#set document(title: "1.5 Visualize Fractions", author: "OpenStax / XYZ Homework")
#set page(width: 8.5in, height: auto, margin: 1in)
#import "@preview/cetz:0.5.2"
#set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en")
#show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math"))
#set par(justify: true, leading: 0.62em, spacing: 0.9em)
#set enum(spacing: 1.1em)   // room between list items so tall inline fractions don't collide
#set list(spacing: 1.1em)
#set table(stroke: 0.5pt + rgb("#c7ccd3"))

#let BLUE = rgb("#183B6F")     // brand navy — section bars + example/solution labels (white on navy 11.09:1)
#let ORANGE = rgb("#A94509")   // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text)
#let RED = rgb("#DC2626")      // brand error-600
#let GREEN = rgb("#059669")    // brand success-600 (decoration only; small green text uses green-text #007942)

#show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt,
  fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt,
  text(fill: white, weight: "bold", size: 19pt, it.body))
#show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt,
  fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt,
  text(fill: white, weight: "bold", size: 12pt, it.body))
#show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body)
#show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body)

#let examplebox(label, title, body) = block(width: 100%, breakable: true,
  fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt,
  above: 12pt, below: 12pt)[
  #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt,
    text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]]
  #body]

// rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade
#let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint,
    stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt),
    above: 11pt, below: 11pt)[
    #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)]
    #linebreak() #body]

#let solutionbox(body) = block(above: 4pt, below: 8pt)[
  #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body]

#let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"),
  stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[
  #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))]

// Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a
// consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the
// print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a
// degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero.
#let _STD_W = 3.5
#let _WIDE_W = 5.6
#let _MAX_H = 3.4
#let _ASPECT_WIDE = 2.2
#let _UPSCALE_MAX = 1.15
#let stdfig(body) = context {
  let m = measure(body)
  let w = calc.max(m.width / 1in, 0.01)
  let h = calc.max(m.height / 1in, 0.01)
  let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W }
  let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX)
  align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body)))
}

#show figure: set block(breakable: false)
#set figure(gap: 8pt)
#show figure.caption: set text(size: 8.5pt, fill: rgb("#555"))


== 1.5#h(0.6em)Visualize Fractions

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A more thorough introduction to the topics covered in this section can be found in the #emph[Prealgebra] chapter, #strong[Fractions].
]


=== Find Equivalent Fractions

#strong[Fractions] are a way to represent parts of a whole. The fraction #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. The fraction #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] represents two of three equal parts. In the fraction #math.equation(block: false, alt: "the fraction 2 over 3 ,")[$frac(2, 3) ,$] the 2 is called the #strong[numerator] and the 3 is called the #strong[denominator].

#figure(figph[Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled “one third” in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.], alt: "Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled “one third” in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.", caption: [The circle on the left has been divided into 3 equal parts. Each part is #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] of the 3 equal parts. In the circle on the right, #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] of the circle is shaded (2 of the 3 equal parts).])

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

A #strong[fraction] is written #math.equation(block: false, alt: "the fraction a over b ,")[$frac(a, b) ,$] where #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$] and

- #emph[a] is the #strong[numerator] and #emph[b] is the #strong[denominator].

A fraction represents parts of a whole. The denominator #emph[b] is the number of equal parts the whole has been divided into, and the numerator #emph[a] indicates how many parts are included.
]

If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate #math.equation(block: false, alt: "the fraction 6 over 6")[$frac(6, 6)$] pieces, or, in other words, one whole pie.

#figure(figph[A circle is shown and is divided into six section. All sections are shaded.], alt: "A circle is shown and is divided into six section. All sections are shaded.", caption: none)

So #math.equation(block: false, alt: "the fraction 6 over 6 equals 1 .")[$frac(6, 6) = 1 .$] This leads us to the property of one that tells us that any number, except zero, divided by itself is 1.

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

#math.equation(block: true, alt: "the fraction a over a equals 1, open parenthesis a not equal to 0 close parenthesis")[$frac(a, a) = 1 &  &  & ( a ≠ 0 )$]Any number, except zero, divided by itself is one.
]

If a pie was cut in #math.equation(block: false, alt: "6")[$6$] pieces and we ate all 6, we ate #math.equation(block: false, alt: "the fraction 6 over 6")[$frac(6, 6)$] pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate #math.equation(block: false, alt: "the fraction 8 over 8")[$frac(8, 8)$] pieces, or one whole pie. We ate the same amount—one whole pie.

The fractions #math.equation(block: false, alt: "the fraction 6 over 6")[$frac(6, 6)$] and #math.equation(block: false, alt: "the fraction 8 over 8")[$frac(8, 8)$] have the same value, 1, and so they are called equivalent fractions. #strong[Equivalent fractions] are fractions that have the same value.

Let’s think of pizzas this time.  shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] is equivalent to #math.equation(block: false, alt: "the fraction 4 over 8 .")[$frac(4, 8) .$] In other words, they are #strong[equivalent fractions].

#figure(figph[A circle is shown that is divided into eight equal wedges by lines. The left side of the circle is a pizza with four sections making up the pizza slices. The right side has four shaded sections. Below the diagram is the fraction four eighths.], alt: "A circle is shown that is divided into eight equal wedges by lines. The left side of the circle is a pizza with four sections making up the pizza slices. The right side has four shaded sections. Below the diagram is the fraction four eighths.", caption: [Since the same amount is of each pizza is shaded, we see that #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] is equivalent to #math.equation(block: false, alt: "the fraction 4 over 8 .")[$frac(4, 8) .$] They are equivalent fractions.])

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

#strong[Equivalent fractions] are fractions that have the same value.
]

How can we use mathematics to change #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] into #math.equation(block: false, alt: "the fraction 4 over 8 ?")[$frac(4, 8) ?$] How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into #math.equation(block: false, alt: "8")[$8$] pieces instead of just 2. Mathematically, what we’ve described could be written like this as #math.equation(block: false, alt: "the fraction 1 times 4 over 2 times 4 equals the fraction 4 over 8 .")[$frac(1 · 4, 2 · 4) = frac(4, 8) .$]

#figure(figph[A circle is shown and is divided in half by a vertical black line. It is further divided into eighths by the addition of dotted red lines.], alt: "A circle is shown and is divided in half by a vertical black line. It is further divided into eighths by the addition of dotted red lines.", caption: [Cutting each half of the pizza into #math.equation(block: false, alt: "4")[$4$] pieces, gives us pizza cut into 8 pieces: #math.equation(block: false, alt: "the fraction 1 times 4 over 2 times 4 equals the fraction 4 over 8 .")[$frac(1 · 4, 2 · 4) = frac(4, 8) .$]])

This model leads to the following property:

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

If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 ,")[$b ≠ 0 , c ≠ 0 ,$] then

#math.equation(block: true, alt: "the fraction a over b equals the fraction a times c over b times c")[$frac(a, b) = frac(a · c, b · c)$]
]

If we had cut the pizza differently, we could get

#figure(figph[An image shows three rows of fractions. In the first row are the fractions “1, times 2, divided by 2, times 2, equals two fourths”. Next to this is the word “so” and the fraction “one half, equals two fourths. The second row reads “1, times 3, divided by 2 times 3, equals three sixths”. Next to this is the word “so” and the fraction “one half equals, three sixths”. The third row reads “1 times 10, divided by 2 times 10, ten twentieths”. Next to this is the word “so” and the fraction “one half equals, ten twentieths”.], alt: "An image shows three rows of fractions. In the first row are the fractions “1, times 2, divided by 2, times 2, equals two fourths”. Next to this is the word “so” and the fraction “one half, equals two fourths. The second row reads “1, times 3, divided by 2 times 3, equals three sixths”. Next to this is the word “so” and the fraction “one half equals, three sixths”. The third row reads “1 times 10, divided by 2 times 10, ten twentieths”. Next to this is the word “so” and the fraction “one half equals, ten twentieths”.", caption: none)

So, we say #math.equation(block: false, alt: "the fraction 1 over 2 , the fraction 2 over 4 , the fraction 3 over 6 , and the fraction 10 over 20")[$frac(1, 2) , frac(2, 4) , frac(3, 6) , #h(0.2em) "and" #h(0.2em) frac(10, 20)$] are equivalent fractions.

#examplebox("Example 1")[][
Find three fractions equivalent to #math.equation(block: false, alt: "the fraction 2 over 5 .")[$frac(2, 5) .$]

#solutionbox[
To find a fraction equivalent to #math.equation(block: false, alt: "the fraction 2 over 5 ,")[$frac(2, 5) ,$] we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let’s multiply them by 2, 3, and then 5. #linebreak() 
#figure(figph[A row of fractions reads “2 times 2, divided by 5 times 2, equals four tenths”. Next to this is “2, times 3, divided by 5 times 3, equals six fifteenths”. Next to this is “2 times 5, divided by 5 times 5, equals ten twenty-fifths”.], alt: "A row of fractions reads “2 times 2, divided by 5 times 2, equals four tenths”. Next to this is “2, times 3, divided by 5 times 3, equals six fifteenths”. Next to this is “2 times 5, divided by 5 times 5, equals ten twenty-fifths”.", caption: none)

 #linebreak() 
So, #math.equation(block: false, alt: "the fraction 4 over 10 , the fraction 6 over 15 , and the fraction 10 over 25")[$frac(4, 10) , frac(6, 15) , #h(0.2em) "and" #h(0.2em) frac(10, 25)$] are equivalent to #math.equation(block: false, alt: "the fraction 2 over 5 .")[$frac(2, 5) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three fractions equivalent to #math.equation(block: false, alt: "the fraction 3 over 5 .")[$frac(3, 5) .$]

#solutionbox[
#math.equation(block: false, alt: "the fraction 6 over 10 , the fraction 9 over 15 , the fraction 12 over 20 ;")[$frac(6, 10) , frac(9, 15) , frac(12, 20) ;$] answers may vary
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three fractions equivalent to #math.equation(block: false, alt: "the fraction 4 over 5 .")[$frac(4, 5) .$]

#solutionbox[
#math.equation(block: false, alt: "the fraction 8 over 10 , the fraction 12 over 15 , the fraction 16 over 20 ;")[$frac(8, 10) , frac(12, 15) , frac(16, 20) ;$] answers may vary
]
]


=== Simplify Fractions

A fraction is considered #strong[simplified] if there are no common factors, other than 1, in its numerator and denominator.

For example,

- #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] is simplified because there are no common factors of 2 and 3.
- #math.equation(block: false, alt: "the fraction 10 over 15")[$frac(10, 15)$] is not simplified because #math.equation(block: false, alt: "5")[$5$] is a common factor of 10 and 15.

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

A fraction is considered #strong[simplified] if there are no common factors in its numerator and denominator.
]

The phrase #emph[reduce a fraction] means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

In , we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.

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

If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 ,")[$b ≠ 0 , c ≠ 0 ,$]

#math.equation(block: true, alt: "then the fraction a over b equals the fraction a times c over b times c and the fraction a times c over b times c equals the fraction a over b")[$"then" #h(1em) frac(a, b) = frac(a · c, b · c) #h(1em) "and" #h(1em) frac(a · c, b · c) = frac(a, b)$]
]

#examplebox("Example 2")[][
Simplify: #math.equation(block: false, alt: "minus the fraction 32 over 56 .")[$− #h(0.2em) frac(32, 56) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 32 over 56")[$− #h(0.2em) frac(32, 56)$]]),
  [Rewrite the numerator and denominator showing the common factors.], [#figure(figph[A mathematical fraction displaying negative (4 multiplied by 8) over (7 multiplied by 8), with the common factor '8' highlighted in red in both numerator and denominator.], alt: "A mathematical fraction displaying negative (4 multiplied by 8) over (7 multiplied by 8), with the common factor '8' highlighted in red in both numerator and denominator.", caption: none)],
  [Simplify using the equivalent fractions property.], [#math.equation(block: false, alt: "minus the fraction 4 over 7")[$− #h(0.2em) frac(4, 7)$]],
))

Notice that the fraction #math.equation(block: false, alt: "minus the fraction 4 over 7")[$− #h(0.2em) frac(4, 7)$] is simplified because there are no more common factors.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 42 over 54 .")[$− #h(0.2em) frac(42, 54) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 7 over 9")[$− #h(0.2em) frac(7, 9)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 45 over 81 .")[$− #h(0.2em) frac(45, 81) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 5 over 9")[$− #h(0.2em) frac(5, 9)$]
]
]

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into #strong[prime number]s. Then divide out the common factors using the equivalent fractions property.

#examplebox("Example 3")[How to Simplify a Fraction][
Simplify: #math.equation(block: false, alt: "minus the fraction 210 over 385 .")[$− #h(0.2em) frac(210, 385) .$]

#solutionbox[
#figure(figph[A table is shown with three columns and three rows. The first row of the left column reads “Step 1. Rewrite the numerator and denominator to show the common factors. If needed, use a factor tree”. Next to this in the middle column, it reads “rewrite 210 and 285 as the product of the primes. Next to this in the right column, it reads “negative 210 divided by 385.” Under this, is the equation “two times three times five times seven.” The five and 7 are blue and red respectively.], alt: "A table is shown with three columns and three rows. The first row of the left column reads “Step 1. Rewrite the numerator and denominator to show the common factors. If needed, use a factor tree”. Next to this in the middle column, it reads “rewrite 210 and 285 as the product of the primes. Next to this in the right column, it reads “negative 210 divided by 385.” Under this, is the equation “two times three times five times seven.” The five and 7 are blue and red respectively.", caption: none)

#figure(figph[The next row down reads “Step 2. Simplify using the equivalent fractions property by dividing out common factors.” Next to this in the middle column, it reads, “Mark the common factors 5 and 7.” Next to this in the right column, it has the equation 2 times, three times five, times seven over 5 times seven times 11. Both the 5 and the 7 are crossed out as common factors. Under this is the equation “negative two times 3 divided by 11.”], alt: "The next row down reads “Step 2. Simplify using the equivalent fractions property by dividing out common factors.” Next to this in the middle column, it reads, “Mark the common factors 5 and 7.” Next to this in the right column, it has the equation 2 times, three times five, times seven over 5 times seven times 11. Both the 5 and the 7 are crossed out as common factors. Under this is the equation “negative two times 3 divided by 11.”", caption: none)

#figure(figph[The next row reads, “Step 3. Multiply the remaining factors, if necessary.” Next to this in the right column is negative six elevenths.], alt: "The next row reads, “Step 3. Multiply the remaining factors, if necessary.” Next to this in the right column is negative six elevenths.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 69 over 120 .")[$− #h(0.2em) frac(69, 120) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 23 over 40")[$− #h(0.2em) frac(23, 40)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 120 over 192 .")[$− #h(0.2em) frac(120, 192) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 5 over 8")[$− #h(0.2em) frac(5, 8)$]
]
]

We now summarize the steps you should follow to simplify fractions.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simplify a Fraction.]

+ Rewrite the numerator and denominator to show the common factors. #linebreak() 
If needed, factor the numerator and denominator into prime numbers first.
+ Simplify using the equivalent fractions property by dividing out common factors.
+ Multiply any remaining factors, if needed.
]

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "the fraction 5 x over 5 y .")[$frac(5 x, 5 y) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 5 x over 5 y")[$frac(5 x, 5 y)$]]),
  [Rewrite showing the common factors, then divide out the common factors.], [#figure(figph[The image illustrates the simplification of the fraction 5x/5y. It shows how to rewrite the expression to identify common factors (5), divide them out, and then simplify to x/y.], alt: "The image illustrates the simplification of the fraction 5x/5y. It shows how to rewrite the expression to identify common factors (5), divide them out, and then simplify to x/y.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "the fraction x over y")[$frac(x, y)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 7 x over 7 y .")[$frac(7 x, 7 y) .$]

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

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

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


=== Multiply Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$]

#figure(figph[A rectangle made up of four squares in a row. The first three squares are shaded.], alt: "A rectangle made up of four squares in a row. The first three squares are shaded.", caption: none)

Now we’ll take #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] of #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$]

#figure(figph[A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.], alt: "A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.", caption: none)

Notice that now, the whole is divided into 8 equal parts. So #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 3 over 4 equals the fraction 3 over 8 .")[$frac(1, 2) · frac(3, 4) = frac(3, 8) .$]

To multiply fractions, we multiply the numerators and multiply the denominators.

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

If #math.equation(block: false, alt: "a , b , c and d")[$a , b , c #h(0.2em) "and" #h(0.2em) d$] are numbers where #math.equation(block: false, alt: "b not equal to 0 and d not equal to 0 ,")[$b ≠ 0 #h(0.2em) "and" #h(0.2em) d ≠ 0 ,$] then

#math.equation(block: true, alt: "the fraction a over b times the fraction c over d equals the fraction a c over b d")[$frac(a, b) · frac(c, d) = frac(a c, b d)$]To multiply fractions, multiply the numerators and multiply the denominators.
]

When #strong[multiplying fractions], the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the #strong[product] as the first step. In , we will multiply negative and a positive, so the product will be negative.

#examplebox("Example 5")[][
Multiply: #math.equation(block: false, alt: "minus the fraction 11 over 12 times the fraction 5 over 7 .")[$− #h(0.2em) frac(11, 12) · frac(5, 7) .$]

#solutionbox[
The first step is to find the sign of the product. Since the signs are the different, the product is negative.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 11 over 12 times the fraction 5 over 7")[$− #h(0.2em) frac(11, 12) · frac(5, 7)$]]),
  [Determine the sign of the product; multiply.], [#math.equation(block: false, alt: "minus the fraction 11 times 5 over 12 times 7")[$− #h(0.2em) frac(11 · 5, 12 · 7)$]],
  [Are there any common factors in the numerator and the demoninator? No.], [#math.equation(block: false, alt: "minus the fraction 55 over 84")[$− #h(0.2em) frac(55, 84)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "minus the fraction 10 over 28 times the fraction 8 over 15 .")[$− #h(0.2em) frac(10, 28) · frac(8, 15) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 4 over 21")[$− #h(0.2em) frac(4, 21)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "minus the fraction 9 over 20 times the fraction 5 over 12 .")[$− #h(0.2em) frac(9, 20) · frac(5, 12) .$]

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

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, #emph[a], can be written as #math.equation(block: false, alt: "the fraction a over 1 .")[$frac(a, 1) .$] So, for example, #math.equation(block: false, alt: "3 equals the fraction 3 over 1 .")[$3 = frac(3, 1) .$]

#examplebox("Example 6")[][
Multiply: #math.equation(block: false, alt: "minus the fraction 12 over 5 open parenthesis −20 x close parenthesis .")[$− #h(0.2em) frac(12, 5) ( −20 x ) .$]

#solutionbox[
Determine the sign of the product. The signs are the same, so the product is positive.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 12 over 5 open parenthesis −20 x close parenthesis")[$− #h(0.2em) frac(12, 5) ( −20 x )$]]),
  [Write #math.equation(block: false, alt: "20 x")[$20 x$] as a fraction.], [#math.equation(block: false, alt: "the fraction 12 over 5 open parenthesis the fraction 20 x over 1 close parenthesis")[$frac(12, 5) ( frac(20 x, 1) )$]],
  [Multiply.], [],
  [Rewrite 20 to show the common factor 5 and divide it out.], [#figure(figph[A mathematical expression showing the fraction (12 \* 4 \* 5x) divided by (5 \* 1), with the '5' in the numerator and denominator canceled out.], alt: "A mathematical expression showing the fraction (12 * 4 * 5x) divided by (5 * 1), with the '5' in the numerator and denominator canceled out.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "48 x")[$48 x$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction 11 over 3 open parenthesis −9 a close parenthesis .")[$frac(11, 3) ( −9 a ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction 13 over 7 open parenthesis −14 b close parenthesis .")[$frac(13, 7) ( −14 b ) .$]

#solutionbox[
#math.equation(block: true, alt: "−26 b")[$−26 b$]
]
]


=== Divide Fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary.

The #strong[reciprocal] of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] is #math.equation(block: false, alt: "the fraction 3 over 2 .")[$frac(3, 2) .$]

Notice that #math.equation(block: false, alt: "the fraction 2 over 3 times the fraction 3 over 2 equals 1 .")[$frac(2, 3) · frac(3, 2) = 1 .$] A number and its reciprocal multiply to 1.

To get a #strong[product] of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of #math.equation(block: false, alt: "minus the fraction 10 over 7")[$− #h(0.2em) frac(10, 7)$] is #math.equation(block: false, alt: "minus the fraction 7 over 10 ,")[$− #h(0.2em) frac(7, 10) ,$] since #math.equation(block: false, alt: "minus the fraction 10 over 7 open parenthesis minus the fraction 7 over 10 close parenthesis equals 1 .")[$− #h(0.2em) frac(10, 7) ( − #h(0.2em) frac(7, 10) ) = 1 .$]

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

The #strong[reciprocal] of #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$] is #math.equation(block: false, alt: "the fraction b over a .")[$frac(b, a) .$]

A number and its reciprocal multiply to one #math.equation(block: false, alt: "the fraction a over b times the fraction b over a equals 1 .")[$frac(a, b) · frac(b, a) = 1 .$]
]

To divide fractions, we multiply the first fraction by the reciprocal of the second.

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

If #math.equation(block: false, alt: "a , b , c and d")[$a , b , c #h(0.2em) "and" #h(0.2em) d$] are numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 and d not equal to 0 ,")[$b ≠ 0 , c ≠ 0 #h(0.2em) "and" #h(0.2em) d ≠ 0 ,$] then

#math.equation(block: true, alt: "the fraction a over b divided by the fraction c over d equals the fraction a over b times the fraction d over c")[$frac(a, b) ÷ frac(c, d) = frac(a, b) · frac(d, c)$]To divide fractions, we multiply the first fraction by the reciprocal of the second.
]

We need to say #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 and d not equal to 0")[$b ≠ 0 , c ≠ 0 #h(0.2em) "and" #h(0.2em) d ≠ 0$] to be sure we don’t divide by zero!

#examplebox("Example 7")[][
Divide: #math.equation(block: false, alt: "minus the fraction 2 over 3 divided by the fraction n over 5 .")[$− #h(0.2em) frac(2, 3) ÷ frac(n, 5) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 2 over 3 divided by the fraction n over 5")[$− #h(0.2em) frac(2, 3) ÷ frac(n, 5)$]]),
  [To divide, multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "minus the fraction 2 over 3 times the fraction 5 over n")[$− #h(0.2em) frac(2, 3) · frac(5, n)$]],
  [Multiply.], [#math.equation(block: false, alt: "minus the fraction 10 over 3 n")[$− #h(0.2em) frac(10, 3 n)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "minus the fraction 3 over 5 divided by the fraction p over 7 .")[$− #h(0.2em) frac(3, 5) ÷ frac(p, 7) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 21 over 5 p")[$− #h(0.2em) frac(21, 5 p)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: #math.equation(block: false, alt: "minus the fraction 5 over 8 divided by the fraction q over 3 .")[$− #h(0.2em) frac(5, 8) ÷ frac(q, 3) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 15 over 8 q")[$− #h(0.2em) frac(15, 8 q)$]
]
]

#examplebox("Example 8")[][
Find the quotient: #math.equation(block: false, alt: "minus the fraction 7 over 18 divided by open parenthesis minus the fraction 14 over 27 close parenthesis .")[$− #h(0.2em) frac(7, 18) ÷ ( − #h(0.2em) frac(14, 27) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 7 over 18 divided by open parenthesis − the fraction 14 over 27 close parenthesis")[$− #h(0.2em) frac(7, 18) ÷ ( − #h(0.2em) frac(14, 27) )$]]),
  [To divide, multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "minus the fraction 7 over 18 times − the fraction 27 over 14")[$− #h(0.2em) frac(7, 18) ⋅ − #h(0.2em) frac(27, 14)$]],
  [Determine the sign of the product, and then multiply..], [#math.equation(block: false, alt: "the fraction 7 times 27 over 18 times 14")[$frac(7 ⋅ 27, 18 ⋅ 14)$]],
  [Rewrite showing common factors.], [#figure(figph[A mathematical fraction is shown where common factors (7 and 9) in the numerator (7\*9\*3) and denominator (9\*2\*7\*2) are crossed out, indicating cancellation to simplify the expression. The remaining terms are 3 in the numerator and 2\*2 in the denominator.], alt: "A mathematical fraction is shown where common factors (7 and 9) in the numerator (7*9*3) and denominator (9*2*7*2) are crossed out, indicating cancellation to simplify the expression. The remaining terms are 3 in the numerator and 2*2 in the denominator.", caption: none)],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 3 over 2 times 2")[$frac(3, 2 ⋅ 2)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "minus the fraction 7 over 27 divided by open parenthesis minus the fraction 35 over 36 close parenthesis .")[$− #h(0.2em) frac(7, 27) ÷ ( − #h(0.2em) frac(35, 36) ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the quotient: #math.equation(block: false, alt: "minus the fraction 5 over 14 divided by open parenthesis minus the fraction 15 over 28 close parenthesis .")[$− #h(0.2em) frac(5, 14) ÷ ( − #h(0.2em) frac(15, 28) ) .$]

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

There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.

- “To multiply fractions, multiply the numerators and multiply the denominators.”
- “To divide fractions, multiply the first fraction by the reciprocal of the second.”

Another way is to keep two examples in mind:

#figure(figph[This is an image with two columns. The first column reads “One fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the center of each one representing one half. The halves are labeled “one half”. Under this is the equation “2 times 1 fourth”. Under this is another equation “two over 1 times 1 fourth.” Under this is the fraction two fourths and under this is the fraction one half. The next column reads “there are eight quarters in two dollars.” Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation “two over one divided by one fourth.” Under this is two over one times four over one. Under this is the answer “8”.], alt: "This is an image with two columns. The first column reads “One fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the center of each one representing one half. The halves are labeled “one half”. Under this is the equation “2 times 1 fourth”. Under this is another equation “two over 1 times 1 fourth.” Under this is the fraction two fourths and under this is the fraction one half. The next column reads “there are eight quarters in two dollars.” Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation “two over one divided by one fourth.” Under this is two over one times four over one. Under this is the answer “8”.", caption: none)

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a #strong[complex fraction].

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

A #strong[complex fraction] is a fraction in which the numerator or the denominator contains a fraction.
]

Some examples of complex fractions are:

#math.equation(block: true, alt: "the fraction the fraction 6 over 7 over 3 the fraction the fraction 3 over 4 over the fraction 5 over 8 the fraction the fraction x over 2 over the fraction 5 over 6")[$frac(frac(6, 7), 3) #h(1em) frac(frac(3, 4), frac(5, 8)) #h(1em) frac(frac(x, 2), frac(5, 6))$]To simplify a complex fraction, we remember that the fraction bar means #strong[division]. For example, the complex fraction #math.equation(block: false, alt: "the fraction the fraction 3 over 4 over the fraction 5 over 8")[$frac(frac(3, 4), frac(5, 8))$] means #math.equation(block: false, alt: "the fraction 3 over 4 divided by the fraction 5 over 8 .")[$frac(3, 4) ÷ frac(5, 8) .$]

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "the fraction the fraction 3 over 4 over the fraction 5 over 8 .")[$frac(frac(3, 4), frac(5, 8)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the fraction 3 over 4 over the fraction 5 over 8")[$frac(frac(3, 4), frac(5, 8))$]]),
  [Rewrite as division.], [#math.equation(block: false, alt: "the fraction 3 over 4 divided by the fraction 5 over 8")[$frac(3, 4) ÷ frac(5, 8)$]],
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 3 over 4 times the fraction 8 over 5")[$frac(3, 4) ⋅ frac(8, 5)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 3 times 8 over 4 times 5")[$frac(3 ⋅ 8, 4 ⋅ 5)$]],
  [Look for common factors.], [#figure(figph[An example of fraction simplification, where the common factor '4' is canceled from both the numerator and denominator of (3\*4\*2)/(4\*5).], alt: "An example of fraction simplification, where the common factor '4' is canceled from both the numerator and denominator of (3*4*2)/(4*5).", caption: none)],
  [Divide out common factors and simplify.], [#math.equation(block: false, alt: "the fraction 6 over 5")[$frac(6, 5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the fraction 2 over 3 over the fraction 5 over 6 .")[$frac(frac(2, 3), frac(5, 6)) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the fraction 3 over 7 over the fraction 6 over 11 .")[$frac(frac(3, 7), frac(6, 11)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 14")[$frac(11, 14)$]
]
]

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "the fraction the fraction x over 2 over the fraction x y over 6 .")[$frac(frac(x, 2), frac(x y, 6)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction the fraction x over 2 over the fraction x y over 6")[$frac(frac(x, 2), frac(x y, 6))$]]),
  [Rewrite as division.], [#math.equation(block: false, alt: "the fraction x over 2 divided by the fraction x y over 6")[$frac(x, 2) ÷ frac(x y, 6)$]],
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction x over 2 times the fraction 6 over x y")[$frac(x, 2) ⋅ frac(6, x y)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction x times 6 over 2 times x y")[$frac(x ⋅ 6, 2 ⋅ x y)$]],
  [Look for common factors.], [#figure(figph[Simplifying a fraction by canceling out common variables 'x' and 'z' from the numerator and denominator, resulting in 3 over y.], alt: "Simplifying a fraction by canceling out common variables 'x' and 'z' from the numerator and denominator, resulting in 3 over y.", caption: none)],
  [Divide out common factors and simplify.], [#math.equation(block: false, alt: "the fraction 3 over y")[$frac(3, y)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the fraction a over 8 over the fraction a b over 6 .")[$frac(frac(a, 8), frac(a b, 6)) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the fraction p over 2 over the fraction p q over 8 .")[$frac(frac(p, 2), frac(p q, 8)) .$]

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


=== Simplify Expressions with a Fraction Bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A #strong[fraction bar] acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression #math.equation(block: false, alt: "the fraction 5 minus 3 over 7 plus 1 ,")[$frac(5 − 3, 7 + 1) ,$] we first simplify the numerator and the denominator separately. Then we divide.

#math.equation(block: true, alt: "the fraction 5 minus 3 over 7 plus 1")[$frac(5 − 3, 7 + 1)$]#math.equation(block: true, alt: "the fraction 2 over 8")[$frac(2, 8)$]#math.equation(block: true, alt: "the fraction 1 over 4")[$frac(1, 4)$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simplify an Expression with a Fraction Bar.]

+ Simplify the expression in the numerator. Simplify the expression in the denominator.
+ Simplify the fraction.
]

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "the fraction 4 minus 2 open parenthesis 3 close parenthesis over 2 squared plus 2 .")[$frac(4 − 2 ( 3 ), 2^(2) + 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 4 minus 2 open parenthesis 3 close parenthesis over 2 squared plus 2")[$frac(4 − 2 ( 3 ), 2^(2) + 2)$]]),
  [Use the order of operations to simpliy the numerator and the denominator.], [#math.equation(block: false, alt: "the fraction 4 minus 6 over 4 plus 2")[$frac(4 − 6, 4 + 2)$]],
  [Simplify the numerator and the denominator.], [#math.equation(block: false, alt: "the fraction −2 over 6")[$frac(−2, 6)$]],
  [Simplify. A negative divided by a positive is negative.], [#math.equation(block: false, alt: "minus the fraction 1 over 3")[$− #h(0.2em) frac(1, 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 6 minus 3 open parenthesis 5 close parenthesis over 3 squared plus 3 .")[$frac(6 − 3 ( 5 ), 3^(2) + 3) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 3 over 4")[$− #h(0.2em) frac(3, 4)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 4 minus 4 open parenthesis 6 close parenthesis over 3 squared plus 3 .")[$frac(4 − 4 ( 6 ), 3^(2) + 3) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 5 over 3")[$− #h(0.2em) frac(5, 3)$]
]
]

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

#math.equation(block: true, alt: "the fraction −1 over 3 equals minus the fraction 1 over 3, the fraction negative over positive equals negative; the fraction 1 over −3 equals minus the fraction 1 over 3, the fraction positive over negative equals negative")[$frac(−1, 3) = − #h(0.2em) frac(1, 3) &  &  &  &  & frac("negative", "positive") = "negative" \ frac(1, −3) = − #h(0.2em) frac(1, 3) &  &  &  &  & frac("positive", "negative") = "negative"$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Placement of Negative Sign in a Fraction]

For any positive numbers #emph[a] and #emph[b],

#math.equation(block: true, alt: "the fraction − a over b equals the fraction a over − b equals minus the fraction a over b")[$frac("−" a, b) = frac(a, "−" b) = − #h(0.2em) frac(a, b)$]
]

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "the fraction 4 open parenthesis −3 close parenthesis plus 6 open parenthesis −2 close parenthesis over −3 open parenthesis 2 close parenthesis minus 2 .")[$frac(4 ( −3 ) + 6 ( −2 ), −3 ( 2 ) − 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 4 open parenthesis −3 close parenthesis plus 6 open parenthesis −2 close parenthesis over −3 open parenthesis 2 close parenthesis minus 2")[$frac(4 ( −3 ) + 6 ( −2 ), −3 ( 2 ) − 2)$]]),
  [Multiply.], [#math.equation(block: false, alt: "the fraction −12 plus open parenthesis −12 close parenthesis over −6 minus 2")[$frac(−12 + ( −12 ), −6 − 2)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction −24 over −8")[$frac(−24, −8)$]],
  [Divide.], [#math.equation(block: false, alt: "3")[$3$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 8 open parenthesis −2 close parenthesis plus 4 open parenthesis −3 close parenthesis over −5 open parenthesis 2 close parenthesis plus 3 .")[$frac(8 ( −2 ) + 4 ( −3 ), −5 ( 2 ) + 3) .$]

#solutionbox[
4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 7 open parenthesis −1 close parenthesis plus 9 open parenthesis −3 close parenthesis over −5 open parenthesis 3 close parenthesis minus 2 .")[$frac(7 ( −1 ) + 9 ( −3 ), −5 ( 3 ) − 2) .$]

#solutionbox[
2
]
]


=== Translate Phrases to Expressions with Fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.

The English words #strong[quotient] and #strong[ratio] are often used to describe fractions. Remember that “quotient” means #strong[division]. The quotient of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] is the result we get from dividing #math.equation(block: false, alt: "a")[$a$] by #math.equation(block: false, alt: "b ,")[$b ,$] or #math.equation(block: false, alt: "the fraction a over b .")[$frac(a, b) .$]

#examplebox("Example 13")[][
Translate the English phrase into an algebraic expression: the quotient of the difference of #emph[m] and #emph[n], and #emph[p].

#solutionbox[
We are looking for the #emph[quotient] #emph[of] the difference of #emph[m] and #emph[n], #emph[and] #emph[p]. This means we want to divide the difference of #math.equation(block: false, alt: "m and n by p .")[$m #h(0.2em) "and" #h(0.2em) n #h(0.2em) "by" #h(0.2em) p .$]

#math.equation(block: true, alt: "the fraction m minus n over p")[$frac(m − n, p)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the English phrase into an algebraic expression: the quotient of the difference of #emph[a] and #emph[b], and #emph[cd].

#solutionbox[
#math.equation(block: true, alt: "the fraction a minus b over c d")[$frac(a − b, c d)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the English phrase into an algebraic expression: the quotient of the sum of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q ,")[$q ,$] and #math.equation(block: false, alt: "r")[$r$]

#solutionbox[
#math.equation(block: true, alt: "the fraction p plus q over r")[$frac(p + q, r)$]
]
]


=== Key Concepts

- #strong[Equivalent Fractions Property:] If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 ,")[$b ≠ 0 , c ≠ 0 ,$] then #linebreak() 
#math.equation(block: false, alt: "the fraction a over b equals the fraction a times c over b times c")[$frac(a, b) = frac(a · c, b · c)$] and #math.equation(block: false, alt: "the fraction a times c over b times c equals the fraction a over b .")[$frac(a · c, b · c) = frac(a, b) .$]
- #strong[Fraction Division:] If #math.equation(block: false, alt: "a , b , c and d")[$a , b , c #h(0.2em) "and" #h(0.2em) d$] are numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 , and d not equal to 0 ,")[$b ≠ 0 , c ≠ 0 , #h(0.2em) "and" #h(0.2em) d ≠ 0 ,$] then #math.equation(block: false, alt: "the fraction a over b divided by the fraction c over d equals the fraction a over b times the fraction d over c .")[$frac(a, b) ÷ frac(c, d) = frac(a, b) · frac(d, c) .$] To divide fractions, multiply the first fraction by the reciprocal of the second.
- #strong[Fraction Multiplication:] If #math.equation(block: false, alt: "a , b , c and d")[$a , b , c #h(0.2em) "and" #h(0.2em) d$] are numbers where #math.equation(block: false, alt: "b not equal to 0 , and d not equal to 0 ,")[$b ≠ 0 , #h(0.2em) "and" #h(0.2em) d ≠ 0 ,$] then #math.equation(block: false, alt: "the fraction a over b times the fraction c over d equals the fraction a c over b d .")[$frac(a, b) · frac(c, d) = frac(a c, b d) .$] To multiply fractions, multiply the numerators and multiply the denominators.
- #strong[Placement of Negative Sign in a Fraction:] For any positive numbers #math.equation(block: false, alt: "a and b ,")[$a #h(0.2em) "and" #h(0.2em) b ,$] #math.equation(block: false, alt: "the fraction minus a over b equals the fraction a over minus b equals minus the fraction a over b .")[$frac(− a, b) = frac(a, − b) = − #h(0.2em) frac(a, b) .$]
- #strong[Property of One:] #math.equation(block: false, alt: "the fraction a over a equals 1 ;")[$frac(a, a) = 1 ;$] Any number, except zero, divided by itself is one.
- #strong[Simplify a Fraction]
+ Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
+ Simplify using the equivalent fractions property by dividing out common factors.
+ Multiply any remaining factors.
- #strong[Simplify an Expression with a Fraction Bar]
+ Simplify the expression in the numerator. Simplify the expression in the denominator.
+ Simplify the fraction.


==== Practice Makes Perfect

#strong[Find Equivalent Fractions]

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

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

#solutionbox[
#math.equation(block: false, alt: "the fraction 6 over 16 , the fraction 9 over 24 , the fraction 12 over 32")[$frac(6, 16) , frac(9, 24) , frac(12, 32)$] answers may vary
]

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

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

#solutionbox[
#math.equation(block: false, alt: "the fraction 10 over 18 , the fraction 15 over 27 , the fraction 20 over 36")[$frac(10, 18) , frac(15, 27) , frac(20, 36)$] answers may vary
]

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

#strong[Simplify Fractions]

In the following exercises, simplify.

#math.equation(block: true, alt: "minus the fraction 40 over 88")[$− #h(0.2em) frac(40, 88)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 5 over 11")[$− #h(0.2em) frac(5, 11)$]
]

#math.equation(block: true, alt: "minus the fraction 63 over 99")[$− #h(0.2em) frac(63, 99)$]

#math.equation(block: true, alt: "minus the fraction 108 over 63")[$− #h(0.2em) frac(108, 63)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 12 over 7")[$− #h(0.2em) frac(12, 7)$]
]

#math.equation(block: true, alt: "minus the fraction 104 over 48")[$− #h(0.2em) frac(104, 48)$]

#math.equation(block: true, alt: "the fraction 120 over 252")[$frac(120, 252)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 10 over 21")[$frac(10, 21)$]
]

#math.equation(block: true, alt: "the fraction 182 over 294")[$frac(182, 294)$]

#math.equation(block: true, alt: "minus the fraction 3 x over 12 y")[$− #h(0.2em) frac(3 x, 12 y)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction x over 4 y")[$− #h(0.2em) frac(x, 4 y)$]
]

#math.equation(block: true, alt: "minus the fraction 4 x over 32 y")[$− #h(0.2em) frac(4 x, 32 y)$]

#math.equation(block: true, alt: "the fraction 14 x squared over 21 y")[$frac(14 x^(2), 21 y)$]

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

#math.equation(block: true, alt: "the fraction 24 a over 32 b squared")[$frac(24 a, 32 b^(2))$]

#strong[Multiply Fractions]

In the following exercises, multiply.

#math.equation(block: true, alt: "the fraction 3 over 4 times the fraction 9 over 10")[$frac(3, 4) · frac(9, 10)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 27 over 40")[$frac(27, 40)$]
]

#math.equation(block: true, alt: "the fraction 4 over 5 times the fraction 2 over 7")[$frac(4, 5) · frac(2, 7)$]

#math.equation(block: true, alt: "minus the fraction 2 over 3 open parenthesis minus the fraction 3 over 8 close parenthesis")[$− #h(0.2em) frac(2, 3) ( − #h(0.2em) frac(3, 8) )$]

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

#math.equation(block: true, alt: "minus the fraction 3 over 4 open parenthesis minus the fraction 4 over 9 close parenthesis")[$− #h(0.2em) frac(3, 4) ( − #h(0.2em) frac(4, 9) )$]

#math.equation(block: true, alt: "minus the fraction 5 over 9 times the fraction 3 over 10")[$− #h(0.2em) frac(5, 9) · frac(3, 10)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 6")[$− #h(0.2em) frac(1, 6)$]
]

#math.equation(block: true, alt: "minus the fraction 3 over 8 times the fraction 4 over 15")[$− #h(0.2em) frac(3, 8) · frac(4, 15)$]

#math.equation(block: true, alt: "open parenthesis minus the fraction 14 over 15 close parenthesis open parenthesis the fraction 9 over 20 close parenthesis")[$( − #h(0.2em) frac(14, 15) ) ( frac(9, 20) )$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 21 over 50")[$− #h(0.2em) frac(21, 50)$]
]

#math.equation(block: true, alt: "open parenthesis minus the fraction 9 over 10 close parenthesis open parenthesis the fraction 25 over 33 close parenthesis")[$( − #h(0.2em) frac(9, 10) ) ( frac(25, 33) )$]

#math.equation(block: true, alt: "open parenthesis minus the fraction 63 over 84 close parenthesis open parenthesis minus the fraction 44 over 90 close parenthesis")[$( − #h(0.2em) frac(63, 84) ) ( − #h(0.2em) frac(44, 90) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 30")[$frac(11, 30)$]
]

#math.equation(block: true, alt: "open parenthesis minus the fraction 33 over 60 close parenthesis open parenthesis minus the fraction 40 over 88 close parenthesis")[$( − #h(0.2em) frac(33, 60) ) ( − #h(0.2em) frac(40, 88) )$]

#math.equation(block: true, alt: "4 times the fraction 5 over 11")[$4 · frac(5, 11)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 20 over 11")[$frac(20, 11)$]
]

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

#math.equation(block: true, alt: "the fraction 3 over 7 times 21 n")[$frac(3, 7) · 21 n$]

#solutionbox[
9#emph[n]
]

#math.equation(block: true, alt: "the fraction 5 over 6 times 30 m")[$frac(5, 6) · 30 m$]

#math.equation(block: true, alt: "−8 open parenthesis the fraction 17 over 4 close parenthesis")[$−8 ( frac(17, 4) )$]

#solutionbox[
#math.equation(block: true, alt: "−34")[$−34$]
]

#math.equation(block: true, alt: "open parenthesis −1 close parenthesis open parenthesis minus the fraction 6 over 7 close parenthesis")[$( −1 ) ( − #h(0.2em) frac(6, 7) )$]

#strong[Divide Fractions]

In the following exercises, divide.

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

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

#math.equation(block: true, alt: "the fraction 4 over 5 divided by the fraction 3 over 4")[$frac(4, 5) ÷ frac(3, 4)$]

#math.equation(block: true, alt: "minus the fraction 7 over 9 divided by open parenthesis minus the fraction 7 over 4 close parenthesis")[$− #h(0.2em) frac(7, 9) ÷ ( − #h(0.2em) frac(7, 4) )$]

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

#math.equation(block: true, alt: "minus the fraction 5 over 6 divided by open parenthesis minus the fraction 5 over 6 close parenthesis")[$− #h(0.2em) frac(5, 6) ÷ ( − #h(0.2em) frac(5, 6) )$]

#math.equation(block: true, alt: "the fraction 3 over 4 divided by the fraction x over 11")[$frac(3, 4) ÷ frac(x, 11)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 33 over 4 x")[$frac(33, 4 x)$]
]

#math.equation(block: true, alt: "the fraction 2 over 5 divided by the fraction y over 9")[$frac(2, 5) ÷ frac(y, 9)$]

#math.equation(block: true, alt: "the fraction 5 over 18 divided by open parenthesis minus the fraction 15 over 24 close parenthesis")[$frac(5, 18) ÷ ( − #h(0.2em) frac(15, 24) )$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 4 over 9")[$− #h(0.2em) frac(4, 9)$]
]

#math.equation(block: true, alt: "the fraction 7 over 18 divided by open parenthesis minus the fraction 14 over 27 close parenthesis")[$frac(7, 18) ÷ ( − #h(0.2em) frac(14, 27) )$]

#math.equation(block: true, alt: "the fraction 8 u over 15 divided by the fraction 12 v over 25")[$frac(8 u, 15) ÷ frac(12 v, 25)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 10 u over 9 v")[$frac(10 u, 9 v)$]
]

#math.equation(block: true, alt: "the fraction 12 r over 25 divided by the fraction 18 s over 35")[$frac(12 r, 25) ÷ frac(18 s, 35)$]

#math.equation(block: true, alt: "−5 divided by the fraction 1 over 2")[$−5 ÷ frac(1, 2)$]

#solutionbox[
#math.equation(block: true, alt: "−10")[$−10$]
]

#math.equation(block: true, alt: "−3 divided by the fraction 1 over 4")[$−3 ÷ frac(1, 4)$]

#math.equation(block: true, alt: "the fraction 3 over 4 divided by open parenthesis −12 close parenthesis")[$frac(3, 4) ÷ ( −12 )$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 16")[$− #h(0.2em) frac(1, 16)$]
]

#math.equation(block: true, alt: "−15 divided by open parenthesis minus the fraction 5 over 3 close parenthesis")[$−15 ÷ ( − #h(0.2em) frac(5, 3) )$]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction minus the fraction 8 over 21 over the fraction 12 over 35")[$frac(− #h(0.2em) frac(8, 21), frac(12, 35))$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 10 over 9")[$− #h(0.2em) frac(10, 9)$]
]

#math.equation(block: true, alt: "the fraction minus the fraction 9 over 16 over the fraction 33 over 40")[$frac(− #h(0.2em) frac(9, 16), frac(33, 40))$]

#math.equation(block: true, alt: "the fraction minus the fraction 4 over 5 over 2")[$frac(− #h(0.2em) frac(4, 5), 2)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 2 over 5")[$− #h(0.2em) frac(2, 5)$]
]

#math.equation(block: true, alt: "the fraction 5 over the fraction 3 over 10")[$frac(5, frac(3, 10))$]

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

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

#math.equation(block: true, alt: "the fraction minus the fraction 3 over 8 over minus the fraction y over 12")[$frac(− #h(0.2em) frac(3, 8), − #h(0.2em) frac(y, 12))$]

#strong[Simplify Expressions Written with a Fraction Bar]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction 22 plus 3 over 10")[$frac(22 + 3, 10)$]

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

#math.equation(block: true, alt: "the fraction 19 minus 4 over 6")[$frac(19 − 4, 6)$]

#math.equation(block: true, alt: "the fraction 48 over 24 minus 15")[$frac(48, 24 − 15)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 16 over 3")[$frac(16, 3)$]
]

#math.equation(block: true, alt: "the fraction 46 over 4 plus 4")[$frac(46, 4 + 4)$]

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

#solutionbox[
0
]

#math.equation(block: true, alt: "the fraction −6 plus 3 over 17 minus 8")[$frac(−6 + 3, 17 − 8)$]

#math.equation(block: true, alt: "the fraction 4 times 3 over 6 times 6")[$frac(4 · 3, 6 · 6)$]

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

#math.equation(block: true, alt: "the fraction 6 times 6 over 9 times 2")[$frac(6 · 6, 9 · 2)$]

#math.equation(block: true, alt: "the fraction 4 squared minus 1 over 25")[$frac(4^(2) − 1, 25)$]

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

#math.equation(block: true, alt: "the fraction 7 squared plus 1 over 60")[$frac(7^(2) + 1, 60)$]

#math.equation(block: true, alt: "the fraction 8 times 3 plus 2 times 9 over 14 plus 3")[$frac(8 · 3 + 2 · 9, 14 + 3)$]

#solutionbox[
#math.equation(block: true, alt: "2 the fraction 8 over 17")[$2 frac(8, 17)$]
]

#math.equation(block: true, alt: "the fraction 9 times 6 minus 4 times 7 over 22 plus 3")[$frac(9 · 6 − 4 · 7, 22 + 3)$]

#math.equation(block: true, alt: "the fraction 5 times 6 minus 3 times 4 over 4 times 5 minus 2 times 3")[$frac(5 · 6 − 3 · 4, 4 · 5 − 2 · 3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 9 over 7")[$frac(9, 7)$]
]

#math.equation(block: true, alt: "the fraction 8 times 9 minus 7 times 6 over 5 times 6 minus 9 times 2")[$frac(8 · 9 − 7 · 6, 5 · 6 − 9 · 2)$]

#math.equation(block: true, alt: "the fraction 5 squared minus 3 squared over 3 minus 5")[$frac(5^(2) − 3^(2), 3 − 5)$]

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

#math.equation(block: true, alt: "the fraction 6 squared minus 4 squared over 4 minus 6")[$frac(6^(2) − 4^(2), 4 − 6)$]

#math.equation(block: true, alt: "the fraction 7 times 4 minus 2 open parenthesis 8 minus 5 close parenthesis over 9 times 3 minus 3 times 5")[$frac(7 · 4 − 2 ( 8 − 5 ), 9 · 3 − 3 · 5)$]

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

#math.equation(block: true, alt: "the fraction 9 times 7 minus 3 open parenthesis 12 minus 8 close parenthesis over 8 times 7 minus 6 times 6")[$frac(9 · 7 − 3 ( 12 − 8 ), 8 · 7 − 6 · 6)$]

#math.equation(block: true, alt: "the fraction 9 open parenthesis 8 minus 2 close parenthesis minus 3 open parenthesis 15 minus 7 close parenthesis over 6 open parenthesis 7 minus 1 close parenthesis minus 3 open parenthesis 17 minus 9 close parenthesis")[$frac(9 ( 8 − 2 ) − 3 ( 15 − 7 ), 6 ( 7 − 1 ) − 3 ( 17 − 9 ))$]

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

#math.equation(block: true, alt: "the fraction 8 open parenthesis 9 minus 2 close parenthesis minus 4 open parenthesis 14 minus 9 close parenthesis over 7 open parenthesis 8 minus 3 close parenthesis minus 3 open parenthesis 16 minus 9 close parenthesis")[$frac(8 ( 9 − 2 ) − 4 ( 14 − 9 ), 7 ( 8 − 3 ) − 3 ( 16 − 9 ))$]

#strong[Translate Phrases to Expressions with Fractions]

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

the quotient of #emph[r] and the sum of #emph[s] and 10

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

the quotient of #emph[A] and the difference of 3 and #emph[B]

the quotient of the difference of #math.equation(block: false, alt: "x and y , and minus 3")[$x #h(0.2em) "and" #h(0.2em) y , #h(0.2em) "and" #h(0.2em) − 3$]

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

the quotient of the sum of #math.equation(block: false, alt: "m and n , and 4 q")[$m #h(0.2em) "and" #h(0.2em) n , #h(0.2em) "and" #h(0.2em) 4 q$]


==== Everyday Math

#strong[Baking.] A recipe for chocolate chip cookies calls for #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] cup brown sugar. Imelda wants to double the recipe. ⓐ How much brown sugar will Imelda need? Show your calculation. ⓑ Measuring cups usually come in sets of #math.equation(block: false, alt: "the fraction 1 over 4 , the fraction 1 over 3 , the fraction 1 over 2 , and 1")[$frac(1, 4) , frac(1, 3) , frac(1, 2) , #h(0.2em) "and" #h(0.2em) 1$] cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis 2 close parenthesis open parenthesis the fraction 3 over 4 close parenthesis ; the fraction 3 over 2")[$( 2 ) ( frac(3, 4) ) ; frac(3, 2)$] cups ⓑ answers will vary
]

#strong[Baking.] Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] cup of condensed milk. ⓐ How much condensed milk will Nina need? Show your calculation. ⓑ Measuring cups usually come in sets of #math.equation(block: false, alt: "the fraction 1 over 4 , the fraction 1 over 3 , the fraction 1 over 2 , and 1")[$frac(1, 4) , frac(1, 3) , frac(1, 2) , #h(0.2em) "and" #h(0.2em) 1$] cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for #math.equation(block: false, alt: "4")[$4$] pans of fudge.

#strong[Portions] Don purchased a bulk package of candy that weighs #math.equation(block: false, alt: "5")[$5$] pounds. He wants to sell the candy in little bags that hold #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] pound. How many little bags of candy can he fill from the bulk package?

#solutionbox[
20 bags
]

#strong[Portions] Kristen has #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] yards of ribbon that she wants to cut into #math.equation(block: false, alt: "6")[$6$] equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be?


==== Writing Exercises

Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning.

#solutionbox[
Answers may vary
]

Give an example from everyday life that demonstrates how #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 2 over 3 is the fraction 1 over 3 .")[$frac(1, 2) · frac(2, 3) #h(0.2em) "is" #h(0.2em) frac(1, 3) .$]

Explain how you find the reciprocal of a fraction.

#solutionbox[
Answers may vary
]

Explain how you find the reciprocal of a negative number.


==== Self Check

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

#figure(figph[A table is shown that is made up of four columns and seven rows. The first row reads “I can…” in the first column, “Confidently” in the second column, “With some help” in the third column and “No – I don’t get it” in the last column. The next row down in the first column reads “find equivalent fractions”, under this reads “simplify fractions”, under this reads “multiply fractions”, under this reads “divide fractions”, under this reads “Simplify expressions written with a fraction bar” and under this reads “translate phrases to expressions with fractions.”], alt: "A table is shown that is made up of four columns and seven rows. The first row reads “I can…” in the first column, “Confidently” in the second column, “With some help” in the third column and “No – I don’t get it” in the last column. The next row down in the first column reads “find equivalent fractions”, under this reads “simplify fractions”, under this reads “multiply fractions”, under this reads “divide fractions”, under this reads “Simplify expressions written with a fraction bar” and under this reads “translate phrases to expressions with fractions.”", caption: none)

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?