)$]
]

total cholesterol of #math.equation(block: false, alt: "215")[$215$] to HDL cholesterol of #math.equation(block: false, alt: "55")[$55$]

#math.equation(block: false, alt: "27")[$27$] inches to #math.equation(block: false, alt: "1")[$1$] foot

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

#math.equation(block: false, alt: "28")[$28$] inches to #math.equation(block: false, alt: "1")[$1$] foot

#strong[Write a Rate as a Fraction]

In the following exercises, write each rate as a fraction.

#math.equation(block: false, alt: "140")[$140$] calories per #math.equation(block: false, alt: "12")[$12$] ounces

#solutionbox[
#math.equation(block: true, alt: "the fraction 35 calories over 3 ounces")[$frac("35 calories", "3 ounces")$]
]

#math.equation(block: false, alt: "180")[$180$] calories per #math.equation(block: false, alt: "16")[$16$] ounces

#math.equation(block: false, alt: "8.2")[$8.2$] pounds per #math.equation(block: false, alt: "3")[$3$] square inches

#solutionbox[
#math.equation(block: true, alt: "the fraction 41 lbs over 15 sq. in .")[$frac("41 lbs", "15 sq. in" .)$]
]

#math.equation(block: false, alt: "9.5")[$9.5$] pounds per #math.equation(block: false, alt: "4")[$4$] square inches

#math.equation(block: false, alt: "488")[$488$] miles in #math.equation(block: false, alt: "7")[$7$] hours

#solutionbox[
#math.equation(block: true, alt: "the fraction 488 miles over 7 hours")[$frac("488 miles", "7 hours")$]
]

#math.equation(block: false, alt: "527")[$527$] miles in #math.equation(block: false, alt: "9")[$9$] hours

#math.equation(block: false, alt: "$595")[$"$595"$] for #math.equation(block: false, alt: "40")[$40$] hours

#solutionbox[
#math.equation(block: true, alt: "the fraction $119 over 8 hours")[$frac("$119", "8 hours")$]
]

#math.equation(block: false, alt: "$798")[$"$798"$] for #math.equation(block: false, alt: "40")[$40$] hours

#strong[Find Unit Rates]

In the following exercises, find the unit rate. Round to two decimal places, if necessary.

#math.equation(block: false, alt: "140")[$140$] calories per #math.equation(block: false, alt: "12")[$12$] ounces

#solutionbox[
11.67 calories/ounce
]

#math.equation(block: false, alt: "180")[$180$] calories per #math.equation(block: false, alt: "16")[$16$] ounces

#math.equation(block: false, alt: "8.2")[$8.2$] pounds per #math.equation(block: false, alt: "3")[$3$] square inches

#solutionbox[
2.73 lbs./sq. in.
]

#math.equation(block: false, alt: "9.5")[$9.5$] pounds per #math.equation(block: false, alt: "4")[$4$] square inches

#math.equation(block: false, alt: "488")[$488$] miles in #math.equation(block: false, alt: "7")[$7$] hours

#solutionbox[
69.71 mph
]

#math.equation(block: false, alt: "527")[$527$] miles in #math.equation(block: false, alt: "9")[$9$] hours

#math.equation(block: false, alt: "$595")[$"$595"$] for #math.equation(block: false, alt: "40")[$40$] hours

#solutionbox[
\$14.88/hour
]

#math.equation(block: false, alt: "$798")[$"$798"$] for #math.equation(block: false, alt: "40")[$40$] hours

#math.equation(block: false, alt: "576")[$576$] miles on #math.equation(block: false, alt: "18")[$18$] gallons of gas

#solutionbox[
32 mpg
]

#math.equation(block: false, alt: "435")[$435$] miles on #math.equation(block: false, alt: "15")[$15$] gallons of gas

#math.equation(block: false, alt: "43")[$43$] pounds in #math.equation(block: false, alt: "16")[$16$] weeks

#solutionbox[
2.69 lbs./week
]

#math.equation(block: false, alt: "57")[$57$] pounds in #math.equation(block: false, alt: "24")[$24$] weeks

#math.equation(block: false, alt: "46")[$46$] beats in #math.equation(block: false, alt: "0.5")[$0.5$] minute

#solutionbox[
92 beats/minute
]

#math.equation(block: false, alt: "54")[$54$] beats in #math.equation(block: false, alt: "0.5")[$0.5$] minute

The bindery at a printing plant assembles #math.equation(block: false, alt: "96,000")[$96,000$] magazines in #math.equation(block: false, alt: "12")[$12$] hours. How many magazines are assembled in one hour?

#solutionbox[
8,000
]

The pressroom at a printing plant prints #math.equation(block: false, alt: "540,000")[$540,000$] sections in #math.equation(block: false, alt: "12")[$12$] hours. How many sections are printed per hour?

#strong[Find Unit Price]

In the following exercises, find the unit price. Round to the nearest cent.

Soap bars at #math.equation(block: false, alt: "8")[$8$] for #math.equation(block: false, alt: "$8.69")[$"$8.69"$]

#solutionbox[
\$1.09/bar
]

Soap bars at #math.equation(block: false, alt: "4")[$4$] for #math.equation(block: false, alt: "$3.39")[$"$3.39"$]

Women’s sports socks at #math.equation(block: false, alt: "6")[$6$] pairs for #math.equation(block: false, alt: "$7.99")[$"$7.99"$]

#solutionbox[
\$1.33/pair
]

Men’s dress socks at #math.equation(block: false, alt: "3")[$3$] pairs for #math.equation(block: false, alt: "$8.49")[$"$8.49"$]

Snack packs of cookies at #math.equation(block: false, alt: "12")[$12$] for #math.equation(block: false, alt: "$5.79")[$"$5.79"$]

#solutionbox[
\$0.48/pack
]

Granola bars at #math.equation(block: false, alt: "5")[$5$] for #math.equation(block: false, alt: "$3.69")[$"$3.69"$]

CD-RW discs at #math.equation(block: false, alt: "25")[$25$] for #math.equation(block: false, alt: "$14.99")[$"$14.99"$]

#solutionbox[
\$0.60/disc
]

CDs at #math.equation(block: false, alt: "50")[$50$] for #math.equation(block: false, alt: "$4.49")[$"$4.49"$]

The grocery store has a special on macaroni and cheese. The price is #math.equation(block: false, alt: "$3.87")[$"$3.87"$] for #math.equation(block: false, alt: "3")[$3$] boxes. How much does each box cost?

#solutionbox[
\$1.29/box
]

The pet store has a special on cat food. The price is #math.equation(block: false, alt: "$4.32")[$"$4.32"$] for #math.equation(block: false, alt: "12")[$12$] cans. How much does each can cost?

In the following exercises, find each unit price and then identify the better buy. Round to three decimal places.

Mouthwash, #math.equation(block: false, alt: "50.7-ounce")[$"50.7-ounce"$] size for #math.equation(block: false, alt: "$6.99")[$"$6.99"$] or #math.equation(block: false, alt: "33.8-ounce")[$"33.8-ounce"$] size for #math.equation(block: false, alt: "$4.79")[$"$4.79"$]

#solutionbox[
The 50.7-ounce size costs \$0.138 per ounce. The 33.8-ounce size costs \$0.142 per ounce. The 50.7-ounce size is the better buy.
]

Toothpaste, #math.equation(block: false, alt: "6")[$6$] ounce size for #math.equation(block: false, alt: "$3.19")[$"$3.19"$] or #math.equation(block: false, alt: "7.8-ounce")[$7.8-"ounce"$] size for #math.equation(block: false, alt: "$5.19")[$"$5.19"$]

Breakfast cereal, #math.equation(block: false, alt: "18")[$18$] ounces for #math.equation(block: false, alt: "$3.99")[$"$3.99"$] or #math.equation(block: false, alt: "14")[$14$] ounces for #math.equation(block: false, alt: "$3.29")[$"$3.29"$]

#solutionbox[
The 18-ounce size costs \$0.222 per ounce. The 14-ounce size costs \$0.235 per ounce. The 18-ounce size is a better buy.
]

Breakfast Cereal, #math.equation(block: false, alt: "10.7")[$10.7$] ounces for #math.equation(block: false, alt: "$2.69")[$"$2.69"$] or #math.equation(block: false, alt: "14.8")[$14.8$] ounces for #math.equation(block: false, alt: "$3.69")[$"$3.69"$]

Ketchup, #math.equation(block: false, alt: "40-ounce")[$"40-ounce"$] regular bottle for #math.equation(block: false, alt: "$2.99")[$"$2.99"$] or #math.equation(block: false, alt: "64-ounce")[$"64-ounce"$] squeeze bottle for #math.equation(block: false, alt: "$4.39")[$"$4.39"$]

#solutionbox[
The regular bottle costs \$0.075 per ounce. The squeeze bottle costs \$0.069 per ounce. The squeeze bottle is a better buy.
]

Mayonnaise #math.equation(block: false, alt: "15-ounce")[$"15-ounce"$] regular bottle for #math.equation(block: false, alt: "$3.49")[$"$3.49"$] or #math.equation(block: false, alt: "22-ounce")[$"22-ounce"$] squeeze bottle for #math.equation(block: false, alt: "$4.99")[$"$4.99"$]

Cheese #math.equation(block: false, alt: "$6.49")[$"$6.49"$] for #math.equation(block: false, alt: "1")[$1$] lb. block or #math.equation(block: false, alt: "$3.39")[$"$3.39"$] for #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] lb. block

#solutionbox[
The half-pound block costs \$6.78/lb, so the 1-lb. block is a better buy.
]

Candy #math.equation(block: false, alt: "$10.99")[$"$10.99"$] for a #math.equation(block: false, alt: "1")[$1$] lb. bag or #math.equation(block: false, alt: "$2.89")[$"$2.89"$] for #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] lb. of loose candy

#strong[Translate Phrases to Expressions with Fractions]

In the following exercises, translate the English phrase into an algebraic expression.

#math.equation(block: false, alt: "793")[$793$] miles per #math.equation(block: false, alt: "p")[$p$] hours

#solutionbox[
#math.equation(block: true, alt: "the fraction 793 miles over p hours")[$frac("793 miles", p #h(0.2em) "hours")$]
]

#math.equation(block: false, alt: "78")[$78$] feet per #math.equation(block: false, alt: "r")[$r$] seconds

#math.equation(block: false, alt: "$3")[$"$3"$] for #math.equation(block: false, alt: "0.5")[$0.5$] lbs.

#solutionbox[
#math.equation(block: true, alt: "the fraction $3 over 0.5 lbs .")[$frac("$3", "0.5 lbs" .)$]
]

#math.equation(block: false, alt: "j")[$j$] beats in #math.equation(block: false, alt: "0.5")[$0.5$] minutes

#math.equation(block: false, alt: "105")[$105$] calories in #math.equation(block: false, alt: "x")[$x$] ounces

#solutionbox[
#math.equation(block: true, alt: "the fraction 105 calories over x ounces")[$frac("105 calories", x #h(0.2em) "ounces")$]
]

#math.equation(block: false, alt: "400")[$400$] minutes for #math.equation(block: false, alt: "m")[$m$] dollars

the ratio of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "5 x")[$5 x$]

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

the ratio of #math.equation(block: false, alt: "12 x")[$12 x$] and #math.equation(block: false, alt: "y")[$y$]


=== Everyday Math

One elementary school in Ohio has #math.equation(block: false, alt: "684")[$684$] students and #math.equation(block: false, alt: "45")[$45$] teachers. Write the student-to-teacher ratio as a unit rate.

#solutionbox[
15.2 students per teacher
]

The average American produces about #math.equation(block: false, alt: "1,600")[$1,600$] pounds of paper trash per year #math.equation(block: false, alt: "(365 days).")[$"(365 days)."$] How many pounds of paper trash does the average American produce each day? (Round to the nearest tenth of a pound.)

A popular fast food burger weighs #math.equation(block: false, alt: "7.5")[$7.5$] ounces and contains #math.equation(block: false, alt: "540")[$540$] calories, #math.equation(block: false, alt: "29")[$29$] grams of fat, #math.equation(block: false, alt: "43")[$43$] grams of carbohydrates, and #math.equation(block: false, alt: "25")[$25$] grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce. Round to two decimal places.

#solutionbox[
+ ⓐ 72 calories/ounce
+ ⓑ 3.87 grams of fat/ounce
+ ⓒ 5.73 grams carbs/ounce
+ ⓓ 3.33 grams protein/ounce
]

A #math.equation(block: false, alt: "16-ounce")[$16-"ounce"$] chocolate mocha coffee with whipped cream contains #math.equation(block: false, alt: "470")[$470$] calories, #math.equation(block: false, alt: "18")[$18$] grams of fat, #math.equation(block: false, alt: "63")[$63$] grams of carbohydrates, and #math.equation(block: false, alt: "15")[$15$] grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce.


=== Writing Exercises

Would you prefer the ratio of your income to your friend’s income to be #math.equation(block: false, alt: "3/1")[$"3/1"$] or #math.equation(block: false, alt: "1/3 ?")[$1/3 ?$] Explain your reasoning.

#solutionbox[
Answers will vary.
]

The parking lot at the airport charges #math.equation(block: false, alt: "$0.75")[$"$0.75"$] for every #math.equation(block: false, alt: "15")[$15$] minutes. ⓐ How much does it cost to park for #math.equation(block: false, alt: "1")[$1$] hour? ⓑ Explain how you got your answer to part ⓐ. Was your reasoning based on the unit cost or did you use another method?

Kathryn ate a #math.equation(block: false, alt: "4-ounce")[$4-"ounce"$] cup of frozen yogurt and then went for a swim. The frozen yogurt had #math.equation(block: false, alt: "115")[$115$] calories. Swimming burns #math.equation(block: false, alt: "422")[$422$] calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning.

#solutionbox[
Kathryn should swim for approximately 16.35 minutes. Explanations will vary.
]

Mollie had a #math.equation(block: false, alt: "16-ounce")[$16-"ounce"$] cappuccino at her neighborhood coffee shop. The cappuccino had #math.equation(block: false, alt: "110")[$110$] calories. If Mollie walks for one hour, she burns #math.equation(block: false, alt: "246")[$246$] calories. For how many minutes must Mollie walk to burn off the calories in the cappuccino? Explain your reasoning.


=== Self Check

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

#figure(figph[Self-assessment checklist for students to evaluate their understanding of ratios, rates, unit prices, and translating phrases to fractions, categorized by confidence level.], alt: "Self-assessment checklist for students to evaluate their understanding of ratios, rates, unit prices, and translating phrases to fractions, categorized by confidence level.", caption: none)

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