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== 4.3#h(0.6em)Multiply and Divide Mixed Numbers and Complex Fractions

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

Divide and reduce, if possible: #math.equation(block: false, alt: "open parenthesis 4 plus 5 close parenthesis divided by open parenthesis 10 minus 7 close parenthesis .")[$( 4 + 5 ) ÷ ( 10 − 7 ) .$] #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 1 over 8 times the fraction 2 over 3")[$frac(1, 8) · frac(2, 3)$]. #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Convert #math.equation(block: false, alt: "2 the fraction 3 over 5")[$2 frac(3, 5)$] into an improper fraction. #linebreak() 
If you missed this problem, review .

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


=== Multiply and Divide Mixed Numbers

In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a #strong[mixed number] to an #strong[improper fraction]. And you learned how to do that in Visualize Fractions.

#examplebox("Example 1")[][
Multiply: #math.equation(block: false, alt: "3 the fraction 1 over 3 times the fraction 5 over 8")[$3 frac(1, 3) · frac(5, 8)$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 the fraction 1 over 3 times the fraction 5 over 8")[$3 frac(1, 3) · frac(5, 8)$]]),
  [Convert #math.equation(block: false, alt: "3 the fraction 1 over 3")[$3 frac(1, 3)$] to an improper fraction.], [#math.equation(block: false, alt: "the fraction 10 over 3 times the fraction 5 over 8")[$frac(10, 3) · frac(5, 8)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 10 times 5 over 3 times 8")[$frac(10 · 5, 3 · 8)$]],
  [Look for common factors.], [#math.equation(block: false, alt: "the fraction 2̸ times 5 times 5 over 3 times 2̸ times 4")[$frac(2̸ · 5 · 5, 3 · 2̸ · 4)$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 5 times 5 over 3 times 4")[$frac(5 · 5, 3 · 4)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 25 over 12")[$frac(25, 12)$]],
))

Notice that we left the answer as an improper fraction, #math.equation(block: false, alt: "the fraction 25 over 12 ,")[$frac(25, 12) ,$] and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between #math.equation(block: false, alt: "2 the fraction 1 over 12")[$2 frac(1, 12)$] and #math.equation(block: false, alt: "2 times the fraction 1 over 12 .")[$2 · frac(1, 12) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write your answer in simplified form: #math.equation(block: false, alt: "5 the fraction 2 over 3 times the fraction 6 over 17 .")[$5 frac(2, 3) · frac(6, 17) .$]

#solutionbox[
2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write your answer in simplified form: #math.equation(block: false, alt: "the fraction 3 over 7 times 5 the fraction 1 over 4 .")[$frac(3, 7) · 5 frac(1, 4) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiply or divide mixed numbers.]

+ Convert the mixed numbers to improper fractions.
+ Follow the rules for fraction multiplication or division.
+ Simplify if possible.
]

#examplebox("Example 2")[][
Multiply, and write your answer in simplified form: #math.equation(block: false, alt: "2 the fraction 4 over 5 open parenthesis minus 1 the fraction 7 over 8 close parenthesis .")[$2 frac(4, 5) #h(0.2em) ( − 1 frac(7, 8) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 the fraction 4 over 5 open parenthesis minus 1 the fraction 7 over 8 close parenthesis")[$2 frac(4, 5) #h(0.2em) ( − 1 frac(7, 8) )$]]),
  [Convert mixed numbers to improper fractions.], [#math.equation(block: false, alt: "the fraction 14 over 5 open parenthesis minus the fraction 15 over 8 close parenthesis")[$frac(14, 5) #h(0.2em) ( − frac(15, 8) )$]],
  [Multiply.], [#math.equation(block: false, alt: "minus the fraction 14 times 15 over 5 times 8")[$− #h(0.2em) frac(14 · 15, 5 · 8)$]],
  [Look for common factors.], [#math.equation(block: false, alt: "minus the fraction 2̸ times 7 times 5̸ times 3 over 5̸ times 2̸ times 4")[$− #h(0.2em) frac(2̸ · 7 · 5̸ · 3, 5̸ · 2̸ · 4)$]],
  [Remove common factors.], [#math.equation(block: false, alt: "minus the fraction 7 times 3 over 4")[$− #h(0.2em) frac(#h(0.2em) 7 · 3, 4)$]],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 21 over 4")[$− #h(0.2em) frac(#h(0.2em) 21, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write your answer in simplified form. #math.equation(block: false, alt: "5 the fraction 5 over 7 open parenthesis minus 2 the fraction 5 over 8 close parenthesis .")[$5 frac(5, 7) #h(0.2em) ( − 2 frac(5, 8) ) .$]

#solutionbox[
−15
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write your answer in simplified form. #math.equation(block: false, alt: "−3 the fraction 2 over 5 times 4 the fraction 1 over 6 .")[$−3 frac(2, 5) · 4 frac(1, 6) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 85 over 6")[$− frac(85, 6)$]
]
]

#examplebox("Example 3")[][
Divide, and write your answer in simplified form: #math.equation(block: false, alt: "3 the fraction 4 over 7 divided by 5 .")[$3 frac(4, 7) #h(0.2em) ÷ #h(0.2em) 5 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 the fraction 4 over 7 divided by 5")[$3 frac(4, 7) #h(0.2em) ÷ #h(0.2em) 5$]]),
  [Convert mixed numbers to improper fractions.], [#math.equation(block: false, alt: "the fraction 25 over 7 divided by the fraction 5 over 1")[$frac(25, 7) #h(0.2em) ÷ #h(0.2em) frac(5, 1)$]],
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 25 over 7 times the fraction 1 over 5")[$frac(25, 7) · frac(1, 5)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 25 times 1 over 7 times 5")[$frac(25 · 1, 7 · 5)$]],
  [Look for common factors.], [#math.equation(block: false, alt: "the fraction 5̸ times 5 times 1 over 7 times 5̸")[$frac(5̸ · 5 · 1, 7 · 5̸)$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 5 times 1 over 7")[$frac(5 · 1, 7)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 5 over 7")[$frac(5, 7)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write your answer in simplified form: #math.equation(block: false, alt: "4 the fraction 3 over 8 divided by 7 .")[$4 frac(3, 8) ÷ 7 .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write your answer in simplified form: #math.equation(block: false, alt: "2 the fraction 5 over 8 divided by 3 .")[$2 frac(5, 8) ÷ 3 .$]

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

#examplebox("Example 4")[][
Divide: #math.equation(block: false, alt: "2 the fraction 1 over 2 divided by 1 the fraction 1 over 4 .")[$2 frac(1, 2) ÷ 1 frac(1, 4) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 the fraction 1 over 2 divided by 1 the fraction 1 over 4")[$2 frac(1, 2) ÷ 1 frac(1, 4)$]]),
  [Convert mixed numbers to improper fractions.], [#math.equation(block: false, alt: "the fraction 5 over 2 divided by the fraction 5 over 4")[$frac(5, 2) ÷ frac(5, 4)$]],
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 5 over 2 times the fraction 4 over 5")[$frac(5, 2) · frac(4, 5)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 5 times 4 over 2 times 5")[$frac(5 · 4, 2 · 5)$]],
  [Look for common factors.], [#math.equation(block: false, alt: "the fraction 5̸ times 2̸ times 2 over 2̸ times 1 times 5̸")[$frac(5̸ · 2̸ · 2, 2̸ · 1 · 5̸)$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 2 over 1")[$frac(2, 1)$]],
  [Simplify.], [#math.equation(block: false, alt: "2")[$2$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write your answer in simplified form: #math.equation(block: false, alt: "2 the fraction 2 over 3 divided by 1 the fraction 1 over 3 .")[$2 frac(2, 3) ÷ 1 frac(1, 3) .$]

#solutionbox[
2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write your answer in simplified form: #math.equation(block: false, alt: "3 the fraction 3 over 4 divided by 1 the fraction 1 over 2 .")[$3 frac(3, 4) ÷ 1 frac(1, 2) .$]

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


=== Translate Phrases to Expressions with Fractions

The words #emph[quotient] and #emph[ratio] are often used to describe fractions. In Subtract Whole Numbers, we defined quotient as the result of division. The quotient of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] is the result you 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 .")[$frac(a, b) .$] Let’s practice translating some phrases into algebraic expressions using these terms.

#examplebox("Example 5")[][
Translate the phrase into an algebraic expression: “the quotient of #math.equation(block: false, alt: "3 x")[$3 x$] and #math.equation(block: false, alt: "8 .”")[$8 .”$]

#solutionbox[
The keyword is #emph[quotient]; it tells us that the operation is division. Look for the words #emph[of] and #emph[and] to find the numbers to divide.

#math.equation(block: true, alt: "The quotient of 3 x and 8 .")[$"The quotient" #h(0.2em) "of" #h(0.2em) 3 x #h(0.2em) "and" #h(0.2em) 8 .$]This tells us that we need to divide #math.equation(block: false, alt: "3 x")[$3 x$] by #math.equation(block: false, alt: "8 .")[$8 .$] #math.equation(block: false, alt: "the fraction 3 x over 8")[$frac(3 x, 8)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the phrase into an algebraic expression: the quotient of #math.equation(block: false, alt: "9 s")[$9 s$] and #math.equation(block: false, alt: "14 .")[$14 .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 9 s over 14")[$frac(9 s, 14)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the phrase into an algebraic expression: the quotient of #math.equation(block: false, alt: "5 y")[$5 y$] and #math.equation(block: false, alt: "6 .")[$6 .$]

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

#examplebox("Example 6")[][
Translate the phrase into an algebraic expression: the quotient of the difference of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n ,")[$n ,$] and #math.equation(block: false, alt: "p .")[$p .$]

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

#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 phrase into an algebraic expression: the quotient of the difference of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b ,")[$b ,$] and #math.equation(block: false, alt: "c d .")[$c d .$]

#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 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")[$frac(p + q, r)$]
]
]


=== Simplify Complex Fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called #strong[complex fraction], which 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")[$frac(#h(0.2em) frac(6, 7) #h(0.2em), #h(0.2em) 3 #h(0.2em)) #h(1.8em) frac(#h(0.2em) frac(3, 4) #h(0.2em), #h(0.2em) frac(5, 8) #h(0.2em)) #h(1.8em) frac(#h(0.2em) frac(x, 2) #h(0.2em), #h(0.2em) frac(5, 6) #h(0.2em))$]To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction #math.equation(block: false, alt: "the fraction the fraction 3 over 4 over the fraction 5 over 8")[$frac(#h(0.2em) frac(3, 4) #h(0.2em), #h(0.2em) frac(5, 8) #h(0.2em))$] can be written as #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 7")[][
Simplify: #math.equation(block: false, alt: "the fraction the fraction 3 over 4 over the fraction 5 over 8 .")[$frac(#h(0.2em) frac(3, 4) #h(0.2em), #h(0.2em) frac(5, 8) #h(0.2em)) .$]

#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")[$frac(#h(0.2em) frac(3, 4) #h(0.2em), #h(0.2em) frac(5, 8) #h(0.2em))$]]),
  [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.], [#math.equation(block: false, alt: "the fraction 3 times 4̸ times 2 over 4̸ times 5")[$frac(3 · 4̸ · 2, 4̸ · 5)$]],
  [Remove 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 .")[$frac(#h(0.2em) frac(2, 3) #h(0.2em), #h(0.2em) frac(5, 6) #h(0.2em)) .$]

#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 .")[$frac(#h(0.2em) frac(3, 7) #h(0.2em), #h(0.2em) frac(6, 11) #h(0.2em)) .$]

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

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

+ Rewrite the complex fraction as a division problem.
+ Follow the rules for dividing fractions.
+ Simplify if possible.
]

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "the fraction minus the fraction 6 over 7 over 3 .")[$frac(− frac(6, 7), 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction minus the fraction 6 over 7 over 3")[$frac(− frac(6, 7), 3)$]]),
  [Rewrite as division.], [#math.equation(block: false, alt: "minus the fraction 6 over 7 divided by 3")[$− frac(6, 7) ÷ 3$]],
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "minus the fraction 6 over 7 times the fraction 1 over 3")[$− frac(6, 7) · frac(1, 3)$]],
  [Multiply; the product will be negative.], [#math.equation(block: false, alt: "minus the fraction 6 times 1 over 7 times 3")[$− frac(6 · 1, 7 · 3)$]],
  [Look for common factors.], [#math.equation(block: false, alt: "minus the fraction 3̸ times 2 times 1 over 7 times 3̸")[$− frac(3̸ · 2 · 1, 7 · 3̸)$]],
  [Remove common factors and simplify.], [#math.equation(block: false, alt: "minus the fraction 2 over 7")[$− frac(2, 7)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction minus the fraction 8 over 7 over 4 .")[$frac(− frac(8, 7), 4) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 3 over the fraction 9 over 10 .")[$− frac(#h(0.2em) 3 #h(0.2em), #h(0.2em) frac(9, 10) #h(0.2em)) .$]

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

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "the fraction the fraction x over 2 over the fraction x y over 6 .")[$frac(#h(0.2em) frac(x, 2) #h(0.2em), #h(0.2em) frac(x y, 6) #h(0.2em)) .$]

#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")[$frac(#h(0.2em) frac(x, 2) #h(0.2em), #h(0.2em) frac(x y, 6) #h(0.2em))$]]),
  [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.], [#math.equation(block: false, alt: "the fraction x̸ times 3 times 2̸ over 2̸ times x̸ times y")[$frac("x̸" · 3 · 2̸, 2̸ · "x̸" · y)$]],
  [Remove 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 .")[$frac(#h(0.2em) frac(a, 8) #h(0.2em), #h(0.2em) frac(a b, 6) #h(0.2em)) .$]

#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 .")[$frac(#h(0.2em) frac(p, 2) #h(0.2em), #h(0.2em) frac(p q, 8) #h(0.2em)) .$]

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

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "the fraction 2 the fraction 3 over 4 over the fraction 1 over 8 .")[$frac(2 frac(3, 4), frac(1, 8)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 2 the fraction 3 over 4 over the fraction 1 over 8")[$frac(2 frac(3, 4), frac(1, 8))$]]),
  [Rewrite as division.], [#math.equation(block: false, alt: "2 the fraction 3 over 4 divided by the fraction 1 over 8")[$2 frac(3, 4) ÷ frac(1, 8)$]],
  [Change the mixed number to an improper fraction.], [#math.equation(block: false, alt: "the fraction 11 over 4 divided by the fraction 1 over 8")[$frac(11, 4) ÷ frac(1, 8)$]],
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 11 over 4 times the fraction 8 over 1")[$frac(11, 4) · frac(8, 1)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 11 times 8 over 4 times 1")[$frac(11 · 8, 4 · 1)$]],
  [Look for common factors.], [#math.equation(block: false, alt: "the fraction 11 times 4̸ times 2 over 4̸ times 1")[$frac(11 · 4̸ · 2, 4̸ · 1)$]],
  [Remove common factors and simplify.], [#math.equation(block: false, alt: "22")[$22$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the fraction 5 over 7 over 1 the fraction 2 over 5 .")[$frac(frac(5, 7), 1 frac(2, 5)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 25 over 49 .")[$frac(25, 49) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the fraction 8 over 5 over 3 the fraction 1 over 5 .")[$frac(frac(8, 5), 3 frac(1, 5)) .$]

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


=== Simplify Expressions with a Fraction Bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction #math.equation(block: false, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$] could be the result of dividing #math.equation(block: false, alt: "the fraction −1 over 3 ,")[$frac(−1, 3) ,$] a negative by a positive, or of dividing #math.equation(block: false, alt: "the fraction 1 over −3 ,")[$frac(1, −3) ,$] a positive by a negative. When the #strong[numerator] and #strong[denominator] have different signs, the quotient is negative.

#figure(figph[Negative 1 over positive 3 is equal to negative one third. Negative over positive equals negative. Positive 1 over negative 3 is equal to negative one third. Positive over negative equals negative.], alt: "Negative 1 over positive 3 is equal to negative one third. Negative over positive equals negative. Positive 1 over negative 3 is equal to negative one third. Positive over negative equals negative.", caption: none)

If #emph[both] the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.

#math.equation(block: true, alt: "the fraction −1 over −3 equals the fraction 1 over 3 the fraction negative over negative equals positive")[$frac(−1, −3) = frac(1, 3) #h(4em) frac("negative", "negative") = "positive"$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Placement of Negative Sign in a Fraction]

For any positive numbers #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b ,")[$b ,$]

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

#examplebox("Example 11")[][
Which of the following fractions are equivalent to #math.equation(block: false, alt: "the fraction 7 over −8 ?")[$frac(7, −8) ?$]

#math.equation(block: true, alt: "the fraction −7 over −8 , the fraction −7 over 8 , the fraction 7 over 8 , minus the fraction 7 over 8")[$#h(0.2em) frac(−7, −8) , frac(−7, 8) , frac(7, 8) , − frac(7, 8)$]#solutionbox[
The quotient of a positive and a negative is a negative, so #math.equation(block: false, alt: "the fraction 7 over −8")[$frac(7, −8)$] is negative. Of the fractions listed, #math.equation(block: false, alt: "the fraction −7 over 8")[$#h(0.2em) frac(−7, 8)$] and #math.equation(block: false, alt: "minus the fraction 7 over 8")[$− frac(7, 8)$] are also negative.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which of the following fractions are equivalent to #math.equation(block: false, alt: "the fraction −3 over 5 ?")[$frac(−3, #h(0.2em) 5) ?$]

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

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which of the following fractions are equivalent to #math.equation(block: false, alt: "minus the fraction 2 over 7 ?")[$− frac(2, 7) ?$]

#math.equation(block: true, alt: "the fraction −2 over −7 , the fraction −2 over 7 , the fraction 2 over 7 , the fraction 2 over −7")[$#h(0.2em) frac(−2, −7) , frac(−2, 7) , frac(2, 7) , frac(2, −7)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction −2 over 7 , the fraction 2 over −7")[$#h(0.2em) frac(−2, 7) , frac(2, −7)$]
]
]

Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, #math.equation(block: false, alt: "the fraction 4 plus 8 over 5 minus 3")[$frac(4 + 8, 5 − 3)$] means #math.equation(block: false, alt: "open parenthesis 4 plus 8 close parenthesis divided by open parenthesis 5 minus 3 close parenthesis .")[$( 4 + 8 ) ÷ ( 5 − 3 ) .$] The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.

We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.

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

#figure(figph[Parentheses, brackets, braces, an absolute value sign, and a fraction bar are shown.], alt: "Parentheses, brackets, braces, an absolute value sign, and a fraction bar are shown.", caption: none)
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Simplify an expression with a fraction bar.]

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

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "the fraction 4 plus 8 over 5 minus 3 .")[$frac(4 + 8, 5 − 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 4 plus 8 over 5 minus 3")[$frac(4 + 8, 5 − 3)$]]),
  [Simplify the expression in the numerator.], [#math.equation(block: false, alt: "the fraction 12 over 5 minus 3")[$frac(12, 5 − 3)$]],
  [Simplify the expression in the denominator.], [#math.equation(block: false, alt: "the fraction 12 over 2")[$frac(12, 2)$]],
  [Simplify the fraction.], [6],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 4 plus 6 over 11 minus 2 .")[$frac(4 + 6, 11 − 2) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 3 plus 5 over 18 minus 2 .")[$frac(3 + 5, 18 − 2) .$]

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

#examplebox("Example 13")[][
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. Multiply in the numerator and use the exponent in 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 the fraction.], [#math.equation(block: false, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$]],
))
]
]

#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: "the fraction −3 over 4")[$frac(−3, 4)$]
]
]

#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 cubed plus 3 .")[$frac(4 − 4 ( 6 ), 3^(3) + 3) .$]

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

#examplebox("Example 14")[][
Simplify: #math.equation(block: false, alt: "the fraction open parenthesis 8 minus 4 close parenthesis squared over 8 squared minus 4 squared .")[$frac(attach(( 8 − 4 ), t: 2), 8^(2) − 4^(2)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction open parenthesis 8 minus 4 close parenthesis squared over 8 squared minus 4 squared")[$frac(attach(( 8 − 4 ), t: 2), 8^(2) − 4^(2))$]]),
  [Use the order of operations (parentheses first, then exponents).], [#math.equation(block: false, alt: "the fraction open parenthesis 4 close parenthesis squared over 64 minus 16")[$frac(attach(( 4 ), t: 2), 64 − 16)$]],
  [Simplify the numerator and denominator.], [#math.equation(block: false, alt: "the fraction 16 over 48")[$frac(16, 48)$]],
  [Simplify the fraction.], [#math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction open parenthesis 11 minus 7 close parenthesis squared over 11 squared minus 7 squared .")[$frac(attach(( 11 − 7 ), t: 2), 11^(2) − 7^(2)) .$]

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

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

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

#examplebox("Example 15")[][
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 −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 −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 −2 .")[$frac(7 ( −1 ) + 9 ( −3 ), −5 ( 3 ) −2) .$]

#solutionbox[
2
]
]

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

- #link("https://www.openstax.org/l/24DivisionMixed")[Division Involving Mixed Numbers]
- #link("https://www.openstax.org/l/24ComplexFrac")[Evaluate a Complex Fraction]
]


=== Key Concepts

- #emph[Multiply or divide mixed numbers.]+ Convert the mixed numbers to improper fractions.
+ Follow the rules for fraction multiplication or division.
+ Simplify if possible.
- #emph[Simplify a complex fraction.]+ Rewrite the complex fraction as a division problem.
+ Follow the rules for dividing fractions.
+ Simplify if possible.
- #emph[Placement of negative sign in a fraction.]- For any positive numbers #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$], #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")[$frac(− a, b) = frac(a, − b) = − frac(a, b)$].
- #emph[Simplify an expression with a fraction bar.]+ Simplify the numerator.
+ Simplify the denominator.
+ Simplify the fraction.


==== Practice Makes Perfect

#strong[Multiply and Divide Mixed Numbers]

In the following exercises, multiply and write the answer in simplified form.

#math.equation(block: true, alt: "4 the fraction 3 over 8 times the fraction 7 over 10")[$4 frac(3, 8) · frac(7, 10)$]

#math.equation(block: true, alt: "2 the fraction 4 over 9 times the fraction 6 over 7")[$2 frac(4, 9) · frac(6, 7)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 44 over 21")[$frac(44, 21)$]
]

#math.equation(block: true, alt: "the fraction 15 over 22 times 3 the fraction 3 over 5")[$frac(15, 22) · 3 frac(3, 5)$]

#math.equation(block: true, alt: "the fraction 25 over 36 times 6 the fraction 3 over 10")[$frac(25, 36) · 6 frac(3, 10)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 35 over 8")[$frac(35, 8)$]
]

#math.equation(block: true, alt: "4 the fraction 2 over 3 open parenthesis −1 the fraction 1 over 8 close parenthesis")[$4 frac(2, 3) #h(0.2em) ( −1 frac(1, 8) )$]

#math.equation(block: true, alt: "2 the fraction 2 over 5 open parenthesis −2 the fraction 2 over 9 close parenthesis")[$2 frac(2, 5) #h(0.2em) ( −2 frac(2, 9) )$]

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

#math.equation(block: true, alt: "−4 the fraction 4 over 9 times 5 the fraction 13 over 16")[$−4 frac(4, 9) · 5 frac(13, 16)$]

#math.equation(block: true, alt: "−1 the fraction 7 over 20 times 2 the fraction 11 over 12")[$−1 frac(7, 20) · 2 frac(11, 12)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 63 over 16")[$− frac(63, 16)$]
]

In the following exercises, divide, and write your answer in simplified form.

#math.equation(block: true, alt: "5 the fraction 1 over 3 divided by 4")[$5 frac(1, 3) ÷ #h(0.2em) 4$]

#math.equation(block: true, alt: "13 the fraction 1 over 2 divided by 9")[$13 frac(1, 2) ÷ #h(0.2em) 9$]

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

#math.equation(block: true, alt: "−12 divided by 3 the fraction 3 over 11")[$−12 ÷ #h(0.2em) 3 frac(3, 11)$]

#math.equation(block: true, alt: "−7 divided by 5 the fraction 1 over 4")[$−7 ÷ #h(0.2em) 5 frac(1, 4)$]

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

#math.equation(block: true, alt: "6 the fraction 3 over 8 divided by 2 the fraction 1 over 8")[$6 frac(3, 8) ÷ #h(0.2em) 2 frac(1, 8)$]

#math.equation(block: true, alt: "2 the fraction 1 over 5 divided by 1 the fraction 1 over 10")[$2 frac(1, 5) ÷ #h(0.2em) 1 frac(1, 10)$]

#solutionbox[
2
]

#math.equation(block: true, alt: "−9 the fraction 3 over 5 divided by open parenthesis −1 the fraction 3 over 5 close parenthesis")[$−9 frac(3, 5) ÷ #h(0.2em) ( −1 frac(3, 5) )$]

#math.equation(block: true, alt: "−18 the fraction 3 over 4 divided by open parenthesis −3 the fraction 3 over 4 close parenthesis")[$−18 frac(3, 4) ÷ #h(0.2em) ( −3 frac(3, 4) )$]

#solutionbox[
5
]

#strong[Translate Phrases to Expressions with Fractions]

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

the quotient of #math.equation(block: false, alt: "5 u")[$5 u$] and #math.equation(block: false, alt: "11")[$11$]

the quotient of #math.equation(block: false, alt: "7 v")[$7 v$] and #math.equation(block: false, alt: "13")[$13$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 v over 13")[$frac(7 v, 13)$]
]

the quotient of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q")[$q$]

the quotient of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]

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

the quotient of #math.equation(block: false, alt: "r")[$r$] and the sum of #math.equation(block: false, alt: "s")[$s$] and #math.equation(block: false, alt: "10")[$10$]

the quotient of #math.equation(block: false, alt: "A")[$A$] and the difference of #math.equation(block: false, alt: "3")[$3$] and #math.equation(block: false, alt: "B")[$B$]

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

#strong[Simplify Complex Fractions]

In the following exercises, simplify the complex fraction.

#math.equation(block: true, alt: "the fraction the fraction 2 over 3 over the fraction 8 over 9")[$frac(#h(0.2em) frac(2, 3) #h(0.2em), #h(0.2em) frac(8, 9) #h(0.2em))$]

#math.equation(block: true, alt: "the fraction the fraction 4 over 5 over the fraction 8 over 15")[$frac(#h(0.2em) frac(4, 5) #h(0.2em), #h(0.2em) frac(8, 15) #h(0.2em))$]

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

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

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

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 15 over 22")[$− frac(15, 22)$]
]

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

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

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

#math.equation(block: true, alt: "the fraction the fraction 2 over 5 over 8")[$frac(#h(0.2em) frac(2, 5) #h(0.2em), #h(0.2em) 8 #h(0.2em))$]

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

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

#math.equation(block: true, alt: "the fraction the fraction m over 3 over the fraction n over 2")[$frac(#h(0.2em) frac(m, 3) #h(0.2em), #h(0.2em) frac(n, 2) #h(0.2em))$]

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

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

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

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

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

#math.equation(block: true, alt: "the fraction 2 the fraction 4 over 5 over the fraction 1 over 10")[$frac(2 frac(4, 5), frac(1, 10))$]

#math.equation(block: true, alt: "the fraction 4 the fraction 2 over 3 over the fraction 1 over 6")[$frac(4 frac(2, 3), frac(1, 6))$]

#solutionbox[
28
]

#math.equation(block: true, alt: "the fraction the fraction 7 over 9 over −2 the fraction 4 over 5")[$frac(frac(7, 9), −2 frac(4, 5))$]

#math.equation(block: true, alt: "the fraction the fraction 3 over 8 over −6 the fraction 3 over 4")[$frac(frac(3, 8), −6 frac(3, 4))$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 18")[$− frac(1, 18)$]
]

#strong[Simplify Expressions with a Fraction Bar]

In the following exercises, identify the equivalent fractions.

Which of the following fractions are equivalent to #math.equation(block: false, alt: "the fraction 5 over −11 ?")[$frac(5, −11) ?$] #linebreak() #math.equation(block: false, alt: "the fraction −5 over −11 , the fraction −5 over 11 , the fraction 5 over 11 , minus the fraction 5 over 11")[$#h(0.2em) frac(−5, −11) , frac(−5, 11) , frac(5, 11) , − frac(5, 11)$]

Which of the following fractions are equivalent to #math.equation(block: false, alt: "the fraction −4 over 9 ?")[$frac(−4, 9) ?$] #linebreak() 
#math.equation(block: false, alt: "the fraction −4 over −9 , the fraction −4 over 9 , the fraction 4 over 9 , minus the fraction 4 over 9")[$#h(0.2em) frac(−4, −9) , frac(−4, 9) , frac(4, 9) , − frac(4, 9)$]

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

Which of the following fractions are equivalent to #math.equation(block: false, alt: "minus the fraction 11 over 3 ?")[$− frac(11, 3) ?$] #linebreak() #math.equation(block: false, alt: "the fraction −11 over 3 , the fraction 11 over 3 , the fraction −11 over −3 , the fraction 11 over −3")[$#h(0.2em) frac(−11, 3) , frac(11, 3) , frac(−11, −3) , #h(0.2em) frac(11, −3)$]

Which of the following fractions are equivalent to #math.equation(block: false, alt: "minus the fraction 13 over 6 ?")[$− frac(13, 6) ?$] #linebreak() #math.equation(block: false, alt: "the fraction 13 over 6 , the fraction 13 over −6 , the fraction −13 over −6 , the fraction −13 over 6")[$#h(0.2em) frac(13, 6) , frac(13, −6) , frac(−13, −6) , frac(−13, 6) #h(0.2em)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over −6 , the fraction −13 over 6")[$#h(0.2em) frac(13, −6) , frac(−13, 6) #h(0.2em)$]
]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction 4 plus 11 over 8")[$frac(4 + 11, 8)$]

#math.equation(block: true, alt: "the fraction 9 plus 3 over 7")[$frac(9 + 3, 7)$]

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

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

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 23 over 4")[$frac(23, 4)$]
]

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

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

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

#math.equation(block: true, alt: "the fraction 22 minus 14 over 19 minus 13")[$frac(22 − 14, 19 − 13)$]

#math.equation(block: true, alt: "the fraction 15 plus 9 over 18 plus 12")[$frac(15 + 9, 18 + 12)$]

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

#math.equation(block: true, alt: "the fraction 5 times 8 over −10")[$frac(5 ⋅ 8, −10)$]

#math.equation(block: true, alt: "the fraction 3 times 4 over −24")[$frac(3 ⋅ 4, −24)$]

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

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

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

#solutionbox[
2
]

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

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 26 over 25")[$frac(26, 25)$]
]

#math.equation(block: true, alt: "the fraction 15 times 5 minus 5 squared over 2 times 10")[$frac(15 ⋅ 5 − 5^(2), 2 ⋅ 10)$]

#math.equation(block: true, alt: "the fraction 12 times 9 minus 3 squared over 3 times 18")[$frac(12 ⋅ 9 − 3^(2), 3 ⋅ 18)$]

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

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

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

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

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

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

#solutionbox[
−10
]

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

#math.equation(block: true, alt: "the fraction 7 plus 3 open parenthesis 5 close parenthesis over −2 minus 3 squared")[$frac(7 + 3 ( 5 ), −2 − 3^(2))$]

#solutionbox[
−2
]

#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")[$frac(7 ⋅ 4 − 2 ( 8 − 5 ), 9 ⋅ 3 − 3 ⋅ 5)$]

#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")[$frac(9 ⋅ 7 − 3 ( 12 − 8 ), 8 ⋅ 7 − 6 ⋅ 6)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 51 over 20")[$frac(51, 20)$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 18 over 7")[$frac(18, 7)$]
]


==== Everyday Math

#strong[Baking] A recipe for chocolate chip cookies calls for #math.equation(block: false, alt: "2 the fraction 1 over 4")[$2 frac(1, 4)$] cups of flour. Graciela wants to double the recipe.

+ ⓐ How much flour will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number.
+ ⓑ Measuring cups usually come in sets with cups for #math.equation(block: false, alt: "the fraction 1 over 8 , the fraction 1 over 4 , the fraction 1 over 3 , the fraction 1 over 2 ,")[$frac(1, 8) , frac(1, 4) , frac(1, 3) , frac(1, 2) ,$] and #math.equation(block: false, alt: "1")[$1$] cup. Draw a diagram to show two different ways that Graciela could measure out the flour needed to double the recipe.

#strong[Baking] A booth at the county fair sells fudge by the pound. Their award winning “Chocolate Overdose” fudge contains #math.equation(block: false, alt: "2 the fraction 2 over 3")[$2 frac(2, 3)$] cups of chocolate chips per pound.

+ ⓐ How many cups of chocolate chips are in a half-pound of the fudge?
+ ⓑ The owners of the booth make the fudge in #math.equation(block: false, alt: "10")[$10$]-pound batches. How many chocolate chips do they need to make a #math.equation(block: false, alt: "10")[$10$]-pound batch? Write your results as improper fractions and as a mixed numbers.

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 4 over 3 equals 1 the fraction 1 over 3")[$frac(4, 3) = 1 frac(1, 3)$] cups
+ ⓑ #math.equation(block: false, alt: "the fraction 80 over 3 equals 26 the fraction 2 over 3")[$frac(80, 3) = 26 frac(2, 3)$] cups
]


==== Writing Exercises

Explain how to find the reciprocal of a mixed number.

Explain how to multiply mixed numbers.

#solutionbox[
Answers will vary.
]

Randy thinks that #math.equation(block: false, alt: "3 the fraction 1 over 2 times 5 the fraction 1 over 4")[$3 frac(1, 2) · 5 frac(1, 4)$] is #math.equation(block: false, alt: "15 the fraction 1 over 8 .")[$15 frac(1, 8) .$] Explain what is wrong with Randy’s thinking.

Explain why #math.equation(block: false, alt: "minus the fraction 1 over 2 , the fraction −1 over 2 ,")[$− frac(1, 2) , frac(−1, 2) ,$] and #math.equation(block: false, alt: "the fraction 1 over −2")[$frac(1, −2)$] are equivalent.

#solutionbox[
Answers will vary.
]


==== Self Check

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

#figure(figph[A self-assessment chart for students to rate their understanding of fraction skills, including multiplying mixed numbers, translating phrases to expressions, and simplifying complex fractions.], alt: "A self-assessment chart for students to rate their understanding of fraction skills, including multiplying mixed numbers, translating phrases to expressions, and simplifying complex fractions.", caption: none)

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