#set document(title: "8.7 Solve Proportion and Similar Figure Applications", author: "OpenStax / XYZ Homework")
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== 8.7#h(0.6em)Solve Proportion and Similar Figure Applications

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

If you miss a problem, go back to the section listed and review the material.

Solve #math.equation(block: false, alt: "the fraction n over 3 equals 30")[$frac(n, 3) = 30$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "n equals 90")[$n = 90$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The perimeter of a triangular window is 23 feet. The lengths of two sides are ten feet and six feet. How long is the third side? #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "7 feet")[$7 #h(0.2em) "feet"$]
]
]

#linebreak()


=== Solve Proportions

When two rational expressions are equal, the equation relating them is called a #emph[proportion].

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

A #strong[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 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 “1 is to 2 as 4 is to 8.”

Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.

Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let #emph[x] be the number of teachers for 60 students and then set up the proportion:

#math.equation(block: true, alt: "the fraction 1 teacher over 20 students equals the fraction x teachers over 60 students")[$frac(1 #h(0.2em) "teacher", 20 #h(0.2em) "students") = frac(x #h(0.2em) "teachers", 60 #h(0.2em) "students")$]We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.

Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solve Rational Equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.

So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed against a white background. The equation shows the fraction 1/20 on the left side, an equals sign in the middle, and the fraction x/60 on the right side.], alt: "A mathematical equation is displayed against a white background. The equation shows the fraction 1/20 on the left side, an equals sign in the middle, and the fraction x/60 on the right side.", caption: none)]),
  [Multiply both sides by the LCD, 60.], [#figure(figph[A mathematical equation showing both sides of an equation multiplied by 60: (1/20) \* 60 = (x/60) \* 60. This step clears the denominator when solving for a variable.], alt: "A mathematical equation showing both sides of an equation multiplied by 60: (1/20) * 60 = (x/60) * 60. This step clears the denominator when solving for a variable.", caption: none)],
  [Simplify.], [#figure(figph[A simple mathematical equation '3 = X' is displayed on a white background, representing the equality between the number three and the variable X.], alt: "A simple mathematical equation '3 = X' is displayed on a white background, representing the equality between the number three and the variable X.", caption: none)],
  [], [The principal needs 3 teachers for 60 students.],
))

Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.

#examplebox("Example 1")[][
Solve the proportion: #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,
  table.header([], [], [#figure(figph[A mathematical equation is displayed, showing a variable 'x' divided by 63, which equals the fraction 4/7. The equation is x/63 = 4/7.], alt: "A mathematical equation is displayed, showing a variable 'x' divided by 63, which equals the fraction 4/7. The equation is x/63 = 4/7.", caption: none)]),
  [To isolate #math.equation(block: false, alt: "x")[$x$], multiply both sides by the LCD, 63.], [], [#figure(figph[An equation showing 63 multiplied by (x/63) on the left side, and 63 multiplied by (4/7) on the right side. The number 63 is highlighted in red on both sides of the equals sign.], alt: "An equation showing 63 multiplied by (x/63) on the left side, and 63 multiplied by (4/7) on the right side. The number 63 is highlighted in red on both sides of the equals sign.", caption: none)],
  [Simplify.], [], [#figure(figph[A mathematical expression showing the equation x = (9  7  4) / 7.], alt: "A mathematical expression showing the equation x = (9 7 4) / 7.", caption: none)],
  [Divide the common factors.], [], [#figure(figph[The image displays a mathematical equation 'x = 36' in a clear, legible font against a plain white background, presenting a simple statement of equality.], alt: "The image displays a mathematical equation 'x = 36' in a clear, legible font against a plain white background, presenting a simple statement of equality.", caption: none)],
  [Check. To check our answer, we substitute into the original proportion.], [], [],
  [], [#figure(figph[A mathematical equation shows a fraction with x over 63, set equal to the fraction 4 over 7, likely intended for solving for x.], alt: "A mathematical equation shows a fraction with x over 63, set equal to the fraction 4 over 7, likely intended for solving for x.", caption: none)], [],
  [#figure(figph[The image shows the text 'Substitute x = 36.'. The number '36' is highlighted in red, indicating a specific value for substitution in a mathematical or algebraic context.], alt: "The image shows the text 'Substitute x = 36.'. The number '36' is highlighted in red, indicating a specific value for substitution in a mathematical or algebraic context.", caption: none)], [#figure(figph[A math problem displays the fractions 36/63 and 4/7, with a question mark next to an equals sign in between, asking if they are equivalent.], alt: "A math problem displays the fractions 36/63 and 4/7, with a question mark next to an equals sign in between, asking if they are equivalent.", caption: none)], [],
  [Show common factors.], [#figure(figph[A mathematical expression displays the fraction (4 \* 9) / (7 \* 9) with a question mark over an equals sign, followed by the simplified fraction 4/7, illustrating the simplification of fractions.], alt: "A mathematical expression displays the fraction (4 * 9) / (7 * 9) with a question mark over an equals sign, followed by the simplified fraction 4/7, illustrating the simplification of fractions.", caption: none)], [],
  [Simplify.], [#figure(figph[The mathematical equation '4/7 = 4/7' is displayed in black text on a white background, followed by a checkmark indicating its correctness.], alt: "The mathematical equation '4/7 = 4/7' is displayed in black text on a white background, followed by a checkmark indicating its correctness.", caption: none)], [],
))
]
]

#notebox("Note", 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[
#math.equation(block: true, alt: "77")[$77$]
]
]

#notebox("Note", 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[
#math.equation(block: true, alt: "104")[$104$]
]
]

When we work with #strong[proportion]s, we exclude values that would make either denominator zero, just like we do for all rational expressions. What value(s) should be excluded for the proportion in the next example?

#examplebox("Example 2")[][
Solve the proportion: #math.equation(block: false, alt: "the fraction 144 over a equals the fraction 9 over 4 .")[$frac(144, a) = frac(9, 4) .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [], [#figure(figph[A mathematical equation displays '144 divided by a equals 9 divided by 4' against a white background.], alt: "A mathematical equation displays '144 divided by a equals 9 divided by 4' against a white background.", caption: none)]),
  [Multiply both sides by the LCD.], [], [#figure(figph[Multiplying both sides of the equation (144/a) = (9/4) by 4a to eliminate the denominators, resulting in (144/a) \* 4a = (9/4) \* 4a.], alt: "Multiplying both sides of the equation (144/a) = (9/4) by 4a to eliminate the denominators, resulting in (144/a) * 4a = (9/4) * 4a.", caption: none)],
  [Remove common factors on each side.], [], [#figure(figph[A mathematical equation shows '4 \* 144 = a \* 9', an algebraic problem to solve for the unknown variable 'a'.], alt: "A mathematical equation shows '4 * 144 = a * 9', an algebraic problem to solve for the unknown variable 'a'.", caption: none)],
  [Simplify.], [], [#figure(figph[A mathematical equation is displayed on a white background, showing 576 equals 9q.], alt: "A mathematical equation is displayed on a white background, showing 576 equals 9q.", caption: none)],
  [Divide both sides by 9.], [], [#figure(figph[A mathematical equation shows '576/9 = 9a/9'. The left side of the equation is the fraction 576 over 9, and the right side is the fraction 9a over 9, with an equals sign in between.], alt: "A mathematical equation shows '576/9 = 9a/9'. The left side of the equation is the fraction 576 over 9, and the right side is the fraction 9a over 9, with an equals sign in between.", caption: none)],
  [Simplify.], [], [#figure(figph[The image shows a mathematical equation displaying the number 64 equal to the variable 'a', presented in a clean, legible font against a white background.], alt: "The image shows a mathematical equation displaying the number 64 equal to the variable 'a', presented in a clean, legible font against a white background.", caption: none)],
  [Check.], [], [],
  [], [#figure(figph[A mathematical equation is displayed, showing the fraction 144 over 'a' set equal to the fraction 9 over 4, representing a problem to solve for the variable 'a'.], alt: "A mathematical equation is displayed, showing the fraction 144 over 'a' set equal to the fraction 9 over 4, representing a problem to solve for the variable 'a'.", caption: none)], [],
  [#figure(figph[The phrase 'Substitute q = 64.' is displayed in gray text, with the number '64' highlighted in red, indicating a mathematical instruction to replace the variable q with the value 64.], alt: "The phrase 'Substitute q = 64.' is displayed in gray text, with the number '64' highlighted in red, indicating a mathematical instruction to replace the variable q with the value 64.", caption: none)], [#figure(figph[A mathematical problem is displayed, comparing the fractions '144/64' and '9/4' with a question mark over an equality sign, asking if they are equivalent. The denominator '64' is highlighted in red.], alt: "A mathematical problem is displayed, comparing the fractions '144/64' and '9/4' with a question mark over an equality sign, asking if they are equivalent. The denominator '64' is highlighted in red.", caption: none)], [],
  [Show common factors.], [#figure(figph[A mathematical problem asking if the fraction (9 \* 16) / (4 \* 16) is equal to 9/4. This visually demonstrates the principle of simplifying fractions by canceling common factors in the numerator and denominator.], alt: "A mathematical problem asking if the fraction (9 * 16) / (4 * 16) is equal to 9/4. This visually demonstrates the principle of simplifying fractions by canceling common factors in the numerator and denominator.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation shows '9/4 = 9/4' with a checkmark, indicating the equality is correct.], alt: "A mathematical equation shows '9/4 = 9/4' with a checkmark, indicating the equality is correct.", caption: none)], [],
))
]
]

#notebox("Note", 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[
#math.equation(block: true, alt: "65")[$65$]
]
]

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

#examplebox("Example 3")[][
Solve the proportion: #math.equation(block: false, alt: "the fraction n over n plus 14 equals the fraction 5 over 7 .")[$frac(n, n + 14) = frac(5, 7) .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [], [#figure(figph[A mathematical equation shows a fraction n over the sum of n and 14, which is set equal to the fraction 5 over 7, appearing in a clean, isolated format on a white background.], alt: "A mathematical equation shows a fraction n over the sum of n and 14, which is set equal to the fraction 5 over 7, appearing in a clean, isolated format on a white background.", caption: none)]),
  [Multiply both sides by the LCD.], [], [#figure(figph[The equation 7(n+14)(n/(n+14)) = 7(n+14)(5/7) is presented, implying that n/(n+14) equals 5/7, demonstrating an algebraic step.], alt: "The equation 7(n+14)(n/(n+14)) = 7(n+14)(5/7) is presented, implying that n/(n+14) equals 5/7, demonstrating an algebraic step.", caption: none)],
  [Remove common factors on each side.], [], [#figure(figph[A mathematical equation is displayed, showing '7n = 5(n + 14)' in black text against a white background. This is a linear equation with one variable 'n'.], alt: "A mathematical equation is displayed, showing '7n = 5(n + 14)' in black text against a white background. This is a linear equation with one variable 'n'.", caption: none)],
  [Simplify.], [], [#figure(figph[A mathematical equation, 7n = 5n + 70, is displayed in a clear, dark font against a plain white background.], alt: "A mathematical equation, 7n = 5n + 70, is displayed in a clear, dark font against a plain white background.", caption: none)],
  [Solve for #math.equation(block: false, alt: "n")[$n$].], [], [#figure(figph[The image displays a simple algebraic equation in black text against a white background, which reads '2n = 70'.], alt: "The image displays a simple algebraic equation in black text against a white background, which reads '2n = 70'.", caption: none)],
  [], [], [#figure(figph[The image displays the text 'n = 35' in a clear, dark gray font against a plain white background, indicating a numerical value or sample size.], alt: "The image displays the text 'n = 35' in a clear, dark gray font against a plain white background, indicating a numerical value or sample size.", caption: none)],
  [Check.], [], [],
  [], [#figure(figph[A mathematical equation shows a fraction with 'n' in the numerator and 'n + 14' in the denominator, set equal to the fraction '5/7'.], alt: "A mathematical equation shows a fraction with 'n' in the numerator and 'n + 14' in the denominator, set equal to the fraction '5/7'.", caption: none)], [],
  [#figure(figph[The text 'Substitute n = 35.' is displayed.], alt: "The text 'Substitute n = 35.' is displayed.", caption: none)], [#figure(figph[A math problem showing the fraction 35 over (35 plus 14) on the left side, which is equal to an unknown numerator (?) over 7 on the right side. The problem asks to solve for the missing numerator.], alt: "A math problem showing the fraction 35 over (35 plus 14) on the left side, which is equal to an unknown numerator (?) over 7 on the right side. The problem asks to solve for the missing numerator.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical problem displaying the fractions 35/49 and 5/7, with a question mark positioned above the equals sign, prompting verification of their equivalence.], alt: "A mathematical problem displaying the fractions 35/49 and 5/7, with a question mark positioned above the equals sign, prompting verification of their equivalence.", caption: none)], [],
  [Show common factors.], [#figure(figph[A mathematical expression showing the fraction (5 multiplied by 7) divided by (7 multiplied by 7), with a question mark above an equals sign, followed by the fraction 5/7. It asks if (5\*7)/(7\*7) is equal to 5/7.], alt: "A mathematical expression showing the fraction (5 multiplied by 7) divided by (7 multiplied by 7), with a question mark above an equals sign, followed by the fraction 5/7. It asks if (5*7)/(7*7) is equal to 5/7.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical expression showing the equality 5/7 = 5/7, with a checkmark indicating correctness.], alt: "A mathematical expression showing the equality 5/7 = 5/7, with a checkmark indicating correctness.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve the proportion: #math.equation(block: false, alt: "the fraction y over y plus 55 equals the fraction 3 over 8 .")[$frac(y, y + 55) = frac(3, 8) .$]

#solutionbox[
#math.equation(block: true, alt: "33")[$33$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve the proportion: #math.equation(block: false, alt: "the fraction z over z minus 84 equals minus the fraction 1 over 5 .")[$frac(z, z − 84) = − frac(1, 5) .$]

#solutionbox[
#math.equation(block: true, alt: "14")[$14$]
]
]

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "the fraction p plus 12 over 9 equals the fraction p minus 12 over 6 .")[$frac(p + 12, 9) = frac(p − 12, 6) .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [], [#figure(figph[A mathematical equation is displayed: the fraction p plus 12 over 9 equals the fraction p minus 12 over 6. This is a linear equation with one variable 'p'.], alt: "A mathematical equation is displayed: the fraction p plus 12 over 9 equals the fraction p minus 12 over 6. This is a linear equation with one variable 'p'.", caption: none)]),
  [Multiply both sides by the LCD, 18.], [], [#figure(figph[A mathematical equation is displayed, showing 18 multiplied by the fraction (p + 12)/9 on the left side, equal to 18 multiplied by the fraction (p - 12)/6 on the right side.], alt: "A mathematical equation is displayed, showing 18 multiplied by the fraction (p + 12)/9 on the left side, equal to 18 multiplied by the fraction (p - 12)/6 on the right side.", caption: none)],
  [Simplify.], [], [#figure(figph[A mathematical equation is displayed with a black text on a white background. The equation is 2(p + 12) = 3(p - 12).], alt: "A mathematical equation is displayed with a black text on a white background. The equation is 2(p + 12) = 3(p - 12).", caption: none)],
  [Distribute.], [], [#figure(figph[A mathematical equation is displayed, reading '2p + 24 = 3p - 36' in black font against a white background. It represents a linear equation with one variable 'p'.], alt: "A mathematical equation is displayed, reading '2p + 24 = 3p - 36' in black font against a white background. It represents a linear equation with one variable 'p'.", caption: none)],
  [Solve for #math.equation(block: false, alt: "p")[$p$].], [], [#figure(figph[A simple mathematical equation displaying '60 = p' on a white background.], alt: "A simple mathematical equation displaying '60 = p' on a white background.", caption: none)],
  [Check.], [], [],
  [], [#figure(figph[An algebraic equation is shown, with the fraction (p+12)/9 on the left side and the fraction (p-12)/6 on the right side, separated by an equals sign.], alt: "An algebraic equation is shown, with the fraction (p+12)/9 on the left side and the fraction (p-12)/6 on the right side, separated by an equals sign.", caption: none)], [],
  [#figure(figph[The text in the image instructs to 'Substitute p = 60.', with the number 60 highlighted in red.], alt: "The text in the image instructs to 'Substitute p = 60.', with the number 60 highlighted in red.", caption: none)], [#figure(figph[A math problem comparing two fractions: (60+12)/9 and (60-12)/6, with a question mark over an equals sign between them. Both sides evaluate to 8, indicating they are equal.], alt: "A math problem comparing two fractions: (60+12)/9 and (60-12)/6, with a question mark over an equals sign between them. Both sides evaluate to 8, indicating they are equal.", caption: none)], [],
  [Simplify.], [#figure(figph[A math problem asking if the fraction 72/9 is equal to 48/6. The expression shows 72 over 9, followed by an equal sign with a question mark on top, and then 48 over 6.], alt: "A math problem asking if the fraction 72/9 is equal to 48/6. The expression shows 72 over 9, followed by an equal sign with a question mark on top, and then 48 over 6.", caption: none)], [],
  [Divide.], [#figure(figph[The equation 8 = 8 is displayed with a checkmark next to it, signifying that the statement is correct or verified as true on a plain white background.], alt: "The equation 8 = 8 is displayed with a checkmark next to it, signifying that the statement is correct or verified as true on a plain white background.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction v plus 30 over 8 equals the fraction v plus 66 over 12 .")[$frac(v + 30, 8) = frac(v + 66, 12) .$]

#solutionbox[
#math.equation(block: true, alt: "42")[$42$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 2 x plus 15 over 9 equals the fraction 7 x plus 3 over 15 .")[$frac(2 x + 15, 9) = frac(7 x + 3, 15) .$]

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

To solve applications with proportions, we will follow our usual strategy for solving applications. But when we set up the proportion, we must make sure to have the units correct—the units in the numerators must match and the units in the denominators must match.

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

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Identify what we are asked to find, and choose a variable to represent it.], [], [How many ml of acetaminophen will the doctor prescribe?]),
  [], [], [Let #math.equation(block: false, alt: "a equals ml")[$a = "ml"$] 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–be careful of the units. #linebreak() #math.equation(block: false, alt: "the fraction ml over pounds equals the fraction ml over pounds")[$frac("ml", "pounds") = frac("ml", "pounds")$]], [], [#linebreak()  #linebreak() #figure(figph[A mathematical equation is shown with the fraction 5/25 equal to the fraction a/80, where 'a' is a variable.], alt: "A mathematical equation is shown with the fraction 5/25 equal to the fraction a/80, where 'a' is a variable.", caption: none)],
  [Multiply both sides by the LCD, 400.], [], [#figure(figph[A mathematical equation showing 400 multiplied by 5/25 equals 400 multiplied by a/80, set against a white background.], alt: "A mathematical equation showing 400 multiplied by 5/25 equals 400 multiplied by a/80, set against a white background.", caption: none)],
  [Remove common factors on each side.], [], [#figure(figph[A mathematical equation is displayed: 25 \* 16 \* (5/25) = 80 \* 5 \* (a/80). The numbers 25 on the left and 80 on the right are shown crossed out, indicating a step in solving or simplifying the equation.], alt: "A mathematical equation is displayed: 25 * 16 * (5/25) = 80 * 5 * (a/80). The numbers 25 on the left and 80 on the right are shown crossed out, indicating a step in solving or simplifying the equation.", caption: none)],
  [Simplify, but don't multiply on the left. Notice what the next step will be.], [], [#figure(figph[A mathematical equation shows '16 \* 5 = 5a' in black text on a white background. This equation can be solved to find the value of 'a'.], alt: "A mathematical equation shows '16 * 5 = 5a' in black text on a white background. This equation can be solved to find the value of 'a'.", caption: none)],
  [Solve for #math.equation(block: false, alt: "a")[$a$].], [], [#figure(figph[A mathematical equation is displayed, showing the fraction 16 multiplied by 5, all divided by 5, which equals 5a divided by 5. The denominator '5' is highlighted in red on both sides of the equation.], alt: "A mathematical equation is displayed, showing the fraction 16 multiplied by 5, all divided by 5, which equals 5a divided by 5. The denominator '5' is highlighted in red on both sides of the equation.", caption: none)],
  [Check.], [], [#figure(figph[A mathematical equation displayed on a white background, showing '16 = a'.], alt: "A mathematical equation displayed on a white background, showing '16 = a'.", caption: none)],
  [Is the answer reasonable?], [], [],
  [Yes, since 80 is about 3 times 25, the medicine should be about 3 times 5. So 16 ml makes sense.], [], [],
  [#figure(figph[The image shows the steps to verify the solution for 'a' in a proportion. 'a = 16' is substituted into 5/25 = a/80. Both sides, 5/25 and 16/80, simplify to 1/5, confirming the equality and validity of the solution with a checkmark.], alt: "The image shows the steps to verify the solution for 'a' in a proportion. 'a = 16' is substituted into 5/25 = a/80. Both sides, 5/25 and 16/80, simplify to 1/5, confirming the equality and validity of the solution with a checkmark.", caption: none)], [], [],
  [Write a complete sentence.], [], [The pediatrician would prescribe 16 ml of acetaminophen to Zoe.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 pounds?

#solutionbox[
#math.equation(block: true, alt: "12 ml")[$12 #h(0.2em) "ml"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For every 1 kilogram (kg) of a child’s weight, pediatricians prescribe 15 milligrams (mg) of a fever reducer. If Isabella weighs 12 kg, how many milligrams of the fever reducer will the pediatrician prescribe?

#solutionbox[
#math.equation(block: true, alt: "180 mg")[$180 #h(0.2em) "mg"$]
]
]

#examplebox("Example 6")[][
A 16-ounce iced caramel macchiato has 230 calories. How many calories are there in a 24-ounce iced caramel macchiato?

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Identify what we are asked to find, and choose a variable to represent it.], [], [How many calories are in a 24 ounce iced caramel macchiato?]),
  [], [], [Let #math.equation(block: false, alt: "c equals calories")[$c = "calories"$] in 24 ounces.],
  [Write a sentence that gives the information to find it.], [], [If there are 230 calories in 16 ounces, then how many calories are in 24 ounces?],
  [Translate into a proportion–be careful of the units. #linebreak() #math.equation(block: false, alt: "the fraction calories over ounce equals the fraction calories over ounce")[$frac("calories", "ounce") = frac("calories", "ounce")$]], [], [#linebreak()  #linebreak() #figure(figph[A mathematical equation shows a proportion: 230 over 16 equals c over 24, where 'c' is an unknown variable.], alt: "A mathematical equation shows a proportion: 230 over 16 equals c over 24, where 'c' is an unknown variable.", caption: none)],
  [Multiply both sides by the LCD, 48.], [], [#figure(figph[A mathematical equation shows '48(230/16) = 48(c/24)' where 48 is in red, illustrating the multiplication of 48 by two different fractions set equal to each other.], alt: "A mathematical equation shows '48(230/16) = 48(c/24)' where 48 is in red, illustrating the multiplication of 48 by two different fractions set equal to each other.", caption: none)],
  [Remove common factors on each side.], [], [#figure(figph[A mathematical equation is displayed where 16 multiplied by 3 and 230/16 equals 24 multiplied by 2 and c/24. The numbers 16 and 24 are struck through in red on both sides of the equation.], alt: "A mathematical equation is displayed where 16 multiplied by 3 and 230/16 equals 24 multiplied by 2 and c/24. The numbers 16 and 24 are struck through in red on both sides of the equation.", caption: none)],
  [Simplify.], [], [#figure(figph[A mathematical equation shows '690 = 2c' centered on a white background, representing a basic algebraic problem where the value of 'c' needs to be determined by dividing 690 by 2.], alt: "A mathematical equation shows '690 = 2c' centered on a white background, representing a basic algebraic problem where the value of 'c' needs to be determined by dividing 690 by 2.", caption: none)],
  [Solve for #math.equation(block: false, alt: "c")[$c$].], [], [#figure(figph[A mathematical equation shows '690/2 = 2c/2' with the denominators, both '2', highlighted in red, indicating division on both sides of the equation to solve for 'c'.], alt: "A mathematical equation shows '690/2 = 2c/2' with the denominators, both '2', highlighted in red, indicating division on both sides of the equation to solve for 'c'.", caption: none)],
  [], [], [#figure(figph[The image displays a mathematical equation], alt: "The image displays a mathematical equation", caption: none)],
  [Check.], [], [],
  [Is the answer reasonable?], [], [],
  [Yes, 345 calories for 24 ounces is more than 290 calories for 16 ounces, but not too much more.], [], [],
  [#figure(figph[This image demonstrates substituting c=345 into the proportion 230/16 = c/24. It verifies the equality by substituting 345 for c, showing 230/16 = 345/24, which simplifies to 115/8 = 115/8, confirmed by a checkmark.], alt: "This image demonstrates substituting c=345 into the proportion 230/16 = c/24. It verifies the equality by substituting 345 for c, showing 230/16 = 345/24, which simplifies to 115/8 = 115/8, confirmed by a checkmark.", caption: none)], [], [],
  [Write a complete sentence.], [], [There are 345 calories in a 24-ounce iced caramel macchiato.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
At a fast-food restaurant, a 22-ounce chocolate shake has 850 calories. How many calories are in their 12-ounce chocolate shake? Round your answer to nearest whole number.

#solutionbox[
#math.equation(block: true, alt: "464 calories")[$464 #h(0.2em) "calories"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Yaneli loves Starburst candies, but wants to keep her snacks to 100 calories. If the candies have 160 calories for 8 pieces, how many pieces can she have in her snack?

#solutionbox[
#math.equation(block: true, alt: "5 pieces")[$5 #h(0.2em) "pieces"$]
]
]

#examplebox("Example 7")[][
Josiah went to Mexico for spring break and changed \$325 dollars into Mexican pesos. At that time, the exchange rate had \$1 US is equal to 12.54 Mexican pesos. How many Mexican pesos did he get for his trip?

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([What are you asked to find?], [], [How many Mexican pesos did Josiah get?]),
  [Assign a variable.], [], [Let #math.equation(block: false, alt: "p equals the number of Mexican pesos.")[$p = "the number of Mexican pesos."$]],
  [Write a sentence that gives the information to find it.], [], [If \$1 US is equal to 12.54 Mexican pesos, then \$325 is how many pesos?],
  [Translate into a proportion–be careful of the units.], [], [],
  [#math.equation(block: false, alt: "the fraction $ over pesos equals the fraction $ over pesos")[$frac("$", "pesos") = frac("$", "pesos")$]], [], [#figure(figph[A mathematical equation is displayed on a white background, reading '1 over 12.54 equals 325 over p'.], alt: "A mathematical equation is displayed on a white background, reading '1 over 12.54 equals 325 over p'.", caption: none)],
  [Multiply both sides by the LCD, #math.equation(block: false, alt: "12.54 p")[$12.54 p$].], [], [#figure(figph[An algebraic equation showing 12.54p multiplied by the reciprocal of 12.54 on the left side, equaling 12.54p multiplied by 325 over p on the right side.], alt: "An algebraic equation showing 12.54p multiplied by the reciprocal of 12.54 on the left side, equaling 12.54p multiplied by 325 over p on the right side.", caption: none)],
  [Remove common factors on each side.], [], [#figure(figph[An example of algebraic simplification where identical factors in the numerator and denominator are cancelled out on both sides of an equation.], alt: "An example of algebraic simplification where identical factors in the numerator and denominator are cancelled out on both sides of an equation.", caption: none)],
  [Simplify.], [], [#figure(figph[The image displays text in a simple, clear font, showing the equation 'p = 4075.5' centered on a white background.], alt: "The image displays text in a simple, clear font, showing the equation 'p = 4075.5' centered on a white background.", caption: none)],
  [Check.], [], [],
  [Is the answer reasonable?], [], [],
  [Yes, \$100 would be 1,254 pesos. \$325 is a little more than 3 times this amount, so our answer of 4075.5 pesos makes sense.], [], [],
  [#figure(figph[A mathematical problem showing the substitution of p = 4075.5 into the proportion 1/12.54 = 325/p, resulting in both sides equaling approximately 0.07874, confirming the equality.], alt: "A mathematical problem showing the substitution of p = 4075.5 into the proportion 1/12.54 = 325/p, resulting in both sides equaling approximately 0.07874, confirming the equality.", caption: none)], [], [],
  [Write a complete sentence.], [], [Josiah got 4075.5 pesos for his spring break trip.],
))
]
]

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

#solutionbox[
#math.equation(block: true, alt: "590.4 Euros")[$590.4 #h(0.2em) "Euros"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Corey and Nicole are traveling to Japan and need to exchange \$600 into Japanese yen. If each dollar is 94.1 yen, how many yen will they get?

#solutionbox[
#math.equation(block: true, alt: "56,460 yen")[$56,460 #h(0.2em) "yen"$]
]
]

In the example above, we related the number of pesos to the number of dollars by using a proportion. We could say the number of pesos #emph[is proportional to] the number of dollars. If two quantities are related by a proportion, we say that they are proportional.

#linebreak()


=== Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with #strong[similar figures]. If two figures have exactly the same shape, but different sizes, they are said to be #emph[similar.] One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.

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

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio.
]

For example, the two triangles in  are similar. Each side of #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is 4 times the length of the corresponding side of #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$].

#figure(figph[Diagram of two triangles, ABC and XYZ. Triangle ABC has side lengths AB equal to 12, BC equal to 16, and AC equal to 20. Triangle XYZ has side lengths XY equal to 3, YZ equal to 4, and XZ equal to 5. The image indicates that angle A is congruent to angle X, angle B is congruent to angle Y, and angle C is congruent to angle Z. Ratios of corresponding sides are shown: AB/XY equals 12/3, BC/YZ equals 16/4, and AC/XZ equals 20/5.], alt: "Diagram of two triangles, ABC and XYZ. Triangle ABC has side lengths AB equal to 12, BC equal to 16, and AC equal to 20. Triangle XYZ has side lengths XY equal to 3, YZ equal to 4, and XZ equal to 5. The image indicates that angle A is congruent to angle X, angle B is congruent to angle Y, and angle C is congruent to angle Z. Ratios of corresponding sides are shown: AB/XY equals 12/3, BC/YZ equals 16/4, and AC/XZ equals 20/5.", caption: none)

This is summed up in the Property of Similar Triangles.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Property of Similar Triangles]

If #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$], then their corresponding angle measure are equal and their corresponding sides are in the same ratio.

#figure(figph[The above figure shows to similar triangles. The larger triangle labeled A B C. The length of A to B is c, The length of B to C is a. The length of C to A is b. The larger triangle is labeled X Y Z. The length of X to Y is z. The length of Y to Z is x. The length of X to Z is y. To the right of the triangles, it states that measure of corresponding angle A is equal to the measure of corresponding angle X, measure of corresponding angle B is equal to the measure of corresponding angle Y, and measure of corresponding angle C is equal to the measure of corresponding angle Z. Therefore, a divided by x equals b divided by y equals c divided by z.], alt: "The above figure shows to similar triangles. The larger triangle labeled A B C. The length of A to B is c, The length of B to C is a. The length of C to A is b. The larger triangle is labeled X Y Z. The length of X to Y is z. The length of Y to Z is x. The length of X to Z is y. To the right of the triangles, it states that measure of corresponding angle A is equal to the measure of corresponding angle X, measure of corresponding angle B is equal to the measure of corresponding angle Y, and measure of corresponding angle C is equal to the measure of corresponding angle Z. Therefore, a divided by x equals b divided by y equals c divided by z.", caption: none)
]

To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve geometry applications.]

+ #strong[Read] the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for by choosing a variable to represent it.
+ #strong[Translate] into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
+ #strong[Solve the equation] using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.
]

#examplebox("Example 8")[][
#math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$]. The lengths of two sides of each triangle are given. Find the lengths of the third sides.

#figure(figph[The above image shows two similar triangles. Two sides are given for each triangle. The larger triangle is labeled A B C. The length of A to B is 4. The length from B to C is a. The length from C to A is 3.2. The smaller triangle is labeled X Y Z. The length from X to Y is 3. The length from Y to Z is 4.5. The length from Z to X is y.], alt: "The above image shows two similar triangles. Two sides are given for each triangle. The larger triangle is labeled A B C. The length of A to B is 4. The length from B to C is a. The length from C to A is 3.2. The smaller triangle is labeled X Y Z. The length from X to Y is 3. The length from Y to Z is 4.5. The length from Z to X is y.", caption: none)

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. Draw the figure and label it with the given information.], [Figure is given.]),
  [#emph[Step 2. Identify] what we are looking for.], [the length of the sides of similar triangles],
  [#emph[Step 3. Name] the variables.], [Let #math.equation(block: false, alt: "a equals")[$#h(0.3em) a =$] length of the third side of #math.equation(block: false, alt: "Δ A B C .")[$"Δ" A B C .$] #linebreak()   #math.equation(block: false, alt: "y equals")[$y =$] length of the third side of #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$]],
  [#emph[Step 4. Translate.]], [Since the triangles are similar, the corresponding sides are proportional.],
  [We need to write an equation that compares the side we are looking for to a known ratio. Since the side #emph[AB] = 4 corresponds to the side #emph[XY] = 3 we know #math.equation(block: false, alt: "the fraction A B over X Y equals the fraction 4 over 3")[$frac(A B, X Y) = frac(4, 3)$]. So we write equations with #math.equation(block: false, alt: "the fraction A B over X Y")[$frac(A B, X Y)$] to find the sides we are looking for. Be careful to match up corresponding sides correctly.], [#math.equation(block: false, alt: "the fraction A B over X Y equals the fraction B C over Y Z equals the fraction A C over X Z .")[$frac(A B, X Y) = frac(B C, Y Z) = frac(A C, X Z) .$] #linebreak() #figure(figph[Formulas for finding unknown lengths 'a' and 'y' in similar triangles. Compares side ratios of a large triangle to a small triangle: AB/XY = BC/YZ for 'a', and AB/XY = AC/XZ for 'y'.], alt: "Formulas for finding unknown lengths 'a' and 'y' in similar triangles. Compares side ratios of a large triangle to a small triangle: AB/XY = BC/YZ for 'a', and AB/XY = AC/XZ for 'y'.", caption: none)],
  [Substitute.], [$#h(11.2em)$#figure(figph[A mathematical equation shows a fraction 4/3 equal to another fraction a/4.5. This represents a proportion where 'a' is an unknown variable to be solved.], alt: "A mathematical equation shows a fraction 4/3 equal to another fraction a/4.5. This represents a proportion where 'a' is an unknown variable to be solved.", caption: none)

$#h(2.7em)$#figure(figph[A mathematical equation shows a proportion where four-thirds equals three point two divided by y.], alt: "A mathematical equation shows a proportion where four-thirds equals three point two divided by y.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [$#h(11.2em)$#figure(figph[A mathematical equation showing 3a equals 4 multiplied by 4.5, written as '3a = 4(4.5)' on a white background.], alt: "A mathematical equation showing 3a equals 4 multiplied by 4.5, written as '3a = 4(4.5)' on a white background.", caption: none)

$#h(2.7em)$#figure(figph[A mathematical equation is displayed, showing '4y = 3(3.2)' in a bold, dark gray font against a light gray background.], alt: "A mathematical equation is displayed, showing '4y = 3(3.2)' in a bold, dark gray font against a light gray background.", caption: none)],
  [], [$#h(11.2em)$#figure(figph[The image displays the mathematical expression 'a = 6' in a plain, clear font on a white background, indicating a variable 'a' being assigned the value of six.], alt: "The image displays the mathematical expression 'a = 6' in a plain, clear font on a white background, indicating a variable 'a' being assigned the value of six.", caption: none)

$#h(2.7em)$#figure(figph[The image displays the equation y = 2.4, presented in a clean, legible font against a plain white background.], alt: "The image displays the equation y = 2.4, presented in a clean, legible font against a plain white background.", caption: none)],
  [#emph[Step 6. Check.] #linebreak() #math.equation(block: false, alt: "the fraction 4 over 3, equals ?, the fraction 6 over 4.5; 4 open parenthesis 4.5 close parenthesis, equals ?, 6 open parenthesis 3 close parenthesis; 18, equals, 18 ✓, the fraction 4 over 3, equals ?, the fraction 3.2 over 2.4; 4 open parenthesis 2.4 close parenthesis, equals ?, 3.2 open parenthesis 3 close parenthesis; 9.6, equals, 9.6 ✓")[$frac(4, 3) & limits(=)^(?) & frac(6, 4.5) \ 4 ( 4.5 ) & limits(=)^(?) & 6 ( 3 ) \ 18 & = & 18 ✓ &  &  &  &  & frac(4, 3) & limits(=)^(?) & frac(3.2, 2.4) \ 4 ( 2.4 ) & limits(=)^(?) & 3.2 ( 3 ) \ 9.6 & = & 9.6 ✓$]], [],
  [#emph[Step 7. Answer] the question.], [The third side of #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is 6 and the third side of #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$] is 2.4.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$]. The lengths of two sides of each triangle are given in the figure.

#figure(figph[The above image shows two similar triangles. The smaller triangle is labeled A B C. The length of two sides is given for the smaller triangle A B C. The length from A to B is 17. The length from B to C is a. The length from C to D is 15. The larger triangle is labeled X Y Z. The length is given for two sides. The length from X to Y is 25.5. The length from Y to Z is 12. The length from Z to X is y.], alt: "The above image shows two similar triangles. The smaller triangle is labeled A B C. The length of two sides is given for the smaller triangle A B C. The length from A to B is 17. The length from B to C is a. The length from C to D is 15. The larger triangle is labeled X Y Z. The length is given for two sides. The length from X to Y is 25.5. The length from Y to Z is 12. The length from Z to X is y.", caption: none)

Find the length of side #math.equation(block: false, alt: "a")[$a$].

#solutionbox[
8
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$]. The lengths of two sides of each triangle are given in the figure.

Find the length of side #math.equation(block: false, alt: "y")[$y$].

#solutionbox[
22.5
]
]

The next example shows how similar triangles are used with maps.

#examplebox("Example 9")[][
On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. If the actual distance from Los Angeles to Las Vegas is 270 miles find the distance from Los Angeles to San Francisco.

#figure(figph[The above image shows two similar triangles and how they are used with maps. The smaller triangle on the left shows San Francisco, Las Vegas and Los Angeles on the three points. San Francisco to Los Angeles is 1.3 inches. Los Angeles to Las Vegas is 1 inch. Las Vegas to San Francisco is 2.1 inches. The second larger triangle shows the same points. The distance from San Francisco to Los Angeles is x. The distance from Los Angeles to Las Vegas is 270 miles. The distance from Las Vegas to San Francisco is not noted.], alt: "The above image shows two similar triangles and how they are used with maps. The smaller triangle on the left shows San Francisco, Las Vegas and Los Angeles on the three points. San Francisco to Los Angeles is 1.3 inches. Los Angeles to Las Vegas is 1 inch. Las Vegas to San Francisco is 2.1 inches. The second larger triangle shows the same points. The distance from San Francisco to Los Angeles is x. The distance from Los Angeles to Las Vegas is 270 miles. The distance from Las Vegas to San Francisco is not noted.", caption: none)

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Read] the problem. Draw the figures and label with the given information.], [The figures are shown above.]),
  [#emph[Identify] what we are looking for.], [The actual distance from Los Angeles to San Francisco.],
  [#emph[Name] the variables.], [Let #math.equation(block: false, alt: "x equals")[$x =$] distance from Los Angeles to San Francisco.],
  [#emph[Translate] into an equation. Since the triangles #linebreak() are similar, the corresponding sides are #linebreak() proportional. We'll make the numerators #linebreak() "miles" and the denominators "inches."], [#figure(figph[A proportion for calculating distance: x miles / 1.3 inches = 270 miles / 1 inch.], alt: "A proportion for calculating distance: x miles / 1.3 inches = 270 miles / 1 inch.", caption: none)],
  [#emph[Solve] the equation.], [#figure(figph[A mathematical equation shows both sides being multiplied by 1.3. The left side is 1.3 times (x divided by 1.3), and the right side is 1.3 times (270 divided by 1).], alt: "A mathematical equation shows both sides being multiplied by 1.3. The left side is 1.3 times (x divided by 1.3), and the right side is 1.3 times (270 divided by 1).", caption: none)],
  [], [#figure(figph[The image displays the mathematical equation 'x = 351' in black text against a plain white background, indicating a variable assigned a numerical value.], alt: "The image displays the mathematical equation 'x = 351' in black text against a plain white background, indicating a variable assigned a numerical value.", caption: none)],
  [#emph[Check.]], [],
  [On the map, the distance from Los Angeles to #linebreak() San Francisco is more than the distance from #linebreak() Los Angeles to Las Vegas. Since 351 is more #linebreak() than 270 the answer makes sense.], [],
  [#figure(figph[A math problem demonstrating how to check if x=351 is the correct solution for the proportion x miles / 1.3 inches = 270 miles / 1 inch. The calculation confirms that 351/1.3 equals 270.], alt: "A math problem demonstrating how to check if x=351 is the correct solution for the proportion x miles / 1.3 inches = 270 miles / 1 inch. The calculation confirms that 351/1.3 equals 270.", caption: none)], [],
  [#emph[Answer] the question.], [The distance from Los Angeles to San Francisco is 351 miles.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

#figure(figph[The above image is a triangle with one side labeled “Seattle, 4.5 inches”. The other side is labeled “Portland 3.5 inches”. The third side is labeled 1.5 inches. The vertex is labeled “Boise.”], alt: "The above image is a triangle with one side labeled “Seattle, 4.5 inches”. The other side is labeled “Portland 3.5 inches”. The third side is labeled 1.5 inches. The vertex is labeled “Boise.”", caption: none)

#solutionbox[
150 miles
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Using the map above, find the distance from Portland to Boise.

#solutionbox[
350 miles
]
]

We can use #strong[similar figures] to find heights that we cannot directly measure.

#examplebox("Example 10")[][
Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Read the problem and draw a figure.], [#figure(figph[Two similar right triangles are shown. The larger triangle has height 'h' and a total base of 24. A vertical segment of length 6 stands on a part of the base, creating a smaller right triangle with a base of 8.], alt: "Two similar right triangles are shown. The larger triangle has height 'h' and a total base of 24. A vertical segment of length 6 stands on a part of the base, creating a smaller right triangle with a base of 8.", caption: none)]),
  [We are looking for #emph[h], the height of the tree.], [],
  [We will use similar triangles to write an equation.], [],
  [The small triangle is similar to the large triangle.], [#figure(figph[A mathematical equation shows h divided by 24 equals 6 divided by 8.], alt: "A mathematical equation shows h divided by 24 equals 6 divided by 8.", caption: none)],
  [Solve the proportion.], [#figure(figph[A mathematical equation shows '24(6/8) = 24(h/24)' where the number 24 is highlighted in red. This equation appears to be setting up a proportion or a problem to solve for 'h'.], alt: "A mathematical equation shows '24(6/8) = 24(h/24)' where the number 24 is highlighted in red. This equation appears to be setting up a proportion or a problem to solve for 'h'.", caption: none)],
  [Simplify.], [#figure(figph[A simple mathematical equation '18 = h' is displayed in gray text on a plain white background.], alt: "A simple mathematical equation '18 = h' is displayed in gray text on a plain white background.", caption: none)],
  [Check.], [],
  [Tyler's height is less than his shadow's length so it makes #linebreak() sense that the tree's height is less than the length of its shadow.], [],
  [#figure(figph[A step-by-step verification of the proportion 6/8 = h/24. The image shows substituting h=18 and simplifying both sides to 3/4 = 3/4, confirming that h=18 is the correct solution.], alt: "A step-by-step verification of the proportion 6/8 = h/24. The image shows substituting h=18 and simplifying both sides to 3/4 = 3/4, confirming that h=18 is the correct solution.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

#solutionbox[
40 feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A pine tree casts a shadow of 80 feet next to a 30-foot tall building which casts a 40 feet shadow. How tall is the pine tree?

#solutionbox[
60 feet
]
]


=== Key Concepts

- #strong[Property of Similar Triangles]
- If #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$], then their corresponding angle measures are equal and their corresponding sides are in the same ratio.
- #strong[Problem Solving Strategy for Geometry Applications]
+ #strong[Read] the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for by choosing a variable to represent it.
+ #strong[Translate] into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
+ #strong[Solve the equation] using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.


==== Practice Makes Perfect

#strong[Solve Proportions]

In the following exercises, solve.

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

#solutionbox[
#math.equation(block: true, alt: "49")[$49$]
]

#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[
#math.equation(block: true, alt: "7")[$7$]
]

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

#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[
#math.equation(block: true, alt: "−11")[$−11$]
]

#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[
#math.equation(block: true, alt: "7")[$7$]
]

#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 2.7 over j equals the fraction 0.9 over 0.2")[$frac(2.7, j) = frac(0.9, 0.2)$]

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

#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 a over a plus 12 equals the fraction 4 over 7")[$frac(a, a + 12) = frac(4, 7)$]

#solutionbox[
#math.equation(block: true, alt: "16")[$16$]
]

#math.equation(block: true, alt: "the fraction b over b minus 16 equals the fraction 11 over 9")[$frac(b, b − 16) = frac(11, 9)$]

#math.equation(block: true, alt: "the fraction c over c minus 104 equals minus the fraction 5 over 8")[$frac(c, c − 104) = − frac(5, 8)$]

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

#math.equation(block: true, alt: "the fraction d over d minus 48 equals minus the fraction 13 over 3")[$frac(d, d − 48) = − frac(13, 3)$]

#math.equation(block: true, alt: "the fraction m plus 90 over 25 equals the fraction m plus 30 over 15")[$frac(m + 90, 25) = frac(m + 30, 15)$]

#solutionbox[
#math.equation(block: true, alt: "60")[$60$]
]

#math.equation(block: true, alt: "the fraction n plus 10 over 4 equals the fraction 40 minus n over 6")[$frac(n + 10, 4) = frac(40 − n, 6)$]

#math.equation(block: true, alt: "the fraction 2 p plus 4 over 8 equals the fraction p plus 18 over 6")[$frac(2 p + 4, 8) = frac(p + 18, 6)$]

#solutionbox[
#math.equation(block: true, alt: "30")[$30$]
]

#math.equation(block: true, alt: "the fraction q minus 2 over 2 equals the fraction 2 q minus 7 over 18")[$frac(q − 2, 2) = frac(2 q − 7, 18)$]

Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 45 pounds?

#solutionbox[
#math.equation(block: true, alt: "9 ml")[$9 #h(0.2em) "ml"$]
]

Brianna, who weighs 6 kg, just received her shots and needs a pain killer. The pain killer is prescribed for children at 15 milligrams (mg) for every 1 kilogram (kg) of the child’s weight. How many milligrams will the doctor prescribe?

A veterinarian prescribed Sunny, a 65 pound dog, an antibacterial medicine in case an infection emerges after her teeth were cleaned. If the dosage is 5 mg for every pound, how much medicine was Sunny given?

#solutionbox[
#math.equation(block: true, alt: "325 mg")[$325 #h(0.2em) "mg"$]
]

Belle, a 13 pound cat, is suffering from joint pain. How much medicine should the veterinarian prescribe if the dosage is 1.8 mg per pound?

A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?

#solutionbox[
#math.equation(block: true, alt: "159 calories")[$159 #h(0.2em) "calories"$]
]

One 12 ounce can of soda has 150 calories. If Josiah drinks the big 32 ounce size from the local mini-mart, how many calories does he get?

A new 7 ounce lemon ice drink is advertised for having only 140 calories. How many ounces could Sally drink if she wanted to drink just 100 calories?

#solutionbox[
#math.equation(block: true, alt: "5 oz")[$5 #h(0.2em) "oz"$]
]

Reese loves to drink healthy green smoothies. A 16 ounce serving of smoothie has 170 calories. Reese drinks 24 ounces of these smoothies in one day. How many calories of smoothie is he consuming in one day?

Janice is traveling to Canada and will change \$250 US dollars into Canadian dollars. At the current exchange rate, \$1 US is equal to \$1.01 Canadian. How many Canadian dollars will she get for her trip?

#solutionbox[
#math.equation(block: true, alt: "252.5 Canadian dollars")[$252.5 #h(0.2em) "Canadian dollars"$]
]

Todd is traveling to Mexico and needs to exchange \$450 into Mexican pesos. If each dollar is worth 12.29 pesos, how many pesos will he get for his trip?

Steve changed \$600 into 480 Euros. How many Euros did he receive for each US dollar?

#solutionbox[
#math.equation(block: true, alt: "0.80 Euros")[$0.80 #h(0.2em) "Euros"$]
]

Martha changed \$350 US into 385 Australian dollars. How many Australian dollars did she receive for each US dollar?

When traveling to Great Britain, Bethany exchanged her \$900 into 570 British pounds. How many pounds did she receive for each American dollar?

#solutionbox[
#math.equation(block: true, alt: "0.63 British pounds")[$0.63 #h(0.2em) "British pounds"$]
]

A missionary commissioned to South Africa had to exchange his \$500 for the South African Rand which is worth 12.63 for every dollar. How many Rand did he have after the exchange?

Ronald needs a morning breakfast drink that will give him at least 390 calories. Orange juice has 130 calories in one cup. How many cups does he need to drink to reach his calorie goal?

#solutionbox[
#math.equation(block: true, alt: "3 cups")[$3 #h(0.2em) "cups"$]
]

Sarah drinks a 32-ounce energy drink containing 80 calories per 12 ounce. How many calories did she drink?

Elizabeth is returning to the United States from Canada. She changes the remaining 300 Canadian dollars she has to \$230.05 in American dollars. What was \$1 worth in Canadian dollars?

#solutionbox[
#math.equation(block: true, alt: "1.30 Canadian dollars")[$1.30 #h(0.2em) "Canadian dollars"$]
]

Ben needs to convert \$1000 to the Japanese Yen. One American dollar is worth 123.3 Yen. How much Yen will he have?

A golden retriever weighing 85 pounds has diarrhea. His medicine is prescribed as 1 teaspoon per 5 pounds. How much medicine should he be given?

#solutionbox[
#math.equation(block: true, alt: "17 tsp")[$17 #h(0.2em) "tsp"$]
]

Five-year-old Lacy was stung by a bee. The dosage for the anti-itch liquid is 150 mg for her weight of 40 pounds. What is the dosage per pound?

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

#solutionbox[
#math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] 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 4 dozen cookies. Hilda needs to make 10 dozen cookies for the bake sale. How many cups of butter will she need?

#strong[Solve Similar Figure Applications]

In the following exercises, #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$]. Find the length of the indicated side.

#figure(figph[The above image shows two triangles. The larger triangle is labeled A B C. The length from A to B is 15. The length from A to C is b. The length from B to C is 9. The smaller triangle is labeled X Y Z. The length from X to Y is 10. The length from X to Z is 8. The length from Y to Z is x.], alt: "The above image shows two triangles. The larger triangle is labeled A B C. The length from A to B is 15. The length from A to C is b. The length from B to C is 9. The smaller triangle is labeled X Y Z. The length from X to Y is 10. The length from X to Z is 8. The length from Y to Z is x.", caption: none)

side #emph[b]

#solutionbox[
#math.equation(block: true, alt: "12")[$12$]
]

side #emph[x]

In the following exercises, #math.equation(block: false, alt: "Δ D E F")[$"Δ" D E F$] is similar to #math.equation(block: false, alt: "Δ N P Q")[$"Δ" N P Q$].

#figure(figph[The above image shows two triangles side by side. The smaller triangle is labeled D E F. The length from E to D is five-halves. The length from D to F is 7. The length from E to F is d. The second triangle is labeled N P Q. the length from P to N is zero. The length from N to Q is 9. The length from Q to P is eleven-halves.], alt: "The above image shows two triangles side by side. The smaller triangle is labeled D E F. The length from E to D is five-halves. The length from D to F is 7. The length from E to F is d. The second triangle is labeled N P Q. the length from P to N is zero. The length from N to Q is 9. The length from Q to P is eleven-halves.", caption: none)

Find the length of side #emph[d].

#solutionbox[
#math.equation(block: true, alt: "the fraction 77 over 18")[$frac(77, 18)$]
]

Find the length of side #emph[q].

In the following two exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle whose sides are shown in the figure below. The actual distance from New York to Chicago is 800 miles.

#figure(figph[The above image shows a triangle. Each angle is labeled, clockwise, “Chicago”, “New York”, and “Memphis”. The side that extends from Chicago to New York is labeled 8 inches. The side that extends from New York to Memphis is labeled 9.5 inches and the side extending from Memphis to Chicago is labeled 5.4 inches.], alt: "The above image shows a triangle. Each angle is labeled, clockwise, “Chicago”, “New York”, and “Memphis”. The side that extends from Chicago to New York is labeled 8 inches. The side that extends from New York to Memphis is labeled 9.5 inches and the side extending from Memphis to Chicago is labeled 5.4 inches.", caption: none)

Find the actual distance from New York to Memphis.

#solutionbox[
950 miles
]

Find the actual distance from Chicago to Memphis.

In the following two exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle whose sides are shown in the figure below. The actual distance from Atlanta to New Orleans is 420 miles.

#figure(figph[The above image shows a triangle. Each angle is labeled, clockwise, “Atlanta”, “Miami”, and “New Orleans”. The side that extends from Atlanta to Miami is labeled 3 inches. The side that extends from Miami to New Orleans is labeled 3.4 inches and the side extending from New Orleans to Atlanta is labeled 2.1 inches.], alt: "The above image shows a triangle. Each angle is labeled, clockwise, “Atlanta”, “Miami”, and “New Orleans”. The side that extends from Atlanta to Miami is labeled 3 inches. The side that extends from Miami to New Orleans is labeled 3.4 inches and the side extending from New Orleans to Atlanta is labeled 2.1 inches.", caption: none)

Find the actual distance from New Orleans to Miami.

#solutionbox[
680 miles
]

Find the actual distance from Atlanta to Miami.

A 2 foot tall dog casts a 3 foot shadow at the same time a cat casts a one foot shadow. How tall is the cat?

#solutionbox[
#math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] foot #math.equation(block: false, alt: "open parenthesis 8 in close parenthesis")[$( 8 #h(0.2em) "in" )$]
]

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?

The tower portion of a windmill is 212 feet tall. A six foot tall person standing next to the tower casts a seven foot shadow. How long is the windmill’s shadow?

#solutionbox[
#math.equation(block: true, alt: "247.3 feet")[$247.3 #h(0.2em) "feet"$]
]

The height of the Statue of Liberty is 305 feet. Nicole, who is standing next to the statue, casts a 6 foot shadow and she is 5 feet tall. How long should the shadow of the statue be?


==== Everyday Math

#strong[Heart Rate] At the gym, Carol takes her pulse for 10 seconds and counts 19 beats.

+ ⓐ How many beats per minute is this?
+ ⓑ Has Carol met her target heart rate of 140 beats per minute?

#solutionbox[
114 beats per minute #linebreak() 
no
]

#strong[Heart Rate] Kevin wants to keep his heart rate at 160 beats per minute while training. During his workout he counts 27 beats in 10 seconds.

+ ⓐ How many beats per minute is this?
+ ⓑ Has Kevin met his target heart rate?

#strong[Cost of a Road Trip] Jesse’s car gets 30 miles per gallon of gas.

+ ⓐ If Las Vegas is 285 miles away, how many gallons of gas are needed to get there and then home?
+ ⓑ If gas is \$3.09 per gallon, what is the total cost of the gas for the trip?

#solutionbox[
19 gallons #linebreak() 
\$58.71
]

#strong[Cost of a Road Trip] Danny wants to drive to Phoenix to see his grandfather. Phoenix is 370 miles from Danny’s home and his car gets 18.5 miles per gallon.

+ ⓐ How many gallons of gas will Danny need to get to and from Phoenix?
+ ⓑ If gas is \$3.19 per gallon, what is the total cost for the gas to drive to see his grandfather?

#strong[Lawn Fertilizer] Phil wants to fertilize his lawn. Each bag of fertilizer covers about 4,000 square feet of lawn. Phil’s lawn is approximately 13,500 square feet. How many bags of fertilizer will he have to buy?

#solutionbox[
4 bags
]

#strong[House Paint] April wants to paint the exterior of her house. One gallon of paint covers about 350 square feet, and the exterior of the house measures approximately 2000 square feet. How many gallons of paint will she have to buy?

#strong[Cooking] Natalia’s pasta recipe calls for 2 pounds of pasta for 1 quart of sauce. How many pounds of pasta should Natalia cook if she has 2.5 quarts of sauce?

#solutionbox[
5 pounds
]

#strong[Heating Oil] A 275 gallon oil tank costs \$400 to fill. How much would it cost to fill a 180 gallon oil tank?


==== Writing Exercises

Marisol solves the proportion #math.equation(block: false, alt: "the fraction 144 over a equals the fraction 9 over 4")[$frac(144, a) = frac(9, 4)$] by ‘cross multiplying’, so her first step looks like #math.equation(block: false, alt: "4 times 144 equals 9 times a .")[$4 · 144 = 9 · a .$] Explain how this differs from the method of solution shown.

#solutionbox[
Answers will vary.
]

Find a printed map and then write and solve an application problem similar to .


==== Self Check

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

#figure(figph[This table has three rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “solve proportions”. The next row reads, “solve similar figure applications”. The remaining columns are blank.], alt: "This table has three rows and four columns. The first row is a header row and it labels each column. The first column is labeled \"I can …\", the second \"Confidently\", the third “With some help” and the last \"No–I don’t get it\". In the “I can…” column the next row reads “solve proportions”. The next row reads, “solve similar figure applications”. The remaining columns are blank.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?