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== 6.5#h(0.6em)Solve Proportions and their Applications

#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 the fraction 1 over 3 over 4.")[$frac(frac(1, 3), 4) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 12")[$frac(1, 12)$]
]
]

#notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction x over 4 equals 20.")[$frac(x, 4) = 20 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "80")[$80$]
]
]

#notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a rate: Sale rode his bike #math.equation(block: false, alt: "24")[$24$] miles in #math.equation(block: false, alt: "2")[$2$] hours. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 24 miles over 2 hours")[$frac(24 " " "miles", 2 " " "hours")$]
]
]


=== Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a #strong[proportion].

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

A proportion is an equation of the form #math.equation(block: false, alt: "the fraction a over b equals the fraction c over d ,")[$frac(a, b) = frac(c, d) ,$] where #math.equation(block: false, alt: "b not equal to 0 , d not equal to 0.")[$b ≠ 0 , d ≠ 0 .$]

The proportion states two ratios or rates are equal. The proportion is read #math.equation(block: false, alt: "“ a")[$"“" a$] is to #math.equation(block: false, alt: "b ,")[$b ,$] as #math.equation(block: false, alt: "c")[$c$] is to #math.equation(block: false, alt: "d ”.")[$d "”."$]
]

The equation #math.equation(block: false, alt: "the fraction 1 over 2 equals the fraction 4 over 8")[$frac(1, 2) = frac(4, 8)$] is a proportion because the two fractions are equal. The proportion #math.equation(block: false, alt: "the fraction 1 over 2 equals the fraction 4 over 8")[$frac(1, 2) = frac(4, 8)$] is read #math.equation(block: false, alt: "“ 1")[$"“" 1$] is to #math.equation(block: false, alt: "2")[$2$] as #math.equation(block: false, alt: "4")[$4$] is to #math.equation(block: false, alt: "8 ”.")[$8 "”."$]

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion #math.equation(block: false, alt: "the fraction 20 students over 1 teacher equals the fraction 60 students over 3 teachers")[$frac("20 students", "1 teacher") = frac("60 students", "3 teachers")$] we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

#examplebox("Example 1")[][
Write each sentence as a proportion:

+ ⓐ #math.equation(block: false, alt: "3")[$#h(0.2em) 3$] is to #math.equation(block: false, alt: "7")[$7$] as #math.equation(block: false, alt: "15")[$15$] is to #math.equation(block: false, alt: "35.")[$35 .$]
+ ⓑ #math.equation(block: false, alt: "5")[$#h(0.2em) 5$] hits in #math.equation(block: false, alt: "8")[$8$] at bats is the same as #math.equation(block: false, alt: "30")[$30$] hits in #math.equation(block: false, alt: "48")[$48$] at-bats.
+ ⓒ #math.equation(block: false, alt: "$1.50")[$#h(0.2em) "$1.50"$] for #math.equation(block: false, alt: "6")[$6$] ounces is equivalent to #math.equation(block: false, alt: "$2.25")[$"$2.25"$] for #math.equation(block: false, alt: "9")[$9$] ounces.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [3 is to 7 as 15 is to 35.],
  [Write as a proportion.], [#math.equation(block: false, alt: "the fraction 3 over 7 equals the fraction 15 over 35")[$frac(3, 7) = frac(15, 35)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [5 hits in 8 at-bats is the same as 30 hits in 48 at-bats.],
  [Write each fraction to compare hits to at-bats.], [#math.equation(block: false, alt: "the fraction hits over at-bats equals the fraction hits over at-bats")[$frac("hits", "at-bats") = frac("hits", "at-bats")$]],
  [Write as a proportion.], [#math.equation(block: false, alt: "the fraction 5 over 8 equals the fraction 30 over 48")[$frac(5, 8) = frac(30, 48)$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [\$1.50 for 6 ounces is equivalent to \$2.25 for 9 ounces.],
  [Write each fraction to compare dollars to ounces.], [#math.equation(block: false, alt: "the fraction $ over ounces equals the fraction $ over ounces")[$frac(\$, "ounces") = frac(\$, "ounces")$]],
  [Write as a proportion.], [#math.equation(block: false, alt: "the fraction 1.50 over 6 equals the fraction 2.25 over 9")[$frac(1.50, 6) = frac(2.25, 9)$]],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each sentence as a proportion:

+ ⓐ #math.equation(block: false, alt: "5")[$#h(0.2em) 5$] is to #math.equation(block: false, alt: "9")[$9$] as #math.equation(block: false, alt: "20")[$20$] is to #math.equation(block: false, alt: "36.")[$36 .$]
+ ⓑ #math.equation(block: false, alt: "7")[$#h(0.2em) 7$] hits in #math.equation(block: false, alt: "11")[$11$] at-bats is the same as #math.equation(block: false, alt: "28")[$28$] hits in #math.equation(block: false, alt: "44")[$44$] at-bats.
+ ⓒ #math.equation(block: false, alt: "$2.50")[$#h(0.2em) "$2.50"$] for #math.equation(block: false, alt: "8")[$8$] ounces is equivalent to #math.equation(block: false, alt: "$3.75")[$"$3.75"$] for #math.equation(block: false, alt: "12")[$12$] ounces.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 5 over 9 equals the fraction 20 over 36")[$#h(0.2em) frac(5, 9) = frac(20, 36) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 7 over 11 equals the fraction 28 over 44")[$#h(0.2em) frac(7, 11) = frac(28, 44) #h(0.2em)$]
+ ⓒ #math.equation(block: false, alt: "the fraction 2.50 over 8 equals the fraction 3.75 over 12")[$frac(2.50, 8) = frac(3.75, 12) #h(0.2em)$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each sentence as a proportion:

+ ⓐ #math.equation(block: false, alt: "6")[$#h(0.2em) 6$] is to #math.equation(block: false, alt: "7")[$7$] as #math.equation(block: false, alt: "36")[$36$] is to #math.equation(block: false, alt: "42.")[$42 .$]
+ ⓑ #math.equation(block: false, alt: "8")[$#h(0.2em) 8$] adults for #math.equation(block: false, alt: "36")[$36$] children is the same as #math.equation(block: false, alt: "12")[$12$] adults for #math.equation(block: false, alt: "54")[$54$] children.
+ ⓒ #math.equation(block: false, alt: "$3.75")[$#h(0.2em) "$3.75"$] for #math.equation(block: false, alt: "6")[$6$] ounces is equivalent to #math.equation(block: false, alt: "$2.50")[$"$2.50"$] for #math.equation(block: false, alt: "4")[$4$] ounces.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 6 over 7 equals the fraction 36 over 42")[$#h(0.2em) frac(6, 7) = frac(36, 42)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 8 over 36 equals the fraction 12 over 54")[$#h(0.2em) frac(8, 36) = frac(12, 54)$]
+ ⓒ #math.equation(block: false, alt: "the fraction 3.75 over 6 equals the fraction 2.50 over 4")[$frac(3.75, 6) = frac(2.50, 4)$]
]
]

Look at the proportions #math.equation(block: false, alt: "the fraction 1 over 2 equals the fraction 4 over 8")[$frac(1, 2) = frac(4, 8)$] and #math.equation(block: false, alt: "the fraction 2 over 3 equals the fraction 6 over 9.")[$frac(2, 3) = frac(6, 9) .$] From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?

To determine if a proportion is true, we find the #strong[cross products] of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross product because of the cross formed. If, and only if, the given proportion is true, that is, the two sides are equal, then the cross products of a proportion will be equal.

#figure(figph[The figure shows cross multiplication of two proportions. There is the proportion 1 is to 2 as 4 is to 8. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 8 · 1 = 8 and 2 · 4 = 8. There is the proportion 2 is to 3 as 6 is to 9. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 9 · 2 = 18 and 3 · 6 = 18.], alt: "The figure shows cross multiplication of two proportions. There is the proportion 1 is to 2 as 4 is to 8. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 8 · 1 = 8 and 2 · 4 = 8. There is the proportion 2 is to 3 as 6 is to 9. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 9 · 2 = 18 and 3 · 6 = 18.", caption: none)

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Cross Products of a Proportion]

For any proportion of the form #math.equation(block: false, alt: "the fraction a over b equals the fraction c over d ,")[$frac(a, b) = frac(c, d) ,$] where #math.equation(block: false, alt: "b not equal to 0 , d not equal to 0 ,")[$b ≠ 0 , d ≠ 0 ,$] its cross products are equal.

#figure(figph[The image demonstrates cross products of a proportion with numerator a multiplying with denominator d, and denominator b multiplying with numerator c.], alt: "The image demonstrates cross products of a proportion with numerator a multiplying with denominator d, and denominator b multiplying with numerator c.", caption: none)
]

Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are both equal, we have a proportion.

#examplebox("Example 2")[][
Determine whether each equation is a proportion:

+ ⓐ #math.equation(block: false, alt: "the fraction 4 over 9 equals the fraction 12 over 28")[$#h(0.2em) frac(4, 9) = frac(12, 28)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 17.5 over 37.5 equals the fraction 7 over 15")[$frac(17.5, 37.5) = frac(7, 15)$]

#solutionbox[
To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#figure(figph[A mathematical equation showing the fractions 4/9 equals 12/28. Both fractions are written in black against a plain white background.], alt: "A mathematical equation showing the fractions 4/9 equals 12/28. Both fractions are written in black against a plain white background.", caption: none)],
  [Find the cross products.], [#math.equation(block: false, alt: "28 times 4 equals 1129 times 12 equals 108")[$28 ⋅ 4 = 112 #h(2em) 9 ⋅ 12 = 108$] #linebreak() #figure(figph[The image illustrates an inequality between two fractions, 4/9 and 12/28, using cross-multiplication with blue arrows to show that 4 multiplied by 28 is not equal to 9 multiplied by 12.], alt: "The image illustrates an inequality between two fractions, 4/9 and 12/28, using cross-multiplication with blue arrows to show that 4 multiplied by 28 is not equal to 9 multiplied by 12.", caption: none)],
))

Since the cross products are not equal, #math.equation(block: false, alt: "28 times 4 not equal to 9 times 12 ,")[$28 · 4 ≠ 9 · 12 ,$] the equation is not a proportion.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#figure(figph[A mathematical equation displays the fraction 17.5/37.5 simplified to 7/15, demonstrating equivalence between the two ratios.], alt: "A mathematical equation displays the fraction 17.5/37.5 simplified to 7/15, demonstrating equivalence between the two ratios.", caption: none)],
  [Find the cross products.], [#math.equation(block: false, alt: "15 times 17.5 equals 262.537.5 times 7 equals 262.5")[$15 ⋅ 17.5 = 262.5 #h(2em) 37.5 ⋅ 7 = 262.5$] #linebreak() 
#figure(figph[An equation showing 17.5 divided by 37.5 is equal to 7 divided by 15, with blue arrows illustrating the concept of cross-multiplication.], alt: "An equation showing 17.5 divided by 37.5 is equal to 7 divided by 15, with blue arrows illustrating the concept of cross-multiplication.", caption: none)],
))

Since the cross products are equal, #math.equation(block: false, alt: "15 times 17.5 equals 37.5 times 7 ,")[$15 · 17.5 = 37.5 · 7 ,$] the equation is a proportion.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each equation is a proportion:

+ ⓐ #math.equation(block: false, alt: "the fraction 7 over 9 equals the fraction 54 over 72")[$#h(0.2em) frac(7, 9) = frac(54, 72)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 24.5 over 45.5 equals the fraction 7 over 13")[$frac(24.5, 45.5) = frac(7, 13)$]

#solutionbox[
+ ⓐ no
+ ⓑ yes
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each equation is a proportion:

+ ⓐ #math.equation(block: false, alt: "the fraction 8 over 9 equals the fraction 56 over 73")[$#h(0.2em) frac(8, 9) = frac(56, 73) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 28.5 over 52.5 equals the fraction 8 over 15")[$#h(0.2em) frac(28.5, 52.5) = frac(8, 15)$]

#solutionbox[
+ ⓐ no
+ ⓑ no
]
]


=== Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the #strong[Least Common Denominator] (LCD) using the #strong[Multiplication Property of Equality].

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "the fraction x over 63 equals the fraction 4 over 7.")[$frac(x, 63) = frac(4, 7) .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed, showing a fraction on the left side, x over 63, which is equal to the fraction 4 over 7 on the right side.], alt: "A mathematical equation is displayed, showing a fraction on the left side, x over 63, which is equal to the fraction 4 over 7 on the right side.", caption: none)], [],
  [To isolate #math.equation(block: false, alt: "x")[$x$], multiply both sides by the LCD, 63.], [#figure(figph[A mathematical equation shows 63 multiplied by the fraction x over 63, which equals 63 multiplied by the fraction 4 over 7. The number 63 is highlighted in red on both sides of the equation.], alt: "A mathematical equation shows 63 multiplied by the fraction x over 63, which equals 63 multiplied by the fraction 4 over 7. The number 63 is highlighted in red on both sides of the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation displays x = (9 \* 7 \* 4) / 7.], alt: "A mathematical equation displays x = (9 * 7 * 4) / 7.", caption: none)], [],
  [Divide the common factors.], [#figure(figph[A mathematical equation displays 'x = 36' in a black serif font on a plain white background, centered within the frame.], alt: "A mathematical equation displays 'x = 36' in a black serif font on a plain white background, centered within the frame.", caption: none)], [],
  [Check: To check our answer, we substitute into the original proportion.], [], [],
  [], [#figure(figph[A mathematical equation shows x divided by 63 is equal to 4 divided by 7.], alt: "A mathematical equation shows x divided by 63 is equal to 4 divided by 7.", caption: none)], [],
  [#figure(figph[The image shows the text 'Substitute x = 36' in blue and red font against a white background.], alt: "The image shows the text 'Substitute x = 36' in blue and red font against a white background.", caption: none)], [#figure(figph[A mathematical equation displays the fraction 36/63, a question mark, and the fraction 4/7, inviting a comparison or to determine if the fractions are equivalent. The number 36 is highlighted in red.], alt: "A mathematical equation displays the fraction 36/63, a question mark, and the fraction 4/7, inviting a comparison or to determine if the fractions are equivalent. The number 36 is highlighted in red.", caption: none)], [],
  [Show common factors.], [#figure(figph[A mathematical expression asks if the fraction (4 times 9) divided by (7 times 9) is equal to 4/7. This illustrates the principle of simplifying fractions by canceling common factors.], alt: "A mathematical expression asks if the fraction (4 times 9) divided by (7 times 9) is equal to 4/7. This illustrates the principle of simplifying fractions by canceling common factors.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation 4/7 = 4/7 with a checkmark, confirming its truth.], alt: "The equation 4/7 = 4/7 with a checkmark, confirming its truth.", caption: none)], [],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve the proportion: #math.equation(block: false, alt: "the fraction n over 84 equals the fraction 11 over 12.")[$frac(n, 84) = frac(11, 12) .$]

#solutionbox[
77
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve the proportion: #math.equation(block: false, alt: "the fraction y over 96 equals the fraction 13 over 12.")[$frac(y, 96) = frac(13, 12) .$]

#solutionbox[
104
]
]

When the variable is in a denominator, we’ll use the fact that the #strong[cross products] of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "the fraction 144 over a equals the fraction 9 over 4.")[$frac(144, a) = frac(9, 4) .$]

#solutionbox[
Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation illustrating cross-multiplication for solving proportions: 144 over a equals 9 over 4, with arrows indicating the cross-multiplication step.], alt: "A mathematical equation illustrating cross-multiplication for solving proportions: 144 over a equals 9 over 4, with arrows indicating the cross-multiplication step.", caption: none)], [],
  [Find the cross products and set them equal.], [#figure(figph[A mathematical equation shows '4 \* 144 = a \* 9'.], alt: "A mathematical equation shows '4 * 144 = a * 9'.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation displays '576 = 9a' in black text on a white background, representing a single linear equation with one unknown variable 'a'.], alt: "A mathematical equation displays '576 = 9a' in black text on a white background, representing a single linear equation with one unknown variable 'a'.", caption: none)], [],
  [Divide both sides by 9.], [#figure(figph[A mathematical equation shows a fraction 576 over 9 equal to a fraction 9a over 9.], alt: "A mathematical equation shows a fraction 576 over 9 equal to a fraction 9a over 9.", caption: none)], [],
  [Simplify.], [#figure(figph[The mathematical equation '64 = a' is displayed in black font on a white background, indicating that the value of 'a' is 64.], alt: "The mathematical equation '64 = a' is displayed in black font on a white background, indicating that the value of 'a' is 64.", caption: none)], [],
  [Check your answer.], [], [],
  [], [#figure(figph[A mathematical equation shows '144 divided by a equals 9 divided by 4' against a white background.], alt: "A mathematical equation shows '144 divided by a equals 9 divided by 4' against a white background.", caption: none)], [],
  [#figure(figph[The image shows the text 'Substitute a = 64' in a dark grey font, with the number '64' highlighted in red.], alt: "The image shows the text 'Substitute a = 64' in a dark grey font, with the number '64' highlighted in red.", caption: none)], [#figure(figph[A mathematical equation shows two fractions, 144/64 and 9/4, with a question mark and an equals sign between them, asking if they are equivalent. The denominator 64 is in red.], alt: "A mathematical equation shows two fractions, 144/64 and 9/4, with a question mark and an equals sign between them, asking if they are equivalent. The denominator 64 is in red.", caption: none)], [],
  [Show common factors..], [#figure(figph[A mathematical equation questions if (9 multiplied by 16) divided by (4 multiplied by 16) is equal to 9 divided by 4, illustrating the concept of simplifying fractions by cancelling common factors.], alt: "A mathematical equation questions if (9 multiplied by 16) divided by (4 multiplied by 16) is equal to 9 divided by 4, illustrating the concept of simplifying fractions by cancelling common factors.", caption: none)], [],
  [Simplify.], [#figure(figph[Mathematical equation 9/4 = 9/4 with a checkmark.], alt: "Mathematical equation 9/4 = 9/4 with a checkmark.", caption: none)], [],
))

Another method to solve this would be to multiply both sides by the LCD, #math.equation(block: false, alt: "4 a .")[$4 a .$] Try it and verify that you get the same solution.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve the proportion: #math.equation(block: false, alt: "the fraction 91 over b equals the fraction 7 over 5.")[$frac(91, b) = frac(7, 5) .$]

#solutionbox[
65
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve the proportion: #math.equation(block: false, alt: "the fraction 39 over c equals the fraction 13 over 8.")[$frac(39, c) = frac(13, 8) .$]

#solutionbox[
24
]
]

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "the fraction 52 over 91 equals the fraction −4 over y .")[$frac(52, 91) = frac(−4, y) .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Find the cross products and set them equal.], [#figure(figph[The equation 52/91 = -4/y, with visual cues for cross-multiplication to solve for y.], alt: "The equation 52/91 = -4/y, with visual cues for cross-multiplication to solve for y.", caption: none)], [],
  [], [#figure(figph[A mathematical equation is displayed, showing 'y \* 52 = 91(-4)'. The equation involves multiplication and a negative number, where 'y' is the unknown variable.], alt: "A mathematical equation is displayed, showing 'y * 52 = 91(-4)'. The equation involves multiplication and a negative number, where 'y' is the unknown variable.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing '52y = -364' in black text against a white background, representing an algebraic problem.], alt: "A mathematical equation is displayed, showing '52y = -364' in black text against a white background, representing an algebraic problem.", caption: none)], [],
  [Divide both sides by 52.], [#figure(figph[The step of dividing both sides of 52y = -364 by 52 to find the value of y.], alt: "The step of dividing both sides of 52y = -364 by 52 to find the value of y.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays a mathematical equation in black text on a white background, stating 'y = -7'.], alt: "The image displays a mathematical equation in black text on a white background, stating 'y = -7'.", caption: none)], [],
  [Check:], [], [],
  [], [#figure(figph[A mathematical equation displays the fraction 52 over 91, which is set equal to the fraction -4 over y.], alt: "A mathematical equation displays the fraction 52 over 91, which is set equal to the fraction -4 over y.", caption: none)], [],
  [#figure(figph[The text reads 'Substitute y = -7', with the word 'Substitute' and 'y = ' in dark gray, and '-7' in red. The text is rendered against a light background.], alt: "The text reads 'Substitute y = -7', with the word 'Substitute' and 'y = ' in dark gray, and '-7' in red. The text is rendered against a light background.", caption: none)], [#figure(figph[A mathematical expression displaying two fractions, 52/91 and -4/-7, separated by an equals sign with a question mark above it, indicating a query about their equivalence.], alt: "A mathematical expression displaying two fractions, 52/91 and -4/-7, separated by an equals sign with a question mark above it, indicating a query about their equivalence.", caption: none)], [],
  [Show common factors.], [#figure(figph[A mathematical equation asking if (13 \* 4) / (13 \* 4) is equal to -4 / -7. The left side simplifies to 1, while the right side simplifies to 4/7, indicating they are not equal.], alt: "A mathematical equation asking if (13 * 4) / (13 * 4) is equal to -4 / -7. The left side simplifies to 1, while the right side simplifies to 4/7, indicating they are not equal.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation 4/7 = 4/7 is displayed with a checkmark, indicating its correctness.], alt: "The equation 4/7 = 4/7 is displayed with a checkmark, indicating its correctness.", caption: none)], [],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve the proportion: #math.equation(block: false, alt: "the fraction 84 over 98 equals the fraction −6 over x .")[$frac(84, 98) = frac(−6, x) .$]

#solutionbox[
−7
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve the proportion: #math.equation(block: false, alt: "the fraction −7 over y equals the fraction 105 over 135.")[$frac(−7, y) = frac(105, 135) .$]

#solutionbox[
−9
]
]


=== Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the #strong[proportion], we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

#examplebox("Example 6")[][
When pediatricians prescribe acetaminophen to children, they prescribe #math.equation(block: false, alt: "5")[$5$] milliliters (ml) of acetaminophen for every #math.equation(block: false, alt: "25")[$25$] pounds of the child’s weight. If Zoe weighs #math.equation(block: false, alt: "80")[$80$] pounds, how many milliliters of acetaminophen will her doctor prescribe?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify what you are asked to find.], [How many ml of acetaminophen the doctor will prescribe]),
  [Choose a variable to represent it.], [Let #math.equation(block: false, alt: "a equals")[$a =$] ml of acetaminophen.],
  [Write a sentence that gives the information to find it.], [If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds?],
  [Translate into a proportion.], [#figure(figph[A simple mathematical identity: ml/pounds = ml/pounds.], alt: "A simple mathematical identity: ml/pounds = ml/pounds.", caption: none)],
  [Substitute given values—be careful of the units.], [#figure(figph[A mathematical equation showing the fraction 5/25 equal to the fraction a/80.], alt: "A mathematical equation showing the fraction 5/25 equal to the fraction a/80.", caption: none)],
  [Multiply both sides by 80.], [#figure(figph[A mathematical equation shows '80 multiplied by the fraction 5 over 25' on the left side, which is set equal to '80 multiplied by the fraction a over 80' on the right side.], alt: "A mathematical equation shows '80 multiplied by the fraction 5 over 25' on the left side, which is set equal to '80 multiplied by the fraction a over 80' on the right side.", caption: none)],
  [Multiply and show common factors.], [#figure(figph[A mathematical equation illustrating fraction simplification: (16 x 5 x 5) / (5 x 5) = 80a / 80.], alt: "A mathematical equation illustrating fraction simplification: (16 x 5 x 5) / (5 x 5) = 80a / 80.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation displays the number 16 equal to the variable 'a', written as '16 = a'.], alt: "A mathematical equation displays the number 16 equal to the variable 'a', written as '16 = a'.", caption: none)],
  [Check if the answer is reasonable.], [],
  [Yes. Since 80 is about 3 times 25, the medicine should be about 3 times 5.], [],
  [Write a complete sentence.], [The pediatrician would prescribe 16 ml of acetaminophen to Zoe.],
))

You could also solve this proportion by setting the cross products equal.
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Pediatricians prescribe #math.equation(block: false, alt: "5")[$5$] milliliters (ml) of acetaminophen for every #math.equation(block: false, alt: "25")[$25$] pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs #math.equation(block: false, alt: "60")[$60$] pounds?

#solutionbox[
12 ml
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For every #math.equation(block: false, alt: "1")[$1$] kilogram (kg) of a child’s weight, pediatricians prescribe #math.equation(block: false, alt: "15")[$15$] milligrams (mg) of a fever reducer. If Isabella weighs #math.equation(block: false, alt: "12")[$12$] kg, how many milligrams of the fever reducer will the pediatrician prescribe?

#solutionbox[
180 mg
]
]

#examplebox("Example 7")[][
One brand of microwave popcorn has #math.equation(block: false, alt: "120")[$120$] calories per serving. A whole bag of this popcorn has #math.equation(block: false, alt: "3.5")[$3.5$] servings. How many calories are in a whole bag of this microwave popcorn?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify what you are asked to find.], [How many calories are in a whole bag of microwave popcorn?]),
  [Choose a variable to represent it.], [Let #math.equation(block: false, alt: "c equals")[$c =$] number of calories.],
  [Write a sentence that gives the information to find it.], [If there are 120 calories per serving, how many calories are in a whole bag with 3.5 servings?],
  [Translate into a proportion.], [#figure(figph[A mathematical equation shows that 'calories / serving' equals 'calories / serving,' representing an identity or a tautology in unit measurement.], alt: "A mathematical equation shows that 'calories / serving' equals 'calories / serving,' representing an identity or a tautology in unit measurement.", caption: none)],
  [Substitute given values.], [#figure(figph[A mathematical equation shows a fraction 120 over 1 equal to the fraction c over 3.5. This represents a proportion where 120 divided by 1 is equal to c divided by 3.5.], alt: "A mathematical equation shows a fraction 120 over 1 equal to the fraction c over 3.5. This represents a proportion where 120 divided by 1 is equal to c divided by 3.5.", caption: none)],
  [Multiply both sides by 3.5.], [#figure(figph[An algebraic equation: (3.5)(120/1) = (3.5)(c/3.5), demonstrating the multiplication of a decimal by a fraction on both sides.], alt: "An algebraic equation: (3.5)(120/1) = (3.5)(c/3.5), demonstrating the multiplication of a decimal by a fraction on both sides.", caption: none)],
  [Multiply.], [#figure(figph[The image displays the mathematical equation '420 = c' in a simple, clear font against a white background.], alt: "The image displays the mathematical equation '420 = c' in a simple, clear font against a white background.", caption: none)],
  [Check if the answer is reasonable.], [],
  [Yes. Since 3.5 is between 3 and 4, the total calories should be between 360 (3⋅120) and 480 (4⋅120).], [],
  [Write a complete sentence.], [The whole bag of microwave popcorn has 420 calories.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Marissa loves the Caramel Macchiato at the coffee shop. The #math.equation(block: false, alt: "16")[$16$] oz. medium size has #math.equation(block: false, alt: "240")[$240$] calories. How many calories will she get if she drinks the large #math.equation(block: false, alt: "20")[$20$] oz. size?

#solutionbox[
300
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Yaneli loves Starburst candies, but wants to keep her snacks to #math.equation(block: false, alt: "100")[$100$] calories. If the candies have #math.equation(block: false, alt: "160")[$160$] calories for #math.equation(block: false, alt: "8")[$8$] pieces, how many pieces can she have in her snack?

#solutionbox[
5 pieces
]
]

#examplebox("Example 8")[][
Josiah went to Mexico for spring break and changed #math.equation(block: false, alt: "$325")[$"$325"$] dollars into Mexican pesos. At that time, the exchange rate had #math.equation(block: false, alt: "$1")[$"$1"$] U.S. is equal to #math.equation(block: false, alt: "12.54")[$12.54$] Mexican pesos. How many Mexican pesos did he get for his trip?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify what you are asked to find.], [How many Mexican pesos did Josiah get?]),
  [Choose a variable to represent it.], [Let #math.equation(block: false, alt: "p equals")[$p =$] number of pesos.],
  [Write a sentence that gives the information to find it.], [If \$1 U.S. is equal to 12.54 Mexican pesos, then \$325 is how many pesos?],
  [Translate into a proportion.], [#figure(figph[A diagram showing the expression "dollars per peso equals dollars per peso." The image illustrates that the ratio of dollars to pesos remains constant.], alt: "A diagram showing the expression \"dollars per peso equals dollars per peso.\" The image illustrates that the ratio of dollars to pesos remains constant.", caption: none)],
  [Substitute given values.], [#figure(figph[A mathematical equation is displayed on a white background, showing 1 divided by 12.54 equals 325 divided by p.], alt: "A mathematical equation is displayed on a white background, showing 1 divided by 12.54 equals 325 divided by p.", caption: none)],
  [The variable is in the denominator, so find the cross products and set them equal.], [#figure(figph[The image displays the mathematical equation p \* 1 = 12.54(325).], alt: "The image displays the mathematical equation p * 1 = 12.54(325).", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation displays 'c = 4,075.5' in black font against a white background.], alt: "A mathematical equation displays 'c = 4,075.5' in black font against a white background.", caption: none)],
  [Check if the answer is reasonable.], [],
  [Yes, \$100 would be \$1,254 pesos. \$325 is a little more than 3 times this amount.], [],
  [Write a complete sentence.], [Josiah has 4075.5 pesos for his spring break trip.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Yurianna is going to Europe and wants to change #math.equation(block: false, alt: "$800")[$"$800"$] dollars into Euros. At the current exchange rate, #math.equation(block: false, alt: "$1")[$"$1"$] US is equal to #math.equation(block: false, alt: "0.738")[$0.738$] Euro. How many Euros will she have for her trip?

#solutionbox[
590 Euros
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Corey and Nicole are traveling to Japan and need to exchange #math.equation(block: false, alt: "$600")[$"$600"$] into Japanese yen. If each dollar is #math.equation(block: false, alt: "94.1")[$94.1$] yen, how many yen will they get?

#solutionbox[
56,460 yen
]
]


=== Write Percent Equations As Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A #strong[percent proportion] is an equation where a percent is equal to an equivalent ratio.

For example, #math.equation(block: false, alt: "60% equals the fraction 60 over 100")[$"60%" = frac(60, 100)$] and we can simplify #math.equation(block: false, alt: "the fraction 60 over 100 equals the fraction 3 over 5.")[$frac(60, 100) = frac(3, 5) .$] Since the equation #math.equation(block: false, alt: "the fraction 60 over 100 equals the fraction 3 over 5")[$frac(60, 100) = frac(3, 5)$] shows a percent equal to an equivalent ratio, we call it a #strong[percent proportion]. Using the vocabulary we used earlier:

#math.equation(block: true, alt: "the fraction amount over base equals the fraction percent over 100")[$frac("amount", "base") = frac("percent", 100)$]#math.equation(block: true, alt: "the fraction 3 over 5 equals the fraction 60 over 100")[$#h(1.2em) frac(3, 5) = frac(60, 100)$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Percent Proportion]

The amount is to the base as the percent is to #math.equation(block: false, alt: "100.")[$100 .$]

#math.equation(block: true, alt: "the fraction amount over base equals the fraction percent over 100")[$frac("amount", "base") = frac("percent", 100)$]
]

If we restate the problem in the words of a proportion, it may be easier to set up the proportion:

#math.equation(block: true, alt: "The amount is to the base as the percent is to one hundred.")[$"The amount is to the base as the percent is to one hundred."$]We could also say:

#math.equation(block: true, alt: "The amount out of the base is the same as the percent out of one hundred.")[$"The amount out of the base is the same as the percent out of one hundred."$]First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

#examplebox("Example 9")[][
Translate to a proportion. What number is #math.equation(block: false, alt: "75%")[$"75%"$] of #math.equation(block: false, alt: "90 ?")[$90 ?$]

#solutionbox[
If you look for the word "of", it may help you identify the base. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the parts of the percent proportion.], [#figure(figph[An image explaining parts of a percentage problem: 'What number is 75% of 90?'. 'What number' is the amount, '75%' is the percent, and '90' is the base. 'Of' is highlighted in red.], alt: "An image explaining parts of a percentage problem: 'What number is 75% of 90?'. 'What number' is the amount, '75%' is the percent, and '90' is the base. 'Of' is highlighted in red.", caption: none)]),
  [Restate as a proportion.], [#figure(figph[A math problem asks: 'What number out of 90 is the same as 75 out of 100?' The number '90' is highlighted in red.], alt: "A math problem asks: 'What number out of 90 is the same as 75 out of 100?' The number '90' is highlighted in red.", caption: none)],
  [Set up the proportion. Let #math.equation(block: false, alt: "n equals number")[$n = "number"$].], [#math.equation(block: false, alt: "the fraction n over 90 equals the fraction 75 over 100")[$frac(n, 90) = frac(75, 100)$]],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to a proportion: What number is #math.equation(block: false, alt: "60%")[$"60%"$] of #math.equation(block: false, alt: "105 ?")[$105 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 105 equals the fraction 60 over 100")[$frac(n, 105) = frac(60, 100)$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to a proportion: What number is #math.equation(block: false, alt: "40%")[$"40%"$] of #math.equation(block: false, alt: "85 ?")[$85 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 85 equals the fraction 40 over 100")[$frac(n, 85) = frac(40, 100)$]
]
]

#examplebox("Example 10")[][
Translate to a proportion. #math.equation(block: false, alt: "19")[$19$] is #math.equation(block: false, alt: "25%")[$"25%"$] of what number?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the parts of the percent proportion.], [#figure(figph[A breakdown of a percentage word problem: '19 is 25% of what number?', labeling '19' as amount, '25%' as percent, and 'what number' as base.], alt: "A breakdown of a percentage word problem: '19 is 25% of what number?', labeling '19' as amount, '25%' as percent, and 'what number' as base.", caption: none)]),
  [Restate as a proportion.], [#figure(figph[A mathematical word problem asks: '19 out of what number is the same as 25 out of 100?' The word 'of' in the first phrase is highlighted in red.], alt: "A mathematical word problem asks: '19 out of what number is the same as 25 out of 100?' The word 'of' in the first phrase is highlighted in red.", caption: none)],
  [Set up the proportion. Let #math.equation(block: false, alt: "n equals number")[$n = "number"$].], [#math.equation(block: false, alt: "the fraction 19 over n equals the fraction 25 over 100")[$frac(19, n) = frac(25, 100)$]],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to a proportion: #math.equation(block: false, alt: "36")[$36$] is #math.equation(block: false, alt: "25%")[$"25%"$] of what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction 36 over n equals the fraction 25 over 100")[$frac(36, n) = frac(25, 100)$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to a proportion: #math.equation(block: false, alt: "27")[$27$] is #math.equation(block: false, alt: "36%")[$"36%"$] of what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction 27 over n equals the fraction 36 over 100")[$frac(27, n) = frac(36, 100)$]
]
]

#examplebox("Example 11")[][
Translate to a proportion. What percent of #math.equation(block: false, alt: "27")[$27$] is #math.equation(block: false, alt: "9 ?")[$9 ?$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the parts of the percent proportion.], [#figure(figph[A math problem asks 'What percent of 27 is 9?' with 'What percent' labeled as 'percent', '27' as 'base', and '9' as 'amount', illustrating how to identify parts of a percentage equation.], alt: "A math problem asks 'What percent of 27 is 9?' with 'What percent' labeled as 'percent', '27' as 'base', and '9' as 'amount', illustrating how to identify parts of a percentage equation.", caption: none)]),
  [Restate as a proportion.], [#figure(figph[A math question asks: '9 out of 27 is the same as what number out of 100?' The word 'of' between 'out' and '27' is highlighted in red, while the rest of the text is in a dark teal color.], alt: "A math question asks: '9 out of 27 is the same as what number out of 100?' The word 'of' between 'out' and '27' is highlighted in red, while the rest of the text is in a dark teal color.", caption: none)],
  [Set up the proportion. Let #math.equation(block: false, alt: "p equals percent")[$p = "percent"$].], [#math.equation(block: false, alt: "the fraction 9 over 27 equals the fraction p over 100")[$frac(9, 27) = frac(p, 100)$]],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to a proportion: What percent of #math.equation(block: false, alt: "52")[$52$] is #math.equation(block: false, alt: "39 ?")[$39 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 100 equals the fraction 39 over 52")[$frac(n, 100) = frac(39, 52)$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to a proportion: What percent of #math.equation(block: false, alt: "92")[$92$] is #math.equation(block: false, alt: "23 ?")[$23 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 100 equals the fraction 23 over 92")[$frac(n, 100) = frac(23, 92)$]
]
]


=== Translate and Solve Percent Proportions

Now that we have written percent equations as proportions, we are ready to solve the equations.

#examplebox("Example 12")[][
Translate and solve using proportions: What number is #math.equation(block: false, alt: "45%")[$"45%"$] of #math.equation(block: false, alt: "80 ?")[$80 ?$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the parts of the percent proportion.], [#figure(figph[A math problem asks 'What number is 45% of 80?' with 'What number' labeled as amount, '45%' as percent, and '80' as base, illustrating how to identify parts of a percentage problem.], alt: "A math problem asks 'What number is 45% of 80?' with 'What number' labeled as amount, '45%' as percent, and '80' as base, illustrating how to identify parts of a percentage problem.", caption: none)]),
  [Restate as a proportion.], [#figure(figph[A math problem asks: 'What number out of 80 is the same as 45 out of 100?'], alt: "A math problem asks: 'What number out of 80 is the same as 45 out of 100?'", caption: none)],
  [Set up the proportion. Let #math.equation(block: false, alt: "n equals")[$n =$] number.], [#figure(figph[A mathematical equation displays the fraction n over 80 set equal to the fraction 45 over 100, which is commonly used to represent a proportion or percentage problem.], alt: "A mathematical equation displays the fraction n over 80 set equal to the fraction 45 over 100, which is commonly used to represent a proportion or percentage problem.", caption: none)],
  [Find the cross products and set them equal.], [#figure(figph[A mathematical equation shows '100 multiplied by n equals 80 multiplied by 45'.], alt: "A mathematical equation shows '100 multiplied by n equals 80 multiplied by 45'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, reading '100n = 3,600'. The numbers and variable 'n' are in a bold, dark font against a white background.], alt: "A mathematical equation is displayed, reading '100n = 3,600'. The numbers and variable 'n' are in a bold, dark font against a white background.", caption: none)],
  [Divide both sides by 100.], [#figure(figph[A mathematical equation showing 100n divided by 100 equals 3,600 divided by 100. This step demonstrates dividing both sides of an equation by 100 to solve for 'n'.], alt: "A mathematical equation showing 100n divided by 100 equals 3,600 divided by 100. This step demonstrates dividing both sides of an equation by 100 to solve for 'n'.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the equation n = 36 in bold, black mathematical text centered against a plain white background. The variable 'n' is followed by an equals sign and the number '36', indicating a simple numerical assignment.], alt: "The image displays the equation n = 36 in bold, black mathematical text centered against a plain white background. The variable 'n' is followed by an equals sign and the number '36', indicating a simple numerical assignment.", caption: none)],
  [Check if the answer is reasonable.], [],
  [Yes. 45 is a little less than half of 100 and 36 is a little less than half 80.], [],
  [Write a complete sentence that answers the question.], [36 is 45% of 80.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve using proportions: What number is #math.equation(block: false, alt: "65%")[$"65%"$] of #math.equation(block: false, alt: "40 ?")[$40 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 40 equals the fraction 65 over 100 ; n equals 26")[$frac(n, 40) = frac(65, 100) ; #h(1em) n = 26$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve using proportions: What number is #math.equation(block: false, alt: "85%")[$"85%"$] of #math.equation(block: false, alt: "40 ?")[$40 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 40 equals the fraction 85 over 100 ; n equals 34")[$frac(n, 40) = frac(85, 100) ; #h(1em) n = 34$]
]
]

In the next example, the percent is more than #math.equation(block: false, alt: "100 ,")[$100 ,$] which is more than one whole. So the unknown number will be more than the base.

#examplebox("Example 13")[][
Translate and solve using proportions: #math.equation(block: false, alt: "125%")[$"125%"$] of #math.equation(block: false, alt: "25")[$25$] is what number?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the parts of the percent proportion.], [#figure(figph[A math problem asking '125% is 25 of what number?', with '125%' labeled as 'percent', '25' as 'base', and 'what number?' as 'amount', demonstrating a percentage calculation.], alt: "A math problem asking '125% is 25 of what number?', with '125%' labeled as 'percent', '25' as 'base', and 'what number?' as 'amount', demonstrating a percentage calculation.", caption: none)]),
  [Restate as a proportion.], [#figure(figph[A mathematical word problem asks: 'What number out of 25 is the same as 125 out of 100?'], alt: "A mathematical word problem asks: 'What number out of 25 is the same as 125 out of 100?'", caption: none)],
  [Set up the proportion. Let #math.equation(block: false, alt: "n equals")[$n =$] number.], [#figure(figph[A mathematical equation showing the proportion n/25 = 125/100, where 'n' is an unknown variable. The equation is presented in a clear, digital format on a white background.], alt: "A mathematical equation showing the proportion n/25 = 125/100, where 'n' is an unknown variable. The equation is presented in a clear, digital format on a white background.", caption: none)],
  [Find the cross products and set them equal.], [#figure(figph[A mathematical equation shows '100 multiplied by n equals 25 multiplied by 125' in black text on a white background.], alt: "A mathematical equation shows '100 multiplied by n equals 25 multiplied by 125' in black text on a white background.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the algebraic equation 100n = 3,125.], alt: "The image displays the algebraic equation 100n = 3,125.", caption: none)],
  [Divide both sides by 100.], [#figure(figph[A mathematical equation shows '100n over 100 equals 3,125 over 100,' demonstrating how to solve for 'n' by dividing both sides by 100, which will result in n = 3,125.], alt: "A mathematical equation shows '100n over 100 equals 3,125 over 100,' demonstrating how to solve for 'n' by dividing both sides by 100, which will result in n = 3,125.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression 'n = 31.25' is displayed in a serif font on a white background.], alt: "A mathematical expression 'n = 31.25' is displayed in a serif font on a white background.", caption: none)],
  [Check if the answer is reasonable.], [],
  [Yes. 125 is more than 100 and 31.25 is more than 25.], [],
  [Write a complete sentence that answers the question.], [125% of 25 is 31.25.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve using proportions: #math.equation(block: false, alt: "125%")[$"125%"$] of #math.equation(block: false, alt: "64")[$64$] is what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 64 equals the fraction 125 over 100 ; n equals 80")[$frac(n, 64) = frac(125, 100) ; #h(1em) n = 80$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve using proportions: #math.equation(block: false, alt: "175%")[$"175%"$] of #math.equation(block: false, alt: "84")[$84$] is what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 84 equals the fraction 175 over 100 ; n equals 147")[$frac(n, 84) = frac(175, 100) ; #h(1em) n = 147$]
]
]

Percents with decimals and money are also used in proportions.

#examplebox("Example 14")[][
Translate and solve: #math.equation(block: false, alt: "6.5%")[$"6.5%"$] of what number is #math.equation(block: false, alt: "$1.56 ?")[$"$1.56" ?$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the parts of the percent proportion.], [#figure(figph[A percentage problem breaking down '6.5% of what number is \$1.56?' into its components: percent, base, and amount.], alt: "A percentage problem breaking down '6.5% of what number is $1.56?' into its components: percent, base, and amount.", caption: none)]),
  [Restate as a proportion.], [#figure(figph[\$1.56 out of what number is the same as 6.5 out of 100?], alt: "$1.56 out of what number is the same as 6.5 out of 100?", caption: none)],
  [Set up the proportion. Let#math.equation(block: false, alt: "n equals")[$n =$] number.], [#figure(figph[A mathematical equation shows a proportion: 1.56 divided by n is equal to 6.5 divided by 100.], alt: "A mathematical equation shows a proportion: 1.56 divided by n is equal to 6.5 divided by 100.", caption: none)],
  [Find the cross products and set them equal.], [#figure(figph[A mathematical equation is displayed, reading 100 multiplied by 1.56 equals 'n' multiplied by 6.5.], alt: "A mathematical equation is displayed, reading 100 multiplied by 1.56 equals 'n' multiplied by 6.5.", caption: none)],
  [Simplify.], [#figure(figph[An image displays the mathematical equation '156 = 6.5n' on a white background. The numbers and symbols are rendered in a clear, dark font.], alt: "An image displays the mathematical equation '156 = 6.5n' on a white background. The numbers and symbols are rendered in a clear, dark font.", caption: none)],
  [Divide both sides by 6.5 to isolate the variable.], [#figure(figph[An equation showing 156 divided by 6.5 equals 6.5n divided by 6.5, illustrating a step in solving for 'n'.], alt: "An equation showing 156 divided by 6.5 equals 6.5n divided by 6.5, illustrating a step in solving for 'n'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation displays '24 = n' in a black font against a white background.], alt: "A mathematical equation displays '24 = n' in a black font against a white background.", caption: none)],
  [Check if the answer is reasonable.], [],
  [Yes. 6.5% is a small amount and \$1.56 is much less than \$24.], [],
  [Write a complete sentence that answers the question.], [6.5% of \$24 is \$1.56.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve using proportions: #math.equation(block: false, alt: "8.5%")[$"8.5%"$] of what number is #math.equation(block: false, alt: "$3.23 ?")[$"$3.23" ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3.23 over n equals the fraction 8.5 over 100 ; n equals 38")[$frac(3.23, n) = frac(8.5, 100) ; #h(1em) n = 38$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve using proportions: #math.equation(block: false, alt: "7.25%")[$"7.25%"$] of what number is #math.equation(block: false, alt: "$4.64 ?")[$"$4.64" ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4.64 over n equals the fraction 7.25 over 100 ; n equals 64")[$frac(4.64, n) = frac(7.25, 100) ; #h(1em) n = 64$]
]
]

#examplebox("Example 15")[][
Translate and solve using proportions: What percent of #math.equation(block: false, alt: "72")[$72$] is #math.equation(block: false, alt: "9 ?")[$9 ?$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the parts of the percent proportion.], [#figure(figph[A percentage problem asks, 'What percent of 72 is 9?' It labels 'What percent' as 'percent', '72' as 'base', and '9' as 'amount', with 'of' highlighted in red to show its role.], alt: "A percentage problem asks, 'What percent of 72 is 9?' It labels 'What percent' as 'percent', '72' as 'base', and '9' as 'amount', with 'of' highlighted in red to show its role.", caption: none)]),
  [Restate as a proportion.], [#figure(figph[A mathematical problem asking to determine what number out of 100 is equivalent to 9 out of 72.], alt: "A mathematical problem asking to determine what number out of 100 is equivalent to 9 out of 72.", caption: none)],
  [Set up the proportion. Let #math.equation(block: false, alt: "n equals")[$n =$] number.], [#figure(figph[A mathematical equation shows the fraction 9/72 equal to the fraction n/100, which can be used to solve for the variable 'n' in a proportion or percentage problem.], alt: "A mathematical equation shows the fraction 9/72 equal to the fraction n/100, which can be used to solve for the variable 'n' in a proportion or percentage problem.", caption: none)],
  [Find the cross products and set them equal.], [#figure(figph[A mathematical equation is displayed, showing '72 times n equals 100 times 9' in a bold, black font on a white background.], alt: "A mathematical equation is displayed, showing '72 times n equals 100 times 9' in a bold, black font on a white background.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing '72n = 900' against a white background.], alt: "A mathematical equation is displayed, showing '72n = 900' against a white background.", caption: none)],
  [Divide both sides by 72.], [#figure(figph[A mathematical equation is displayed, showing '72n over 72 equals 900 over 72' with fractions represented by horizontal lines.], alt: "A mathematical equation is displayed, showing '72n over 72 equals 900 over 72' with fractions represented by horizontal lines.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation n = 12.5 is displayed, showing the variable 'n' assigned a numerical value of twelve and a half.], alt: "The mathematical equation n = 12.5 is displayed, showing the variable 'n' assigned a numerical value of twelve and a half.", caption: none)],
  [Check if the answer is reasonable.], [],
  [Yes. 9 is #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] of 72 and #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] is 12.5%.], [],
  [Write a complete sentence that answers the question.], [12.5% of 72 is 9.],
))
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve using proportions: What percent of #math.equation(block: false, alt: "72")[$72$] is #math.equation(block: false, alt: "27 ?")[$27 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 27 over 72 equals the fraction n over 100 ; n equals 37.5%")[$frac(27, 72) = frac(n, 100) ; #h(1em) n = 37.5%$]
]
]

#notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve using proportions: What percent of #math.equation(block: false, alt: "92")[$92$] is #math.equation(block: false, alt: "23 ?")[$23 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 23 over 92 equals the fraction n over 100 ; n equals 25%")[$frac(23, 92) = frac(n, 100) ; #h(1em) n = 25%$]
]
]


==== Key Concepts

- #emph[Proportion]
- A proportion is an equation of the form #math.equation(block: false, alt: "the fraction a over b equals the fraction c over d")[$frac(a, b) = frac(c, d)$], where #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], #math.equation(block: false, alt: "d not equal to 0")[$d ≠ 0$]\.The proportion states two ratios or rates are equal. The proportion is read “#math.equation(block: false, alt: "a")[$a$] is to #math.equation(block: false, alt: "b")[$b$], as #math.equation(block: false, alt: "c")[$c$] is to  #math.equation(block: false, alt: "d")[$d$]”.
- #emph[Cross Products of a Proportion]
- For any proportion of the form #math.equation(block: false, alt: "the fraction a over b equals the fraction c over d")[$frac(a, b) = frac(c, d)$], where #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], its cross products are equal: #math.equation(block: false, alt: "a times d equals b times c")[$a ⋅ d = b ⋅ c$].
- #emph[Percent Proportion]
- The amount is to the base as the percent is to 100. #math.equation(block: false, alt: "the fraction amount over base equals the fraction percent over 100")[$frac("amount", "base") = frac("percent", 100)$]


=== Section Exercises


==== Practice Makes Perfect

#strong[Use the Definition of Proportion]

In the following exercises, write each sentence as a proportion.

#math.equation(block: false, alt: "4")[$4$] is to #math.equation(block: false, alt: "15")[$15$] as #math.equation(block: false, alt: "36")[$36$] is to #math.equation(block: false, alt: "135.")[$135 .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 15 equals the fraction 36 over 135")[$frac(4, 15) = frac(36, 135)$]
]

#math.equation(block: false, alt: "7")[$7$] is to #math.equation(block: false, alt: "9")[$9$] as #math.equation(block: false, alt: "35")[$35$] is to #math.equation(block: false, alt: "45.")[$45 .$]

#math.equation(block: false, alt: "12")[$12$] is to #math.equation(block: false, alt: "5")[$5$] as #math.equation(block: false, alt: "96")[$96$] is to #math.equation(block: false, alt: "40.")[$40 .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 12 over 5 equals the fraction 96 over 40")[$frac(12, 5) = frac(96, 40)$]
]

#math.equation(block: false, alt: "15")[$15$] is to #math.equation(block: false, alt: "8")[$8$] as #math.equation(block: false, alt: "75")[$75$] is to #math.equation(block: false, alt: "40.")[$40 .$]

#math.equation(block: false, alt: "5")[$5$] wins in #math.equation(block: false, alt: "7")[$7$] games is the same as #math.equation(block: false, alt: "115")[$115$] wins in #math.equation(block: false, alt: "161")[$161$] games.

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 7 equals the fraction 115 over 161")[$frac(5, 7) = frac(115, 161)$]
]

#math.equation(block: false, alt: "4")[$4$] wins in #math.equation(block: false, alt: "9")[$9$] games is the same as #math.equation(block: false, alt: "36")[$36$] wins in #math.equation(block: false, alt: "81")[$81$] games.

#math.equation(block: false, alt: "8")[$8$] campers to #math.equation(block: false, alt: "1")[$1$] counselor is the same as #math.equation(block: false, alt: "48")[$48$] campers to #math.equation(block: false, alt: "6")[$6$] counselors.

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 over 1 equals the fraction 48 over 6")[$frac(8, 1) = frac(48, 6)$]
]

#math.equation(block: false, alt: "6")[$6$] campers to #math.equation(block: false, alt: "1")[$1$] counselor is the same as #math.equation(block: false, alt: "48")[$48$] campers to #math.equation(block: false, alt: "8")[$8$] counselors.

#math.equation(block: false, alt: "$9.36")[$"$9.36"$] for #math.equation(block: false, alt: "18")[$18$] ounces is the same as #math.equation(block: false, alt: "$2.60")[$"$2.60"$] for #math.equation(block: false, alt: "5")[$5$] ounces.

#solutionbox[
#math.equation(block: true, alt: "the fraction 9.36 over 18 equals the fraction 2.60 over 5")[$frac(9.36, 18) = frac(2.60, 5)$]
]

#math.equation(block: false, alt: "$3.92")[$"$3.92"$] for #math.equation(block: false, alt: "8")[$8$] ounces is the same as #math.equation(block: false, alt: "$1.47")[$"$1.47"$] for #math.equation(block: false, alt: "3")[$3$] ounces.

#math.equation(block: false, alt: "$18.04")[$"$18.04"$] for #math.equation(block: false, alt: "11")[$11$] pounds is the same as #math.equation(block: false, alt: "$4.92")[$"$4.92"$] for #math.equation(block: false, alt: "3")[$3$] pounds.

#solutionbox[
#math.equation(block: true, alt: "the fraction 18.04 over 11 equals the fraction 4.92 over 3")[$frac(18.04, 11) = frac(4.92, 3)$]
]

#math.equation(block: false, alt: "$12.42")[$"$12.42"$] for #math.equation(block: false, alt: "27")[$27$] pounds is the same as #math.equation(block: false, alt: "$5.52")[$"$5.52"$] for #math.equation(block: false, alt: "12")[$12$] pounds.

In the following exercises, determine whether each equation is a proportion.

#math.equation(block: true, alt: "the fraction 7 over 15 equals the fraction 56 over 120")[$frac(7, 15) = frac(56, 120)$]

#solutionbox[
yes
]

#math.equation(block: true, alt: "the fraction 5 over 12 equals the fraction 45 over 108")[$frac(5, 12) = frac(45, 108)$]

#math.equation(block: true, alt: "the fraction 11 over 6 equals the fraction 21 over 16")[$frac(11, 6) = frac(21, 16)$]

#solutionbox[
no
]

#math.equation(block: true, alt: "the fraction 9 over 4 equals the fraction 39 over 34")[$frac(9, 4) = frac(39, 34)$]

#math.equation(block: true, alt: "the fraction 12 over 18 equals the fraction 4.99 over 7.56")[$frac(12, 18) = frac(4.99, 7.56)$]

#solutionbox[
no
]

#math.equation(block: true, alt: "the fraction 9 over 16 equals the fraction 2.16 over 3.89")[$frac(9, 16) = frac(2.16, 3.89)$]

#math.equation(block: true, alt: "the fraction 13.5 over 8.5 equals the fraction 31.05 over 19.55")[$frac(13.5, 8.5) = frac(31.05, 19.55)$]

#solutionbox[
yes
]

#math.equation(block: true, alt: "the fraction 10.1 over 8.4 equals the fraction 3.03 over 2.52")[$frac(10.1, 8.4) = frac(3.03, 2.52)$]

#strong[Solve Proportions]

In the following exercises, solve each proportion.

#math.equation(block: true, alt: "the fraction x over 56 equals the fraction 7 over 8")[$frac(x, 56) = frac(7, 8)$]

#solutionbox[
#emph[x] = 49
]

#math.equation(block: true, alt: "the fraction n over 91 equals the fraction 8 over 13")[$frac(n, 91) = frac(8, 13)$]

#math.equation(block: true, alt: "the fraction 49 over 63 equals the fraction z over 9")[$frac(49, 63) = frac(z, 9)$]

#solutionbox[
#emph[z] = 7
]

#math.equation(block: true, alt: "the fraction 56 over 72 equals the fraction y over 9")[$frac(56, 72) = frac(y, 9)$]

#math.equation(block: true, alt: "the fraction 5 over a equals the fraction 65 over 117")[$frac(5, a) = frac(65, 117)$]

#solutionbox[
#emph[a] = 9
]

#math.equation(block: true, alt: "the fraction 4 over b equals the fraction 64 over 144")[$frac(4, b) = frac(64, 144)$]

#math.equation(block: true, alt: "the fraction 98 over 154 equals the fraction −7 over p")[$frac(98, 154) = frac(−7, p)$]

#solutionbox[
#emph[p] = −11
]

#math.equation(block: true, alt: "the fraction 72 over 156 equals the fraction −6 over q")[$frac(72, 156) = frac(−6, q)$]

#math.equation(block: true, alt: "the fraction a over −8 equals the fraction −42 over 48")[$frac(a, −8) = frac(−42, 48)$]

#solutionbox[
#emph[a] = 7
]

#math.equation(block: true, alt: "the fraction b over −7 equals the fraction −30 over 42")[$frac(b, −7) = frac(−30, 42)$]

#math.equation(block: true, alt: "the fraction 2.6 over 3.9 equals the fraction c over 3")[$frac(2.6, 3.9) = frac(c, 3)$]

#solutionbox[
#emph[c] = 2
]

#math.equation(block: true, alt: "the fraction 2.7 over 3.6 equals the fraction d over 4")[$frac(2.7, 3.6) = frac(d, 4)$]

#math.equation(block: true, alt: "the fraction 2.7 over j equals the fraction 0.9 over 0.2")[$frac(2.7, j) = frac(0.9, 0.2)$]

#solutionbox[
#emph[j] = 0.6
]

#math.equation(block: true, alt: "the fraction 2.8 over k equals the fraction 2.1 over 1.5")[$frac(2.8, k) = frac(2.1, 1.5)$]

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

#solutionbox[
#emph[m] = 4
]

#math.equation(block: true, alt: "the fraction the fraction 1 over 3 over 3 equals the fraction 9 over n")[$frac(frac(1, 3), 3) = frac(9, n)$]

#strong[Solve Applications Using Proportions]

In the following exercises, solve the proportion problem.

Pediatricians prescribe #math.equation(block: false, alt: "5")[$5$] milliliters (ml) of acetaminophen for every #math.equation(block: false, alt: "25")[$25$] pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs #math.equation(block: false, alt: "45")[$45$] pounds?

#solutionbox[
9 ml
]

Brianna, who weighs #math.equation(block: false, alt: "6")[$6$] kg, just received her shots and needs a pain killer. The pain killer is prescribed for children at #math.equation(block: false, alt: "15")[$15$] milligrams (mg) for every #math.equation(block: false, alt: "1")[$1$] kilogram (kg) of the child’s weight. How many milligrams will the doctor prescribe?

At the gym, Carol takes her pulse for #math.equation(block: false, alt: "10")[$10$] sec and counts #math.equation(block: false, alt: "19")[$19$] beats. How many beats per minute is this? Has Carol met her target heart rate of #math.equation(block: false, alt: "140")[$140$] beats per minute?

#solutionbox[
114 beats/minute. Carol has not met her target heart rate.
]

Kevin wants to keep his heart rate at #math.equation(block: false, alt: "160")[$160$] beats per minute while training. During his workout he counts #math.equation(block: false, alt: "27")[$27$] beats in #math.equation(block: false, alt: "10")[$10$] seconds. How many beats per minute is this? Has Kevin met his target heart rate?

A new energy drink advertises #math.equation(block: false, alt: "106")[$106$] calories for #math.equation(block: false, alt: "8")[$8$] ounces. How many calories are in #math.equation(block: false, alt: "12")[$12$] ounces of the drink?

#solutionbox[
159 cal
]

One #math.equation(block: false, alt: "12")[$12$] ounce can of soda has #math.equation(block: false, alt: "150")[$150$] calories. If Josiah drinks the big #math.equation(block: false, alt: "32")[$32$] ounce size from the local mini-mart, how many calories does he get?

Karen eats #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] cup of oatmeal that counts for #math.equation(block: false, alt: "2")[$2$] points on her weight loss program. Her husband, Joe, can have #math.equation(block: false, alt: "3")[$3$] points of oatmeal for breakfast. How much oatmeal can he have?

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 4 cup")[$frac(3, 4) #h(0.3em) "cup"$]
]

An oatmeal cookie recipe calls for #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] cup of butter to make #math.equation(block: false, alt: "4")[$4$] dozen cookies. Hilda needs to make #math.equation(block: false, alt: "10")[$10$] dozen cookies for the bake sale. How many cups of butter will she need?

Janice is traveling to Canada and will change #math.equation(block: false, alt: "$250")[$"$250"$] US dollars into Canadian dollars. At the current exchange rate, #math.equation(block: false, alt: "$1")[$"$1"$] US is equal to #math.equation(block: false, alt: "$1.01")[$"$1.01"$] Canadian. How many Canadian dollars will she get for her trip?

#solutionbox[
\$252.50
]

Todd is traveling to Mexico and needs to exchange #math.equation(block: false, alt: "$450")[$"$450"$] into Mexican pesos. If each dollar is worth #math.equation(block: false, alt: "12.29")[$12.29$] pesos, how many pesos will he get for his trip?

Steve changed #math.equation(block: false, alt: "$600")[$"$600"$] into #math.equation(block: false, alt: "480")[$480$] Euros. How many Euros did he receive per US dollar?

#solutionbox[
0.8 Euros
]

Martha changed #math.equation(block: false, alt: "$350")[$"$350"$] US into #math.equation(block: false, alt: "385")[$385$] Australian dollars. How many Australian dollars did she receive per US dollar?

At the laundromat, Lucy changed #math.equation(block: false, alt: "$12.00")[$"$12.00"$] into quarters. How many quarters did she get?

#solutionbox[
48 quarters
]

When she arrived at a casino, Gerty changed #math.equation(block: false, alt: "$20")[$"$20"$] into nickels. How many nickels did she get?

Jesse’s car gets #math.equation(block: false, alt: "30")[$30$] miles per gallon of gas. If Las Vegas is #math.equation(block: false, alt: "285")[$285$] miles away, how many gallons of gas are needed to get there and then home? If gas is #math.equation(block: false, alt: "$3.09")[$"$3.09"$] per gallon, what is the total cost of the gas for the trip?

#solutionbox[
19 gallons, \$58.71
]

Danny wants to drive to Phoenix to see his grandfather. Phoenix is #math.equation(block: false, alt: "370")[$370$] miles from Danny’s home and his car gets #math.equation(block: false, alt: "18.5")[$18.5$] miles per gallon. How many gallons of gas will Danny need to get to and from Phoenix? If gas is #math.equation(block: false, alt: "$3.19")[$"$3.19"$] per gallon, what is the total cost for the gas to drive to see his grandfather?

Hugh leaves early one morning to drive from his home in Chicago to go to Mount Rushmore, #math.equation(block: false, alt: "812")[$812$] miles away. After #math.equation(block: false, alt: "3")[$3$] hours, he has gone #math.equation(block: false, alt: "190")[$190$] miles. At that rate, how long will the whole drive take?

#solutionbox[
12.8 hours
]

Kelly leaves her home in Seattle to drive to Spokane, a distance of #math.equation(block: false, alt: "280")[$280$] miles. After #math.equation(block: false, alt: "2")[$2$] hours, she has gone #math.equation(block: false, alt: "152")[$152$] miles. At that rate, how long will the whole drive take?

Phil wants to fertilize his lawn. Each bag of fertilizer covers about #math.equation(block: false, alt: "4,000")[$4,000$] square feet of lawn. Phil’s lawn is approximately #math.equation(block: false, alt: "13,500")[$13,500$] square feet. How many bags of fertilizer will he have to buy?

#solutionbox[
4 bags
]

April wants to paint the exterior of her house. One gallon of paint covers about #math.equation(block: false, alt: "350")[$350$] square feet, and the exterior of the house measures approximately #math.equation(block: false, alt: "2000")[$2000$] square feet. How many gallons of paint will she have to buy?

#strong[Write Percent Equations as Proportions]

In the following exercises, translate to a proportion.

What number is #math.equation(block: false, alt: "35%")[$"35%"$] of #math.equation(block: false, alt: "250 ?")[$250 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 250 equals the fraction 35 over 100")[$frac(n, 250) = frac(35, 100)$]
]

What number is #math.equation(block: false, alt: "75%")[$"75%"$] of #math.equation(block: false, alt: "920 ?")[$920 ?$]

What number is #math.equation(block: false, alt: "110%")[$"110%"$] of #math.equation(block: false, alt: "47 ?")[$47 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 47 equals the fraction 110 over 100")[$frac(n, 47) = frac(110, 100)$]
]

What number is #math.equation(block: false, alt: "150%")[$"150%"$] of #math.equation(block: false, alt: "64 ?")[$64 ?$]

#math.equation(block: false, alt: "45")[$45$] is #math.equation(block: false, alt: "30%")[$"30%"$] of what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction 45 over n equals the fraction 30 over 100")[$frac(45, n) = frac(30, 100)$]
]

#math.equation(block: false, alt: "25")[$25$] is #math.equation(block: false, alt: "80%")[$"80%"$] of what number?

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 90 over n equals the fraction 150 over 100")[$frac(90, n) = frac(150, 100)$]
]

#math.equation(block: false, alt: "77")[$77$] is #math.equation(block: false, alt: "110%")[$"110%"$] of what number?

What percent of #math.equation(block: false, alt: "85")[$85$] is #math.equation(block: false, alt: "17 ?")[$17 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 17 over 85 equals the fraction p over 100")[$frac(17, 85) = frac(p, 100)$]
]

What percent of #math.equation(block: false, alt: "92")[$92$] is #math.equation(block: false, alt: "46 ?")[$46 ?$]

What percent of #math.equation(block: false, alt: "260")[$260$] is #math.equation(block: false, alt: "340 ?")[$340 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 340 over 260 equals the fraction p over 100")[$frac(340, 260) = frac(p, 100)$]
]

What percent of #math.equation(block: false, alt: "180")[$180$] is #math.equation(block: false, alt: "220 ?")[$220 ?$]

#strong[Translate and Solve Percent Proportions]

In the following exercises, translate and solve using proportions.

What number is #math.equation(block: false, alt: "65%")[$"65%"$] of #math.equation(block: false, alt: "180 ?")[$180 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 180 equals the fraction 65 over 100")[$frac(n, 180) = frac(65, 100)$]; #math.equation(block: true, alt: "117")[$117$]
]

What number is #math.equation(block: false, alt: "55%")[$"55%"$] of #math.equation(block: false, alt: "300 ?")[$300 ?$]

#math.equation(block: false, alt: "18%")[$"18%"$] of #math.equation(block: false, alt: "92")[$92$] is what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 92 equals the fraction 18 over 100")[$frac(n, 92) = frac(18, 100)$]; #math.equation(block: true, alt: "16.56")[$16.56$]
]

#math.equation(block: false, alt: "22%")[$"22%"$] of #math.equation(block: false, alt: "74")[$74$] is what number?

#math.equation(block: false, alt: "175%")[$"175%"$] of #math.equation(block: false, alt: "26")[$26$] is what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 26 equals the fraction 175 over 100")[$frac(n, 26) = frac(175, 100)$]; #math.equation(block: true, alt: "45.5")[$45.5$]
]

#math.equation(block: false, alt: "250%")[$"250%"$] of #math.equation(block: false, alt: "61")[$61$] is what number?

What is #math.equation(block: false, alt: "300%")[$"300%"$] of #math.equation(block: false, alt: "488 ?")[$488 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 488 equals the fraction 300 over 100")[$frac(n, 488) = frac(300, 100)$]; #math.equation(block: true, alt: "1464")[$1464$]
]

What is #math.equation(block: false, alt: "500%")[$"500%"$] of #math.equation(block: false, alt: "315 ?")[$315 ?$]

#math.equation(block: false, alt: "17%")[$"17%"$] of what number is #math.equation(block: false, alt: "$7.65 ?")[$"$7.65" ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7.65 over n equals the fraction 17 over 100")[$frac(7.65, n) = frac(17, 100)$]; #math.equation(block: true, alt: "45")[$45$]
]

#math.equation(block: false, alt: "19%")[$"19%"$] of what number is #math.equation(block: false, alt: "$6.46 ?")[$"$6.46" ?$]

#math.equation(block: false, alt: "$13.53")[$"$13.53"$] is #math.equation(block: false, alt: "8.25%")[$"8.25%"$] of what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction 13.53 over n equals the fraction 8.25 over 100")[$frac(13.53, n) = frac(8.25, 100)$]; #math.equation(block: true, alt: "164")[$164$]
]

#math.equation(block: false, alt: "$18.12")[$"$18.12"$] is #math.equation(block: false, alt: "7.55%")[$"7.55%"$] of what number?

What percent of #math.equation(block: false, alt: "56")[$56$] is #math.equation(block: false, alt: "14 ?")[$14 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 14 over 56 equals the fraction p over 100")[$frac(14, 56) = frac(p, 100)$]; #math.equation(block: true, alt: "25 %")[$25 %$]
]

What percent of #math.equation(block: false, alt: "80")[$80$] is #math.equation(block: false, alt: "28 ?")[$28 ?$]

What percent of #math.equation(block: false, alt: "96")[$96$] is #math.equation(block: false, alt: "12 ?")[$12 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 12 over 96 equals the fraction p over 100")[$frac(12, 96) = frac(p, 100)$]; #math.equation(block: true, alt: "12.5 %")[$12.5 %$]
]

What percent of #math.equation(block: false, alt: "120")[$120$] is #math.equation(block: false, alt: "27 ?")[$27 ?$]


==== Everyday Math

#strong[Mixing a concentrate] Sam bought a large bottle of concentrated cleaning solution at the warehouse store. He must mix the concentrate with water to make a solution for washing his windows. The directions tell him to mix #math.equation(block: false, alt: "3")[$3$] ounces of concentrate with #math.equation(block: false, alt: "5")[$5$] ounces of water. If he puts #math.equation(block: false, alt: "12")[$12$] ounces of concentrate in a bucket, how many ounces of water should he add? How many ounces of the solution will he have altogether?

#solutionbox[
He must add 20 oz of water to obtain a final solution of 32 oz.
]

#strong[Mixing a concentrate] Travis is going to wash his car. The directions on the bottle of car wash concentrate say to mix #math.equation(block: false, alt: "2")[$2$] ounces of concentrate with #math.equation(block: false, alt: "15")[$15$] ounces of water. If Travis puts #math.equation(block: false, alt: "6")[$6$] ounces of concentrate in a bucket, how much water must he mix with the concentrate?


==== Writing Exercises

To solve “what number is #math.equation(block: false, alt: "45%")[$"45%"$] of #math.equation(block: false, alt: "350 ”")[$350 "”"$] do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

#solutionbox[
Answers will vary.
]

To solve “what percent of #math.equation(block: false, alt: "125")[$125$] is #math.equation(block: false, alt: "25 ”")[$25 "”"$] do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.


==== Self Check

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

#figure(figph[A self-assessment checklist for students to rate their understanding of proportions and percentages, with options: Confidently, With some help, or No-I don't get it!], alt: "A self-assessment checklist for students to rate their understanding of proportions and percentages, with options: Confidently, With some help, or No-I don't get it!", caption: none)

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


=== Chapter Review Exercises


==== Understand Percent

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

#math.equation(block: false, alt: "32%")[$"32%"$] admission rate for the university

#solutionbox[
#math.equation(block: true, alt: "the fraction 32 over 100")[$frac(32, 100)$]
]

#math.equation(block: false, alt: "53.3%")[$"53.3%"$] rate of college students with student loans

In the following exercises, write as a ratio and then as a percent.

#math.equation(block: false, alt: "13")[$13$] out of #math.equation(block: false, alt: "100")[$100$] architects are women.

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

#math.equation(block: false, alt: "9")[$9$] out of every #math.equation(block: false, alt: "100")[$100$] nurses are men.

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

#math.equation(block: true, alt: "48%")[$"48%"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 12 over 25")[$frac(12, 25)$]
]

#math.equation(block: true, alt: "175%")[$"175%"$]

#math.equation(block: true, alt: "64.1%")[$"64.1%"$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 641 over 1000")[$frac(641, 1000)$]
]

#math.equation(block: true, alt: "8 the fraction 1 over 4 %")[$8 frac(1, 4) %$]

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

#math.equation(block: true, alt: "6%")[$"6%"$]

#solutionbox[
0.06
]

#math.equation(block: true, alt: "23%")[$"23%"$]

#math.equation(block: true, alt: "128%")[$"128%"$]

#solutionbox[
1.28
]

4.9%

In the following exercises, convert each percent to ⓐ a simplified fraction and ⓑ a decimal.

In #math.equation(block: false, alt: "2012 , 13.5%")[$2012 , "13.5%"$] of the United States population was age #math.equation(block: false, alt: "65")[$65$] or over. (Source: www.census.gov)

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 27 over 200")[$#h(0.2em) frac(27, 200)$]
+ ⓑ #math.equation(block: false, alt: "0.135")[$#h(0.2em) 0.135$]
]

In #math.equation(block: false, alt: "2012 , 6.5%")[$2012 , "6.5%"$] of the United States population was under #math.equation(block: false, alt: "5")[$5$] years old. (Source: www.census.gov)

When a die is tossed, the probability it will land with an even number of dots on the top side is #math.equation(block: false, alt: "50% .")[$"50%" .$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 1 over 2")[$#h(0.2em) frac(1, 2)$]
+ ⓑ #math.equation(block: false, alt: "0.5")[$#h(0.2em) 0.5$]
]

A couple plans to have three children. The probability they will all be girls is #math.equation(block: false, alt: "12.5% .")[$"12.5%" .$]

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

#math.equation(block: true, alt: "0.04")[$0.04$]

#solutionbox[
4%
]

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

#math.equation(block: true, alt: "2.82")[$2.82$]

#solutionbox[
282%
]

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

#math.equation(block: true, alt: "0.003")[$0.003$]

#solutionbox[
0.3%
]

#math.equation(block: true, alt: "1.395")[$1.395$]

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

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

#solutionbox[
75%
]

#math.equation(block: true, alt: "the fraction 11 over 5")[$frac(11, 5)$]

#math.equation(block: true, alt: "3 the fraction 5 over 8")[$3 frac(5, 8)$]

#solutionbox[
362.5%
]

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

According to the Centers for Disease Control, #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$] of adults do not take a vitamin or supplement.

#solutionbox[
40%
]

According to the Centers for Disease Control, among adults who do take a vitamin or supplement, #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] take a multivitamin.

In the following exercises, translate and solve.

What number is #math.equation(block: false, alt: "46%")[$"46%"$] of #math.equation(block: false, alt: "350 ?")[$350 ?$]

#solutionbox[
161
]

#math.equation(block: false, alt: "120%")[$"120%"$] of #math.equation(block: false, alt: "55")[$55$] is what number?

#math.equation(block: false, alt: "84")[$84$] is #math.equation(block: false, alt: "35%")[$"35%"$] of what number?

#solutionbox[
240
]

#math.equation(block: false, alt: "15")[$15$] is #math.equation(block: false, alt: "8%")[$"8%"$] of what number?

#math.equation(block: false, alt: "200%")[$"200%"$] of what number is #math.equation(block: false, alt: "50 ?")[$50 ?$]

#solutionbox[
25
]

#math.equation(block: false, alt: "7.9%")[$"7.9%"$] of what number is #math.equation(block: false, alt: "$4.74 ?")[$"$4.74" ?$]

What percent of #math.equation(block: false, alt: "120")[$120$] is #math.equation(block: false, alt: "81.6 ?")[$81.6 ?$]

#solutionbox[
68%
]

What percent of #math.equation(block: false, alt: "340")[$340$] is #math.equation(block: false, alt: "595 ?")[$595 ?$]


==== Solve General Applications of Percents

In the following exercises, solve.

When Aurelio and his family ate dinner at a restaurant, the bill was #math.equation(block: false, alt: "$83.50.")[$"$83.50" .$] Aurelio wants to leave #math.equation(block: false, alt: "20%")[$"20%"$] of the total bill as a tip. How much should the tip be?

#solutionbox[
\$16.70
]

One granola bar has #math.equation(block: false, alt: "2")[$2$] grams of fiber, which is #math.equation(block: false, alt: "8%")[$"8%"$] of the recommended daily amount. What is the total recommended daily amount of fiber?

The nutrition label on a package of granola bars says that each granola bar has #math.equation(block: false, alt: "190")[$190$] calories, and #math.equation(block: false, alt: "54")[$54$] calories are from fat. What percent of the total calories is from fat?

#solutionbox[
28.4%
]

Elsa gets paid #math.equation(block: false, alt: "$4,600")[$"$4,600"$] per month. Her car payment is #math.equation(block: false, alt: "$253.")[$"$253" .$] What percent of her monthly pay goes to her car payment?

In the following exercises, solve.

Jorge got a raise in his hourly pay, from #math.equation(block: false, alt: "$19.00")[$"$19.00"$] to #math.equation(block: false, alt: "$19.76.")[$"$19.76" .$] Find the percent increase.

#solutionbox[
4%
]

Last year Bernard bought a new car for #math.equation(block: false, alt: "$30,000.")[$"$30,000" .$] This year the car is worth #math.equation(block: false, alt: "$24,000.")[$"$24,000" .$] Find the percent decrease.


==== Solve Sales Tax, Commission, and Discount Applications

In the following exercises, find ⓐ the sales tax ⓑ the total cost.

The cost of a lawn mower was #math.equation(block: false, alt: "$750.")[$"$750" .$] The sales tax rate is #math.equation(block: false, alt: "6%")[$"6%"$] of the purchase price.

#solutionbox[
+ ⓐ \$45
+ ⓑ \$795
]

The cost of a water heater is #math.equation(block: false, alt: "$577.")[$"$577" .$] The sales tax rate is #math.equation(block: false, alt: "8.75%")[$"8.75%"$] of the purchase price.

In the following exercises, find the sales tax rate.

Andy bought a piano for #math.equation(block: false, alt: "$4,600.")[$"$4,600" .$] The sales tax on the purchase was #math.equation(block: false, alt: "$333.50.")[$"$333.50" .$]

#solutionbox[
7.25%
]

Nahomi bought a purse for #math.equation(block: false, alt: "$200.")[$"$200" .$] The sales tax on the purchase was #math.equation(block: false, alt: "$16.75.")[$"$16.75" .$]

In the following exercises, find the commission.

Ginny is a realtor. She receives #math.equation(block: false, alt: "3%")[$"3%"$] commission when she sells a house. How much commission will she receive for selling a house for #math.equation(block: false, alt: "$380,000 ?")[$"$380,000" ?$]

#solutionbox[
\$11,400
]

Jackson receives #math.equation(block: false, alt: "16.5%")[$"16.5%"$] commission when he sells a dinette set. How much commission will he receive for selling a dinette set for #math.equation(block: false, alt: "$895 ?")[$"$895" ?$]

In the following exercises, find the rate of commission.

Ruben received #math.equation(block: false, alt: "$675")[$"$675"$] commission when he sold a #math.equation(block: false, alt: "$4,500")[$"$4,500"$] painting at the art gallery where he works. What was the rate of commission?

#solutionbox[
15%
]

Tori received #math.equation(block: false, alt: "$80.75")[$"$80.75"$] for selling a #math.equation(block: false, alt: "$950")[$"$950"$] membership at her gym. What was her rate of commission?

In the following exercises, find the sale price.

Aya bought a pair of shoes that was on sale for #math.equation(block: false, alt: "$30")[$"$30"$] off. The original price of the shoes was #math.equation(block: false, alt: "$75.")[$"$75" .$]

#solutionbox[
\$45
]

Takwanna saw a cookware set she liked on sale for #math.equation(block: false, alt: "$145")[$"$145"$] off. The original price of the cookware was #math.equation(block: false, alt: "$312.")[$"$312" .$]

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

Nga bought a microwave for her office. The microwave was discounted #math.equation(block: false, alt: "30%")[$"30%"$] from an original price of #math.equation(block: false, alt: "$84.90.")[$"$84.90" .$]

#solutionbox[
+ ⓐ \$25.47
+ ⓑ \$59.43
]

Jarrett bought a tie that was discounted #math.equation(block: false, alt: "65%")[$"65%"$] from an original price of #math.equation(block: false, alt: "$45.")[$"$45" .$]

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

Hilda bought a bedspread on sale for #math.equation(block: false, alt: "$37.")[$"$37" .$] The original price of the bedspread was #math.equation(block: false, alt: "$50.")[$"$50" .$]

#solutionbox[
+ ⓐ \$13
+ ⓑ 26%
]

Tyler bought a phone on sale for #math.equation(block: false, alt: "$49.99.")[$"$49.99" .$] The original price of the phone was #math.equation(block: false, alt: "$79.99.")[$"$79.99" .$]

In the following exercises, find

+ ⓐ the amount of the mark-up
+ ⓑ the list price

Manny paid #math.equation(block: false, alt: "$0.80")[$"$0.80"$] a pound for apples. He added #math.equation(block: false, alt: "60%")[$"60%"$] mark-up before selling them at his produce stand. What price did he charge for the apples?

#solutionbox[
+ ⓐ \$0.48
+ ⓑ \$1.28
]

It cost Noelle #math.equation(block: false, alt: "$17.40")[$"$17.40"$] for the materials she used to make a purse. She added a #math.equation(block: false, alt: "325%")[$"325%"$] mark-up before selling it at her friend’s store. What price did she ask for the purse?


==== Solve Simple Interest Applications

In the following exercises, solve the simple interest problem.

Find the simple interest earned after #math.equation(block: false, alt: "4")[$4$] years on #math.equation(block: false, alt: "$2,250")[$"$2,250"$] invested at an interest rate of #math.equation(block: false, alt: "5% .")[$"5%" .$]

#solutionbox[
\$450
]

Find the simple interest earned after #math.equation(block: false, alt: "7")[$7$] years on #math.equation(block: false, alt: "$12,000")[$"$12,000"$] invested at an interest rate of #math.equation(block: false, alt: "8.5% .")[$"8.5%" .$]

Find the principal invested if #math.equation(block: false, alt: "$660")[$"$660"$] interest was earned in #math.equation(block: false, alt: "5")[$5$] years at an interest rate of #math.equation(block: false, alt: "3% .")[$"3%" .$]

#solutionbox[
\$4400
]

Find the interest rate if #math.equation(block: false, alt: "$2,898")[$"$2,898"$] interest was earned from a principal of #math.equation(block: false, alt: "$23,000")[$"$23,000"$] invested for #math.equation(block: false, alt: "3")[$3$] years.

Kazuo deposited #math.equation(block: false, alt: "$10,000")[$"$10,000"$] in a bank account with interest rate #math.equation(block: false, alt: "4.5% .")[$"4.5%" .$] How much interest was earned in #math.equation(block: false, alt: "2")[$2$] years?

#solutionbox[
\$900
]

Brent invested #math.equation(block: false, alt: "$23,000")[$"$23,000"$] in a friend’s business. In #math.equation(block: false, alt: "5")[$5$] years the friend paid him the #math.equation(block: false, alt: "$23,000")[$"$23,000"$] plus #math.equation(block: false, alt: "$9,200")[$"$9,200"$] interest. What was the rate of interest?

Fresia lent her son #math.equation(block: false, alt: "$5,000")[$"$5,000"$] for college expenses. Three years later he repaid her the #math.equation(block: false, alt: "$5,000")[$"$5,000"$] plus #math.equation(block: false, alt: "$375")[$"$375"$] interest. What was the rate of interest?

#solutionbox[
2.5%
]

In #math.equation(block: false, alt: "6")[$6$] years, a bond that paid #math.equation(block: false, alt: "5.5%")[$"5.5%"$] earned #math.equation(block: false, alt: "$594")[$"$594"$] interest. What was the principal of the bond?


==== Solve Proportions and their Applications

In the following exercises, write each sentence as a proportion.

#math.equation(block: false, alt: "3")[$3$] is to #math.equation(block: false, alt: "8")[$8$] as #math.equation(block: false, alt: "12")[$12$] is to #math.equation(block: false, alt: "32.")[$32 .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 8 equals the fraction 12 over 32")[$frac(3, 8) = frac(12, 32)$]
]

#math.equation(block: false, alt: "95")[$95$] miles to #math.equation(block: false, alt: "3")[$3$] gallons is the same as #math.equation(block: false, alt: "475")[$475$] miles to #math.equation(block: false, alt: "15")[$15$] gallons.

#math.equation(block: false, alt: "1")[$1$] teacher to #math.equation(block: false, alt: "18")[$18$] students is the same as #math.equation(block: false, alt: "23")[$23$] teachers to #math.equation(block: false, alt: "414")[$414$] students.

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 18 equals the fraction 23 over 414")[$frac(1, 18) = frac(23, 414)$]
]

#math.equation(block: false, alt: "$7.35")[$"$7.35"$] for #math.equation(block: false, alt: "15")[$15$] ounces is the same as #math.equation(block: false, alt: "$2.94")[$"$2.94"$] for #math.equation(block: false, alt: "6")[$6$] ounces.

In the following exercises, determine whether each equation is a proportion.

#math.equation(block: true, alt: "the fraction 5 over 13 equals the fraction 30 over 78")[$frac(5, 13) = frac(30, 78)$]

#solutionbox[
yes
]

#math.equation(block: true, alt: "the fraction 16 over 7 equals the fraction 48 over 23")[$frac(16, 7) = frac(48, 23)$]

#math.equation(block: true, alt: "the fraction 12 over 18 equals the fraction 6.99 over 10.99")[$frac(12, 18) = frac(6.99, 10.99)$]

#solutionbox[
no
]

#math.equation(block: true, alt: "the fraction 11.6 over 9.2 equals the fraction 37.12 over 29.44")[$frac(11.6, 9.2) = frac(37.12, 29.44)$]

In the following exercises, solve each proportion.

#math.equation(block: true, alt: "the fraction x over 36 equals the fraction 5 over 9")[$frac(x, 36) = frac(5, 9)$]

#solutionbox[
20
]

#math.equation(block: true, alt: "the fraction 7 over a equals the fraction −6 over 84")[$frac(7, a) = frac(−6, 84)$]

#math.equation(block: true, alt: "the fraction 1.2 over 1.8 equals the fraction d over 6")[$frac(1.2, 1.8) = frac(d, 6)$]

#solutionbox[
4
]

#math.equation(block: true, alt: "the fraction the fraction 1 over 2 over 2 equals the fraction m over 20")[$frac(frac(1, 2), 2) = frac(m, 20)$]

In the following exercises, solve the proportion problem.

The children’s dosage of acetaminophen is #math.equation(block: false, alt: "5")[$5$] milliliters (ml) for every #math.equation(block: false, alt: "25")[$25$] pounds of a child’s weight. How many milliliters of acetaminophen will be prescribed for a #math.equation(block: false, alt: "60")[$60$] pound child?

#solutionbox[
12 ml
]

After a workout, Dennis takes his pulse for #math.equation(block: false, alt: "10")[$10$] sec and counts #math.equation(block: false, alt: "21")[$21$] beats. How many beats per minute is this?

An #math.equation(block: false, alt: "8")[$8$] ounce serving of ice cream has #math.equation(block: false, alt: "272")[$272$] calories. If Lavonne eats #math.equation(block: false, alt: "10")[$10$] ounces of ice cream, how many calories does she get?

#solutionbox[
340 calories
]

Alma is going to Europe and wants to exchange #math.equation(block: false, alt: "$1,200")[$"$1,200"$] into Euros. If each dollar is #math.equation(block: false, alt: "0.75")[$0.75$] Euros, how many Euros will Alma get?

Zack wants to drive from Omaha to Denver, a distance of #math.equation(block: false, alt: "494")[$494$] miles. If his car gets #math.equation(block: false, alt: "38")[$38$] miles to the gallon, how many gallons of gas will Zack need to get to Denver?

#solutionbox[
13 gallons
]

Teresa is planning a party for #math.equation(block: false, alt: "100")[$100$] people. Each gallon of punch will serve #math.equation(block: false, alt: "18")[$18$] people. How many gallons of punch will she need?

In the following exercises, translate to a proportion.

What number is #math.equation(block: false, alt: "62%")[$"62%"$] of #math.equation(block: false, alt: "395 ?")[$395 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 395 equals the fraction 62 over 100")[$frac(n, 395) = frac(62, 100)$]
]

#math.equation(block: false, alt: "42")[$42$] is #math.equation(block: false, alt: "70%")[$"70%"$] of what number?

What percent of #math.equation(block: false, alt: "1,000")[$1,000$] is #math.equation(block: false, alt: "15 ?")[$15 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 15 over 1000 equals the fraction p over 100")[$frac(15, 1000) = frac(p, 100)$]
]

What percent of #math.equation(block: false, alt: "140")[$140$] is #math.equation(block: false, alt: "210 ?")[$210 ?$]

In the following exercises, translate and solve using proportions.

What number is #math.equation(block: false, alt: "85%")[$"85%"$] of #math.equation(block: false, alt: "900 ?")[$900 ?$]

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 900 equals the fraction 85 over 100 , 765")[$frac(n, 900) = frac(85, 100) , 765$]
]

#math.equation(block: false, alt: "6%")[$"6%"$] of what number is #math.equation(block: false, alt: "$24 ?")[$"$24" ?$]

#math.equation(block: false, alt: "$3.51")[$"$3.51"$] is #math.equation(block: false, alt: "4.5%")[$"4.5%"$] of what number?

#solutionbox[
#math.equation(block: true, alt: "the fraction 3.51 over n equals the fraction 4.5 over 100 , $78")[$frac(3.51, n) = frac(4.5, 100) , \$78$]
]

What percent of #math.equation(block: false, alt: "3,100")[$3,100$] is #math.equation(block: false, alt: "930 ?")[$930 ?$]

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

#math.equation(block: true, alt: "24%")[$"24%"$]

#solutionbox[
#math.equation(block: true, alt: "0.24 , the fraction 6 over 25")[$0.24 , frac(6, 25)$]
]

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

#math.equation(block: true, alt: "350%")[$"350%"$]

#solutionbox[
#math.equation(block: true, alt: "3.5 , the fraction 7 over 2")[$3.5 , frac(7, 2)$]
]

In the following exercises, convert each fraction to a percent. (Round to #math.equation(block: false, alt: "3")[$3$] decimal places if needed.)

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

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

#solutionbox[
#math.equation(block: true, alt: "33.3 bar % o r 33 the fraction 1 over 3 %")[$33. overline(3) "%" #h(0.2em) o r #h(0.2em) 33 frac(1, 3) "%"$]
]

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

In the following exercises, solve the percent problem.

#math.equation(block: false, alt: "65")[$65$] is what percent of #math.equation(block: false, alt: "260 ?")[$260 ?$]

#solutionbox[
25%
]

What number is #math.equation(block: false, alt: "27%")[$"27%"$] of #math.equation(block: false, alt: "3,000 ?")[$3,000 ?$]

#math.equation(block: false, alt: "150%")[$"150%"$] of what number is #math.equation(block: false, alt: "60 ?")[$60 ?$]

#solutionbox[
40
]

Yuki’s monthly paycheck is #math.equation(block: false, alt: "$3,825.")[$"$3,825" .$] She pays #math.equation(block: false, alt: "$918")[$"$918"$] for rent. What percent of her paycheck goes to rent?

The total number of vehicles on one freeway dropped from #math.equation(block: false, alt: "84,000")[$"84,000"$] to #math.equation(block: false, alt: "74,000.")[$"74,000" .$] Find the percent decrease (round to the nearest tenth of a percent).

#solutionbox[
11.9%
]

Kyle bought a bicycle in Denver where the sales tax was #math.equation(block: false, alt: "7.72%")[$"7.72%"$] of the purchase price. The purchase price of the bicycle was #math.equation(block: false, alt: "$600.")[$"$600" .$] What was the total cost?

Mara received #math.equation(block: false, alt: "$31.80")[$"$31.80"$] commission when she sold a #math.equation(block: false, alt: "$795")[$"$795"$] suit. What was her rate of commission?

#solutionbox[
4%
]

Kiyoshi bought a television set on sale for #math.equation(block: false, alt: "$899.")[$"$899" .$] The original price was #math.equation(block: false, alt: "$1,200.")[$"$1,200" .$] Find:
 #linebreak()

+ ⓐ the amount of discount
+ ⓑ the discount rate (round to the nearest tenth of a percent)

Oxana bought a dresser at a garage sale for #math.equation(block: false, alt: "$20.")[$"$20" .$] She refinished it, then added a #math.equation(block: false, alt: "250%")[$"250%"$] markup before advertising it for sale. What price did she ask for the dresser?

#solutionbox[
\$70
]

Find the simple interest earned after #math.equation(block: false, alt: "5")[$5$] years on #math.equation(block: false, alt: "$3000")[$"$3000"$] invested at an interest rate of #math.equation(block: false, alt: "4.2% .")[$"4.2%" .$]

Brenda borrowed #math.equation(block: false, alt: "$400")[$"$400"$] from her brother. Two years later, she repaid the #math.equation(block: false, alt: "$400")[$"$400"$] plus #math.equation(block: false, alt: "$50")[$"$50"$] interest. What was the rate of interest?

#solutionbox[
6.25%
]

Write as a proportion: #math.equation(block: false, alt: "4")[$4$] gallons to #math.equation(block: false, alt: "144")[$144$] miles is the same as #math.equation(block: false, alt: "10")[$10$] gallons to #math.equation(block: false, alt: "360")[$360$] miles.

Solve for a: #math.equation(block: false, alt: "the fraction 12 over a equals the fraction −15 over 65")[$frac(12, a) = frac(−15, 65)$]

#solutionbox[
−52
]

Vin read #math.equation(block: false, alt: "10")[$10$] pages of a book in #math.equation(block: false, alt: "12")[$12$] minutes. At that rate, how long will it take him to read #math.equation(block: false, alt: "35")[$35$] pages?