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== 4.2#h(0.6em)Multiply and Divide Fractions

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

Find the prime factorization of #math.equation(block: false, alt: "48 .")[$48 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2 times 2 times 2 times 2 times 3")[$2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Draw a model of the fraction #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#figure(figph[A circle is shown divided into 4 pieces, of which 3 are shaded.], alt: "A circle is shown divided into 4 pieces, of which 3 are shaded.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find two fractions equivalent to #math.equation(block: false, alt: "the fraction 5 over 6 .")[$frac(5, 6) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
Answers may vary. Acceptable answers include #math.equation(block: false, alt: "the fraction 10 over 12 , the fraction 15 over 18 , the fraction 50 over 60 ,")[$frac(10, 12) , frac(15, 18) , frac(50, 60) ,$] etc.
]
]


=== Simplify Fractions

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in #emph[simplified] form.

A fraction is considered simplified if there are no common factors, other than #math.equation(block: false, alt: "1 ,")[$1 ,$] in the #strong[numerator] and #strong[denominator]. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

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

A fraction is considered simplified if there are no common factors in the 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 #math.equation(block: false, alt: "2")[$2$] and #math.equation(block: false, alt: "3 .")[$3 .$]
- #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 #math.equation(block: false, alt: "10")[$10$] and #math.equation(block: false, alt: "15 .")[$15 .$]

The process of simplifying a fraction is often called #emph[reducing the fraction]. In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We 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 ,")[$b ≠ 0 , c ≠ 0 ,$] then

#math.equation(block: true, alt: "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 .")[$frac(a, b) = frac(a · c, b · c) #h(1.0em) "and" #h(1.0em) frac(a · c, b · c) = frac(a, b) .$]
]

Notice that #math.equation(block: false, alt: "c")[$c$] is a common factor in the #strong[numerator] and #strong[denominator]. Anytime we have a common factor in the numerator and denominator, it can be removed.

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

+ Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
+ Simplify, using the equivalent fractions property, by removing common factors.
+ Multiply any remaining factors.
]

#examplebox("Example 1")[][
Simplify: #math.equation(block: false, alt: "the fraction 10 over 15 .")[$frac(10, 15) .$]

#solutionbox[
To simplify the fraction, we look for any common factors in the numerator and the denominator.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Notice that 5 is a factor of both 10 and 15.], [#math.equation(block: false, alt: "the fraction 10 over 15")[$frac(10, 15)$]]),
  [Factor the numerator and denominator.], [#figure(figph[A fraction showing (2 multiplied by 5) divided by (3 multiplied by 5), where the common factor '5' is highlighted in red, demonstrating potential simplification.], alt: "A fraction showing (2 multiplied by 5) divided by (3 multiplied by 5), where the common factor '5' is highlighted in red, demonstrating potential simplification.", caption: none)],
  [Remove the common factors.], [#figure(figph[This image illustrates the simplification of a fraction by canceling out a common factor 'x' from both the numerator (2x) and the denominator (3x), resulting in 2/3.], alt: "This image illustrates the simplification of a fraction by canceling out a common factor 'x' from both the numerator (2x) and the denominator (3x), resulting in 2/3.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 8 over 12")[$frac(8, 12)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 12 over 16")[$frac(12, 16)$].

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

To simplify a negative fraction, we use the same process as in . Remember to keep the negative sign.

#examplebox("Example 2")[][
Simplify: #math.equation(block: false, alt: "minus the fraction 18 over 24 .")[$− frac(18, 24) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([We notice that 18 and 24 both have factors of 6.], [#math.equation(block: false, alt: "minus the fraction 18 over 24")[$− frac(18, 24)$]]),
  [Rewrite the numerator and denominator showing the common factor.], [#figure(figph[A mathematical fraction: -(3\*6)/(4\*6). The '6' in the numerator and denominator is highlighted in red, demonstrating a common factor for simplification.], alt: "A mathematical fraction: -(3*6)/(4*6). The '6' in the numerator and denominator is highlighted in red, demonstrating a common factor for simplification.", caption: none)],
  [Remove common factors.], [#figure(figph[Simplifying a fraction by canceling out the common factor 'x' (represented by the crossed-out 6s) from both the numerator and denominator, resulting in -3/4. This illustrates algebraic cancellation.], alt: "Simplifying a fraction by canceling out the common factor 'x' (represented by the crossed-out 6s) from both the numerator and denominator, resulting in -3/4. This illustrates algebraic cancellation.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 21 over 28 .")[$− frac(21, 28) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 16 over 24 .")[$− frac(16, 24) .$]

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

After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: #emph[a fraction is considered simplified if there are no common factors in the numerator and denominator].

When we simplify an improper fraction, there is no need to change it to a mixed number.

#examplebox("Example 3")[][
Simplify: #math.equation(block: false, alt: "minus the fraction 56 over 32 .")[$− frac(56, 32) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 56 over 32")[$− frac(56, 32)$]]),
  [Rewrite the numerator and denominator, showing the common factors, 8.], [#figure(figph[A mathematical fraction with 7 multiplied by 8 in the numerator and 4 multiplied by 8 in the denominator, with the number 8 highlighted in red.], alt: "A mathematical fraction with 7 multiplied by 8 in the numerator and 4 multiplied by 8 in the denominator, with the number 8 highlighted in red.", caption: none)],
  [Remove common factors.], [#figure(figph[A visual example of fraction simplification, where the common factor '8' is canceled from both the numerator and denominator, leaving 7/4.], alt: "A visual example of fraction simplification, where the common factor '8' is canceled from both the numerator and denominator, leaving 7/4.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 7 over 4")[$− frac(7, 4)$]],
))
]
]

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

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

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

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

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

+ Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
+ Simplify, using the equivalent fractions property, by removing common factors.
+ Multiply any remaining factors
]

Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "the fraction 210 over 385 .")[$frac(210, 385) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Use factor trees to factor the numerator and denominator.], [#math.equation(block: false, alt: "the fraction 210 over 385")[$frac(210, 385)$] #linebreak() #figure(figph[Two factor trees demonstrating the prime factorization of 210 (2x3x5x7) and 385 (5x7x11), visually breaking down composite numbers into their prime components.], alt: "Two factor trees demonstrating the prime factorization of 210 (2x3x5x7) and 385 (5x7x11), visually breaking down composite numbers into their prime components.", caption: none)]),
  [Rewrite the numerator and denominator as the product of the primes.], [#math.equation(block: false, alt: "the fraction 210 over 385 equals the fraction 2 times 3 times 5 times 7 over 5 times 7 times 11")[$frac(210, 385) = frac(2 ⋅ 3 ⋅ 5 ⋅ 7, 5 ⋅ 7 ⋅ 11)$]],
  [Remove the common factors.], [#figure(figph[Simplification of a mathematical fraction where common factors 5 and 7 are crossed out in both the numerator (2\*3\*5\*7) and denominator (5\*7\*11).], alt: "Simplification of a mathematical fraction where common factors 5 and 7 are crossed out in both the numerator (2*3*5*7) and denominator (5*7*11).", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 2 times 3 over 11")[$frac(2 ⋅ 3, 11)$]],
  [Multiply any remaining factors.], [#math.equation(block: false, alt: "the fraction 6 over 11")[$frac(6, 11)$]],
))
]
]

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

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

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

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

We can also simplify fractions containing variables. If a variable is a common factor in the #strong[numerator] and #strong[denominator], we remove it just as we do with an integer factor.

#examplebox("Example 5")[][
Simplify: #math.equation(block: false, alt: "the fraction 5 x y over 15 x .")[$frac(5 x y, 15 x) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 5 x y over 15 x")[$frac(5 x y, 15 x)$]]),
  [Rewrite numerator and denominator showing common factors.], [#math.equation(block: false, alt: "the fraction 5 times x times y over 3 times 5 times x")[$frac(5 · x · y, 3 · 5 · x)$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 5 times x times y over 3 times 5 times x")[$frac(cancel(5) · cancel(x) · y, 3 · cancel(5) · cancel(x))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction y over 3")[$frac(y, 3)$]],
))
]
]

#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 9 a over 9 b .")[$frac(9 a, 9 b) .$]

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


=== Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 3 over 4 .")[$frac(1, 2) · frac(3, 4) .$] To multiply #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] and #math.equation(block: false, alt: "the fraction 3 over 4 ,")[$frac(3, 4) ,$] think #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] of #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$]

Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] tiles evenly into two parts, we exchange them for smaller tiles.

#figure(figph[A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths.], alt: "A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths.", caption: none)

We see #math.equation(block: false, alt: "the fraction 6 over 8")[$frac(6, 8)$] is equivalent to #math.equation(block: false, alt: "the fraction 3 over 4 .")[$frac(3, 4) .$] Taking half of the six #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] tiles gives us three #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] tiles, which is #math.equation(block: false, alt: "the fraction 3 over 8 .")[$frac(3, 8) .$]

Therefore,

#math.equation(block: true, alt: "the fraction 1 over 2 times the fraction 3 over 4 equals the fraction 3 over 8")[$frac(1, 2) · frac(3, 4) = frac(3, 8)$]#examplebox("Example 6")[][
Use a diagram to model #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 3 over 4 .")[$frac(1, 2) · frac(3, 4) .$]

#solutionbox[
First shade in #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] of the rectangle.

#figure(figph[A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded.], alt: "A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded.", caption: none)

We will take #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] of this #math.equation(block: false, alt: "the fraction 3 over 4 ,")[$frac(3, 4) ,$] so we heavily shade #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] of the shaded region.

#figure(figph[A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded.], alt: "A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded.", caption: none)

Notice that #math.equation(block: false, alt: "3")[$3$] out of the #math.equation(block: false, alt: "8")[$8$] pieces are heavily shaded. This means that #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] of the rectangle is heavily shaded.

Therefore, #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] of #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] is #math.equation(block: false, alt: "the fraction 3 over 8 ,")[$frac(3, 8) ,$] or #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 3 over 4 equals the fraction 3 over 8 .")[$frac(1, 2) · frac(3, 4) = frac(3, 8) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use a diagram to model: #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 3 over 5 .")[$frac(1, 2) · frac(3, 5) .$]

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

#figure(figph[A rectangle is shown, divided vertically into five equal pieces. Three of the pieces are shaded. The first three pieces of the rectangle are divided by a horizontal line, creating six equal pieces. Three of the six pieces are darkly shaded.], alt: "A rectangle is shown, divided vertically into five equal pieces. Three of the pieces are shaded. The first three pieces of the rectangle are divided by a horizontal line, creating six equal pieces. Three of the six pieces are darkly shaded.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use a diagram to model: #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 5 over 6 .")[$frac(1, 2) · frac(5, 6) .$]

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

#figure(figph[A rectangle is shown, divided vertically into six equal pieces. Five of the pieces are shaded.], alt: "A rectangle is shown, divided vertically into six equal pieces. Five of the pieces are shaded.", caption: none)
]
]

Look at the result we got from the model in . We found that #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 3 over 4 equals the fraction 3 over 8 .")[$frac(1, 2) · frac(3, 4) = frac(3, 8) .$] Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 3 over 4")[$frac(1, 2) · frac(3, 4)$]]),
  [Multiply the numerators, and multiply the denominators.], [#math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 3 over 4")[$frac(1, 2) · frac(3, 4)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$]],
))

This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

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

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

#examplebox("Example 7")[][
Multiply, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 3 over 4 times the fraction 1 over 5 .")[$frac(3, 4) · frac(1, 5) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 3 over 4 times the fraction 1 over 5")[$frac(3, 4) · frac(1, 5)$]]),
  [Multiply the numerators; multiply the denominators.], [#math.equation(block: false, alt: "the fraction 3 times 1 over 4 times 5")[$frac(3 · 1, 4 · 5)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 3 over 20")[$frac(3, 20)$]],
))

There are no common factors, so the fraction is simplified.
]
]

#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 3 times the fraction 2 over 5 .")[$frac(1, 3) · frac(2, 5) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 15")[$frac(2, 15)$]
]
]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 21 over 40")[$frac(21, 40)$]
]
]

When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In Example 4.26 we will multiply two negatives, so the product will be positive.

#examplebox("Example 8")[][
Multiply, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 5 over 8 open parenthesis minus the fraction 2 over 3 close parenthesis .")[$− frac(5, 8) ( − frac(2, 3) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 5 over 8 open parenthesis minus the fraction 2 over 3 close parenthesis")[$− frac(5, 8) ( − frac(2, 3) )$]]),
  [The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators.], [#math.equation(block: false, alt: "the fraction 5 times 2 over 8 times 3")[$frac(5 ⋅ 2, 8 ⋅ 3)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 10 over 24")[$frac(10, 24)$]],
  [Look for common factors in the numerator and denominator. Rewrite showing common factors.], [#figure(figph[The fraction 5x/12x, illustrating the simplification of an algebraic expression by canceling out the common factor 'x' highlighted in red in both the numerator and denominator.], alt: "The fraction 5x/12x, illustrating the simplification of an algebraic expression by canceling out the common factor 'x' highlighted in red in both the numerator and denominator.", caption: none)],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 5 over 12")[$frac(5, 12)$]],
))

Another way to find this product involves removing common factors earlier.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 5 over 8 open parenthesis minus the fraction 2 over 3 close parenthesis")[$− frac(5, 8) ( − frac(2, 3) )$]]),
  [Determine the sign of the product. Multiply.], [#math.equation(block: false, alt: "the fraction 5 times 2 over 8 times 3")[$frac(5 ⋅ 2, 8 ⋅ 3)$]],
  [Show common factors and then remove them.], [#figure(figph[A mathematical fraction with a numerator of 5 multiplied by 2, and a denominator of 4 multiplied by 2 multiplied by 3. The '2' in both the numerator and denominator is crossed out in red, illustrating cancellation.], alt: "A mathematical fraction with a numerator of 5 multiplied by 2, and a denominator of 4 multiplied by 2 multiplied by 3. The '2' in both the numerator and denominator is crossed out in red, illustrating cancellation.", caption: none)],
  [Multiply remaining factors.], [#math.equation(block: false, alt: "the fraction 5 over 12")[$frac(5, 12)$]],
))

We get the same result.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 4 over 7 open parenthesis minus the fraction 5 over 8 close parenthesis .")[$− frac(4, 7) ( − frac(5, 8) ) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 14")[$frac(5, 14)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 7 over 12 open parenthesis minus the fraction 8 over 9 close parenthesis .")[$− frac(7, 12) ( − frac(8, 9) ) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 14 over 27")[$frac(14, 27)$]
]
]

#examplebox("Example 9")[][
Multiply, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 14 over 15 times the fraction 20 over 21 .")[$− frac(14, 15) · frac(20, 21) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 14 over 15 times the fraction 20 over 21")[$− frac(14, 15) · frac(20, 21)$]]),
  [Determine the sign of the product; multiply.], [#math.equation(block: false, alt: "minus the fraction 14 over 15 times the fraction 20 over 21")[$− frac(14, 15) · frac(20, 21)$]],
  [Are there any common factors in the numerator and the denominator? #linebreak() $#h(0.2em)$We know that 7 is a factor of 14 and 21, and 5 is a factor of 20 and 15.], [],
  [Rewrite showing common factors.], [#figure(figph[A fraction with a negative sign, where common factors 7 and 5 are shown crossed out from the numerator (2\*7\*4\*5) and the denominator (3\*5\*3\*7), illustrating simplification.], alt: "A fraction with a negative sign, where common factors 7 and 5 are shown crossed out from the numerator (2*7*4*5) and the denominator (3*5*3*7), illustrating simplification.", caption: none)],
  [Remove the common factors.], [#math.equation(block: false, alt: "minus the fraction 2 times 4 over 3 times 3")[$− frac(2 · 4, 3 · 3)$]],
  [Multiply the remaining factors.], [#math.equation(block: false, alt: "minus the fraction 8 over 9")[$− frac(8, 9)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 10 over 28 times the fraction 8 over 15 .")[$− frac(10, 28) · frac(8, 15) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 9 over 20 times the fraction 5 over 12 .")[$− frac(9, 20) · frac(5, 12) .$]

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

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

#examplebox("Example 10")[][
Multiply, and write the answer in simplified form:

ⓐ #math.equation(block: false, alt: "the fraction 1 over 7 times 56")[$frac(1, 7) · 56$]

ⓑ #math.equation(block: false, alt: "the fraction 12 over 5 open parenthesis −20 x close parenthesis")[$frac(12, 5) ( −20 x )$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "the fraction 1 over 7 times 56")[$frac(1, 7) · 56$]],
  [Write 56 as a fraction.], [#math.equation(block: false, alt: "the fraction 1 over 7 times the fraction 56 over 1")[$frac(1, 7) · frac(56, 1)$]],
  [Determine the sign of the product; multiply.], [#math.equation(block: false, alt: "the fraction 56 over 7")[$frac(56, 7)$]],
  [Simplify.], [#math.equation(block: false, alt: "8")[$8$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "the fraction 12 over 5 open parenthesis −20 x close parenthesis")[$frac(12, 5) ( −20 x )$]],
  [Write −20x as a fraction.], [#math.equation(block: false, alt: "the fraction 12 over 5 open parenthesis the fraction −20 x over 1 close parenthesis")[$frac(12, 5) ( frac(−20 x, 1) )$]],
  [Determine the sign of the product; multiply.], [#math.equation(block: false, alt: "minus the fraction 12 times 20 times x over 5 times 1")[$− frac(12 · 20 · x, 5 · 1)$]],
  [Show common factors and then remove them.], [#figure(figph[A mathematical expression showing a fraction with 12, 4, and 5x multiplied in the numerator, and 5 and 1 multiplied in the denominator, with the '5's in both numerator and denominator canceled out, preceded by a negative sign.], alt: "A mathematical expression showing a fraction with 12, 4, and 5x multiplied in the numerator, and 5 and 1 multiplied in the denominator, with the '5's in both numerator and denominator canceled out, preceded by a negative sign.", caption: none)],
  [Multiply remaining factors; simplify.], [−48x],
))
]
]

#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 72")[$#h(0.2em) frac(1, 8) · 72 #h(1.0em)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 11 over 3 open parenthesis −9 a close parenthesis")[$#h(0.2em) frac(11, 3) ( −9 a )$]

#solutionbox[
+ ⓐ 9
+ ⓑ −33#emph[a]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply, and write the answer in simplified form:

+ ⓐ #math.equation(block: false, alt: "the fraction 3 over 8 times 64")[$frac(3, 8) · 64$]
+ ⓑ #math.equation(block: false, alt: "16 x times the fraction 11 over 12")[$16 x · frac(11, 12)$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "24")[$24$]
+ ⓑ #math.equation(block: false, alt: "the fraction 44 x over 3")[$frac(44 x, 3)$]
]
]


=== Find Reciprocals

The fractions #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] and #math.equation(block: false, alt: "the fraction 3 over 2")[$frac(3, 2)$] are related to each other in a special way. So are #math.equation(block: false, alt: "minus the fraction 10 over 7")[$− frac(10, 7)$] and #math.equation(block: false, alt: "minus the fraction 7 over 10 .")[$− frac(7, 10) .$] Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be #math.equation(block: false, alt: "1 .")[$1 .$]

#math.equation(block: true, alt: "the fraction 2 over 3 times the fraction 3 over 2 equals 1 and minus the fraction 10 over 7 open parenthesis minus the fraction 7 over 10 close parenthesis equals 1")[$frac(2, 3) · frac(3, 2) = 1 #h(1.0em) "and" #h(1.0em) − frac(10, 7) ( − frac(7, 10) ) = 1$]Such pairs of numbers are called reciprocals.

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

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

A number and its reciprocal have a product of #math.equation(block: false, alt: "1 .")[$1 .$]

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

To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.

To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

#figure(figph[“a” over “b” multiplied by “b” over “a” equals positive one.], alt: "“a” over “b” multiplied by “b” over “a” equals positive one.", caption: none)

To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to #math.equation(block: false, alt: "1 .")[$1 .$] Is there any number #math.equation(block: false, alt: "r")[$r$] so that #math.equation(block: false, alt: "0 times r equals 1 ?")[$0 · r = 1 ?$] No. So, the number #math.equation(block: false, alt: "0")[$0$] does not have a reciprocal.

#examplebox("Example 11")[][
Find the reciprocal of each number. Then check that the product of each number and its reciprocal is #math.equation(block: false, alt: "1 .")[$1 .$]

+ ⓐ #math.equation(block: false, alt: "the fraction 4 over 9")[$frac(4, 9)$]
+ ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 6")[$− frac(1, 6)$]
+ ⓒ #math.equation(block: false, alt: "minus the fraction 14 over 5")[$− frac(14, 5)$]
+ ⓓ #math.equation(block: false, alt: "7")[$7$]

#solutionbox[
To find the reciprocals, we keep the sign and invert the fractions.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [Find the reciprocal of #math.equation(block: false, alt: "the fraction 4 over 9")[$frac(4, 9)$].], [The reciprocal of #math.equation(block: false, alt: "the fraction 4 over 9")[$frac(4, 9)$] is #math.equation(block: false, alt: "the fraction 9 over 4")[$frac(9, 4)$].],
  [Check:], [],
  [Multiply the number and its reciprocal.], [#math.equation(block: false, alt: "the fraction 4 over 9 times the fraction 9 over 4")[$frac(4, 9) ⋅ frac(9, 4)$]],
  [Multiply numerators and denominators.], [#math.equation(block: false, alt: "the fraction 36 over 36")[$frac(36, 36)$]],
  [Simplify.], [#math.equation(block: false, alt: "1 ✓")[$1 ✓$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [Find the reciprocal of #math.equation(block: false, alt: "minus the fraction 1 over 6")[$− frac(1, 6)$].], [#math.equation(block: false, alt: "minus the fraction 6 over 1")[$− frac(6, 1)$]],
  [Simplify.], [#math.equation(block: false, alt: "minus 6")[$− 6$]],
  [Check:], [#math.equation(block: false, alt: "minus the fraction 1 over 6 times open parenthesis minus 6 close parenthesis")[$− frac(1, 6) ⋅ ( − 6 )$]],
  [], [#math.equation(block: false, alt: "1 ✓")[$1 ✓$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [Find the reciprocal of #math.equation(block: false, alt: "minus the fraction 14 over 5")[$− frac(14, 5)$].], [#math.equation(block: false, alt: "minus the fraction 5 over 14")[$− frac(5, 14)$]],
  [Check:], [#math.equation(block: false, alt: "minus the fraction 14 over 5 times open parenthesis minus the fraction 5 over 14 close parenthesis")[$− frac(14, 5) ⋅ ( − frac(5, 14) )$]],
  [], [#math.equation(block: false, alt: "the fraction 70 over 70")[$frac(70, 70)$]],
  [], [#math.equation(block: false, alt: "1 ✓")[$1 ✓$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓓ], []),
  [Find the reciprocal of #math.equation(block: false, alt: "7")[$7$].], [],
  [Write #math.equation(block: false, alt: "7")[$7$] as a fraction.], [#math.equation(block: false, alt: "the fraction 7 over 1")[$frac(7, 1)$]],
  [Write the reciprocal of #math.equation(block: false, alt: "the fraction 7 over 1")[$frac(7, 1)$].], [#math.equation(block: false, alt: "the fraction 1 over 7")[$frac(1, 7)$]],
  [Check:], [#math.equation(block: false, alt: "7 times open parenthesis the fraction 1 over 7 close parenthesis")[$7 ⋅ ( frac(1, 7) )$]],
  [], [#math.equation(block: false, alt: "1 ✓")[$1 ✓$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the reciprocal:

+ ⓐ #math.equation(block: false, alt: "the fraction 5 over 7")[$frac(5, 7)$]
+ ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 8")[$− frac(1, 8)$]
+ ⓒ #math.equation(block: false, alt: "minus the fraction 11 over 4")[$− frac(11, 4)$]
+ ⓓ #math.equation(block: false, alt: "14")[$14$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 7 over 5")[$frac(7, 5)$]
+ ⓑ #math.equation(block: false, alt: "−8")[$−8$]
+ ⓒ #math.equation(block: false, alt: "minus the fraction 4 over 11")[$− frac(4, 11)$]
+ ⓓ #math.equation(block: false, alt: "the fraction 1 over 14")[$frac(1, 14)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the reciprocal:

+ ⓐ #math.equation(block: false, alt: "the fraction 3 over 7")[$frac(3, 7)$]
+ ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 12")[$− frac(1, 12)$]
+ ⓒ #math.equation(block: false, alt: "minus the fraction 14 over 9")[$− frac(14, 9)$]
+ ⓓ #math.equation(block: false, alt: "21")[$21$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 7 over 3")[$frac(7, 3)$]
+ ⓑ #math.equation(block: false, alt: "minus 12")[$− 12$]
+ ⓒ #math.equation(block: false, alt: "minus the fraction 9 over 14")[$− frac(9, 14)$]
+ ⓓ #math.equation(block: false, alt: "the fraction 1 over 21")[$frac(1, 21)$]
]
]

In a previous chapter, we worked with opposites and absolute values.  compares opposites, absolute values, and reciprocals.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Opposite], [Absolute Value], [Reciprocal]),
  [has opposite sign], [is never negative], [has same sign, fraction inverts],
))

#examplebox("Example 12")[][
Fill in the chart for each fraction in the left column:

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([Number], [Opposite], [Absolute Value], [Reciprocal]),
  [#math.equation(block: false, alt: "minus the fraction 3 over 8")[$− frac(3, 8)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 9 over 5")[$frac(9, 5)$]], [], [], [],
  [#math.equation(block: false, alt: "−5")[$−5$]], [], [], [],
))

#solutionbox[
To find the opposite, change the sign. To find the absolute value, leave the positive numbers the same, but take the opposite of the negative numbers. To find the reciprocal, keep the sign the same and #strong[invert] the fraction.

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([Number], [Opposite], [Absolute Value], [Reciprocal]),
  [#math.equation(block: false, alt: "minus the fraction 3 over 8")[$− frac(3, 8)$]], [#math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$]], [#math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$]], [#math.equation(block: false, alt: "minus the fraction 8 over 3")[$− frac(8, 3)$]],
  [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "2")[$2$]],
  [#math.equation(block: false, alt: "the fraction 9 over 5")[$frac(9, 5)$]], [#math.equation(block: false, alt: "minus the fraction 9 over 5")[$− frac(9, 5)$]], [#math.equation(block: false, alt: "the fraction 9 over 5")[$frac(9, 5)$]], [#math.equation(block: false, alt: "the fraction 5 over 9")[$frac(5, 9)$]],
  [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "minus the fraction 1 over 5")[$− frac(1, 5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Fill in the chart for each number given:

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([Number], [Opposite], [Absolute Value], [Reciprocal]),
  [#math.equation(block: false, alt: "minus the fraction 5 over 8")[$− frac(5, 8)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 8 over 3")[$frac(8, 3)$]], [], [], [],
  [#math.equation(block: false, alt: "−8")[$−8$]], [], [], [],
))

#solutionbox[
#figure(figph[The image is of a table with four columns and 5 rows. The top header row reads: number, opposite, absolute value, and reciprocal. The rest of the cells contain answers to the problem.], alt: "The image is of a table with four columns and 5 rows. The top header row reads: number, opposite, absolute value, and reciprocal. The rest of the cells contain answers to the problem.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Fill in the chart for each number given:

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([Number], [Opposite], [Absolute Value], [Reciprocal]),
  [#math.equation(block: false, alt: "minus the fraction 4 over 7")[$− frac(4, 7)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 9 over 4")[$frac(9, 4)$]], [], [], [],
  [#math.equation(block: false, alt: "−1")[$−1$]], [], [], [],
))

#solutionbox[
#figure(figph[This table demonstrates how to find the opposite, absolute value, and reciprocal for various numbers, including fractions and integers, as fundamental mathematical concepts.], alt: "This table demonstrates how to find the opposite, absolute value, and reciprocal for various numbers, including fractions and integers, as fundamental mathematical concepts.", caption: none)
]
]


=== Divide Fractions

Why is #math.equation(block: false, alt: "12 divided by 3 equals 4 ?")[$12 ÷ 3 = 4 ?$] We previously modeled this with counters. How many groups of #math.equation(block: false, alt: "3")[$3$] counters can be made from a group of #math.equation(block: false, alt: "12")[$12$] counters?

#figure(figph[Four red ovals are shown. Inside each oval are three grey circles.], alt: "Four red ovals are shown. Inside each oval are three grey circles.", caption: none)

There are #math.equation(block: false, alt: "4")[$4$] groups of #math.equation(block: false, alt: "3")[$3$] counters. In other words, there are four #math.equation(block: false, alt: "3")[$3$]s in #math.equation(block: false, alt: "12 .")[$12 .$] So, #math.equation(block: false, alt: "12 divided by 3 equals 4 .")[$12 ÷ 3 = 4 .$]

What about dividing fractions? Suppose we want to find the quotient: #math.equation(block: false, alt: "the fraction 1 over 2 divided by the fraction 1 over 6 .")[$frac(1, 2) ÷ frac(1, 6) .$] We need to figure out how many #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$]s there are in #math.equation(block: false, alt: "the fraction 1 over 2 .")[$frac(1, 2) .$] We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown. Notice, there are three #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] tiles in #math.equation(block: false, alt: "the fraction 1 over 2 ,")[$frac(1, 2) ,$] so #math.equation(block: false, alt: "the fraction 1 over 2 divided by the fraction 1 over 6 equals 3 .")[$frac(1, 2) ÷ frac(1, 6) = 3 .$]

#figure(figph[A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.], alt: "A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.", caption: none)

#examplebox("Example 13")[][
Model: #math.equation(block: false, alt: "the fraction 1 over 4 divided by the fraction 1 over 8 .")[$frac(1, 4) ÷ frac(1, 8) .$]

#solutionbox[
We want to determine how many #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]s are in #math.equation(block: false, alt: "the fraction 1 over 4 .")[$frac(1, 4) .$] Start with one #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] tile. Line up #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] tiles underneath the #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] tile.

#figure(figph[A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.], alt: "A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.", caption: none)

There are two #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]s in #math.equation(block: false, alt: "the fraction 1 over 4 .")[$frac(1, 4) .$]

So, #math.equation(block: false, alt: "the fraction 1 over 4 divided by the fraction 1 over 8 equals 2 .")[$frac(1, 4) ÷ frac(1, 8) = 2 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model: #math.equation(block: false, alt: "the fraction 1 over 3 divided by the fraction 1 over 6 .")[$frac(1, 3) ÷ frac(1, 6) .$]

#solutionbox[
2

#figure(figph[A rectangle is shown, labeled as one third. Below it is an identical rectangle split into two equal pieces, each labeled as one sixth.], alt: "A rectangle is shown, labeled as one third. Below it is an identical rectangle split into two equal pieces, each labeled as one sixth.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model: #math.equation(block: false, alt: "the fraction 1 over 2 divided by the fraction 1 over 4 .")[$frac(1, 2) ÷ frac(1, 4) .$]

#solutionbox[
2

#figure(figph[A rectangle is shown, labeled as one half. Below it is an identical rectangle split into two equal pieces, each labeled as one fourth.], alt: "A rectangle is shown, labeled as one half. Below it is an identical rectangle split into two equal pieces, each labeled as one fourth.", caption: none)
]
]

#examplebox("Example 14")[][
Model: #math.equation(block: false, alt: "2 divided by the fraction 1 over 4 .")[$2 ÷ frac(1, 4) .$]

#solutionbox[
We are trying to determine how many #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]s there are in #math.equation(block: false, alt: "2 .")[$2 .$] We can model this as shown.

#figure(figph[Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.], alt: "Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.", caption: none)

Because there are eight #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]s in #math.equation(block: false, alt: "2 , 2 divided by the fraction 1 over 4 equals 8 .")[$2 , 2 ÷ frac(1, 4) = 8 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model: #math.equation(block: false, alt: "2 divided by the fraction 1 over 3")[$2 ÷ frac(1, 3)$]

#solutionbox[
6

#figure(figph[Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into three pieces. Each of the six pieces is labeled as one third.], alt: "Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into three pieces. Each of the six pieces is labeled as one third.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model: #math.equation(block: false, alt: "3 divided by the fraction 1 over 2")[$3 ÷ frac(1, 2)$]

#solutionbox[
6

#figure(figph[Three rectangles are shown, each labeled as 1. Below are three identical rectangles, each split into 2 equal pieces. Each of these six pieces is labeled as one half.], alt: "Three rectangles are shown, each labeled as 1. Below are three identical rectangles, each split into 2 equal pieces. Each of these six pieces is labeled as one half.", caption: none)
]
]

Let’s use money to model #math.equation(block: false, alt: "2 divided by the fraction 1 over 4")[$2 ÷ frac(1, 4)$] in another way. We often read #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown. So we can think of #math.equation(block: false, alt: "2 divided by the fraction 1 over 4")[$2 ÷ frac(1, 4)$] as, “How many quarters are there in two dollars?” One dollar is #math.equation(block: false, alt: "4")[$4$] quarters, so #math.equation(block: false, alt: "2")[$2$] dollars would be #math.equation(block: false, alt: "8")[$8$] quarters. So again, #math.equation(block: false, alt: "2 divided by the fraction 1 over 4 equals 8 .")[$2 ÷ frac(1, 4) = 8 .$]

#figure(figph[A picture of a United States quarter is shown.], alt: "A picture of a United States quarter is shown.", caption: [The U.S. coin called a quarter is worth one-fourth of a dollar.])

Using fraction tiles, we showed that #math.equation(block: false, alt: "the fraction 1 over 2 divided by the fraction 1 over 6 equals 3 .")[$frac(1, 2) ÷ frac(1, 6) = 3 .$] Notice that #math.equation(block: false, alt: "the fraction 1 over 2 times the fraction 6 over 1 equals 3")[$frac(1, 2) · frac(6, 1) = 3$] also. How are #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] and #math.equation(block: false, alt: "the fraction 6 over 1")[$frac(6, 1)$] related? They are reciprocals. This leads us to the procedure for fraction division.

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

If #math.equation(block: false, alt: "a , b , c ,")[$a , b , c ,$] and #math.equation(block: false, alt: "d")[$d$] are numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 ,")[$b ≠ 0 , c ≠ 0 ,$] and #math.equation(block: false, alt: "d not equal to 0 ,")[$d ≠ 0 ,$] 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, 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")[$b ≠ 0 , c ≠ 0$] and #math.equation(block: false, alt: "d not equal to 0")[$d ≠ 0$] to be sure we don’t divide by zero.

#examplebox("Example 15")[][
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 2 over 5 divided by open parenthesis minus the fraction 3 over 7 close parenthesis .")[$frac(2, 5) ÷ ( − frac(3, 7) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 2 over 5 divided by open parenthesis minus the fraction 3 over 7 close parenthesis")[$frac(2, 5) ÷ ( − frac(3, 7) )$]]),
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 2 over 5 open parenthesis minus the fraction 7 over 3 close parenthesis")[$frac(2, 5) ( − frac(7, 3) )$]],
  [Multiply. The product is negative.], [#math.equation(block: false, alt: "minus the fraction 14 over 15")[$− frac(14, 15)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 3 over 7 divided by open parenthesis minus the fraction 2 over 3 close parenthesis .")[$frac(3, 7) ÷ ( − frac(2, 3) ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 2 over 3 divided by open parenthesis minus the fraction 7 over 5 close parenthesis .")[$frac(2, 3) ÷ ( − frac(7, 5) ) .$]

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

#examplebox("Example 16")[][
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 2 over 3 divided by the fraction n over 5 .")[$frac(2, 3) ÷ frac(n, 5) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 3 over 5 divided by the fraction p over 7 .")[$frac(3, 5) ÷ frac(p, 7) .$]

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

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

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

#examplebox("Example 17")[][
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 3 over 4 divided by open parenthesis minus the fraction 7 over 8 close parenthesis .")[$− frac(3, 4) ÷ ( − frac(7, 8) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 3 over 4 divided by open parenthesis minus the fraction 7 over 8 close parenthesis")[$− frac(3, 4) ÷ ( − frac(7, 8) )$]]),
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "minus the fraction 3 over 4 times open parenthesis minus the fraction 8 over 7 close parenthesis")[$− frac(3, 4) · ( − frac(8, 7) )$]],
  [Multiply. Remember to determine the sign first.], [#math.equation(block: false, alt: "the fraction 3 times 8 over 4 times 7")[$frac(3 · 8, 4 · 7)$]],
  [Rewrite to show common factors.], [#math.equation(block: false, alt: "the fraction 3 times 4 times 2 over 4 times 7")[$frac(3 · cancel(4) · 2, cancel(4) · 7)$]],
  [Remove common factors and simplify.], [#math.equation(block: false, alt: "the fraction 6 over 7")[$frac(6, 7)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 2 over 3 divided by open parenthesis minus the fraction 5 over 6 close parenthesis .")[$− frac(2, 3) ÷ ( − frac(5, 6) ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "minus the fraction 5 over 6 divided by open parenthesis minus the fraction 2 over 3 close parenthesis .")[$− frac(5, 6) ÷ ( − frac(2, 3) ) .$]

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

#examplebox("Example 18")[][
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 7 over 18 divided by the fraction 14 over 27 .")[$frac(7, 18) ÷ frac(14, 27) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 7 over 18 divided by the fraction 14 over 27")[$frac(7, 18) ÷ frac(14, 27)$]]),
  [Multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "the fraction 7 over 18 times the fraction 27 over 14")[$frac(7, 18) · frac(27, 14)$]],
  [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 displays the cancellation of common factors. The terms 7 and 9 are crossed out from both the numerator and denominator, simplifying the expression.], alt: "A mathematical fraction displays the cancellation of common factors. The terms 7 and 9 are crossed out from both the numerator and denominator, simplifying the expression.", 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"))[
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 7 over 27 divided by the fraction 35 over 36 .")[$frac(7, 27) ÷ frac(35, 36) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide, and write the answer in simplified form: #math.equation(block: false, alt: "the fraction 5 over 14 divided by the fraction 15 over 28 .")[$frac(5, 14) ÷ frac(15, 28) .$]

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

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

- #link("https://www.openstax.org/l/24SimplifyFrac")[Simplifying Fractions]
- #link("https://www.openstax.org/l/24MultiplyFrac")[Multiplying Fractions (Positive Only)]
- #link("https://www.openstax.org/l/24MultSigned")[Multiplying Signed Fractions]
- #link("https://www.openstax.org/l/24DivideFrac")[Dividing Fractions (Positive Only)]
- #link("https://www.openstax.org/l/24DivideSign")[Dividing Signed Fractions]
]


=== Key Concepts

- #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")[$b ≠ 0$], #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$], then #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")[$frac(a ⋅ c, b ⋅ c) = frac(a, b)$].
- #emph[Simplify a fraction.]+ Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
+ Simplify, using the equivalent fractions property, by removing common factors.
+ Multiply any remaining factors.
- #emph[Fraction Multiplication]- If #math.equation(block: false, alt: "a , b , c ,")[$a , b , c ,$] and #math.equation(block: false, alt: "d")[$d$] are numbers where #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$]and #math.equation(block: false, alt: "d not equal to 0")[$d ≠ 0$], 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")[$frac(a, b) ⋅ frac(c, d) = frac(a c, b d)$].
- #emph[Reciprocal]- A number and its reciprocal have a product of #math.equation(block: false, alt: "1")[$1$]. #math.equation(block: false, alt: "the fraction a over b times the fraction b over a equals 1")[$frac(a, b) ⋅ frac(b, a) = 1$]
- #figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Opposite]], [#emph[Absolute Value]], [#emph[Reciprocal]]),
  [has opposite sign], [is never negative], [has same sign, fraction inverts],
))
- #emph[Fraction Division]- If #math.equation(block: false, alt: "a , b , c ,")[$a , b , c ,$] and #math.equation(block: false, alt: "d")[$d$] are numbers where #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$] and #math.equation(block: false, alt: "d not equal to 0")[$d ≠ 0$] , then
 #linebreak() #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, multiply the first fraction by the reciprocal of the second.


==== Practice Makes Perfect

#strong[Simplify Fractions]

In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.

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

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

#math.equation(block: true, alt: "the fraction 8 over 24")[$frac(8, 24)$]

#math.equation(block: true, alt: "the fraction 15 over 20")[$frac(15, 20)$]

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

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

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

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

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

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

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

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

#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 168 over 192")[$− frac(168, 192)$]

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

#math.equation(block: true, alt: "minus the fraction 140 over 224")[$− frac(140, 224)$]

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

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

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

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

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

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

#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, use a diagram to model.

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 3")[$frac(1, 3)$]
 #linebreak() 
#figure(figph[The image shows green blocks to demonstrate the fraction problem given.], alt: "The image shows green blocks to demonstrate the fraction problem given.", caption: none)
]

#math.equation(block: true, alt: "the fraction 1 over 2 times the fraction 5 over 8")[$frac(1, 2) · frac(5, 8)$]

#math.equation(block: true, alt: "the fraction 1 over 3 times the fraction 5 over 6")[$frac(1, 3) · frac(5, 6)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 18")[$frac(5, 18)$]
 #linebreak() 
#figure(figph[The image shows green blocks to demonstrate the fraction problem given.], alt: "The image shows green blocks to demonstrate the fraction problem given.", caption: none)
]

#math.equation(block: true, alt: "the fraction 1 over 3 times the fraction 2 over 5")[$frac(1, 3) · frac(2, 5)$]

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

#math.equation(block: true, alt: "the fraction 2 over 5 times the fraction 1 over 3")[$frac(2, 5) · frac(1, 3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 15")[$frac(2, 15)$]
]

#math.equation(block: true, alt: "the fraction 1 over 2 times the fraction 3 over 8")[$frac(1, 2) · frac(3, 8)$]

#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")[$− frac(2, 3) ( − frac(3, 8) )$]

#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")[$− frac(3, 4) ( − frac(4, 9) )$]

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

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

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

#math.equation(block: true, alt: "the fraction 7 over 12 open parenthesis minus the fraction 8 over 21 close parenthesis")[$frac(7, 12) ( − frac(8, 21) )$]

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

#math.equation(block: true, alt: "the fraction 5 over 12 open parenthesis minus the fraction 8 over 15 close parenthesis")[$frac(5, 12) ( − frac(8, 15) )$]

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

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

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

#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")[$( − frac(63, 84) ) ( − frac(44, 90) )$]

#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")[$( − frac(33, 60) ) ( − frac(40, 88) )$]

#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: "−28 p open parenthesis minus the fraction 1 over 4 close parenthesis")[$−28 p ( − frac(1, 4) )$]

#solutionbox[
7#emph[p]
]

#math.equation(block: true, alt: "−51 q open parenthesis minus the fraction 1 over 3 close parenthesis")[$−51 q ( − frac(1, 3) )$]

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

#solutionbox[
−34
]

#math.equation(block: true, alt: "the fraction 14 over 5 open parenthesis −15 close parenthesis")[$frac(14, 5) ( −15 )$]

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

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

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

#math.equation(block: true, alt: "open parenthesis the fraction 2 over 3 close parenthesis cubed")[$attach(( frac(2, 3) ), t: 3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 over 27")[$frac(8, 27)$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 4 over 5 close parenthesis squared")[$attach(( frac(4, 5) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis the fraction 6 over 5 close parenthesis to the power 4")[$attach(( frac(6, 5) ), t: 4)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1296 over 625")[$frac(1296, 625)$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 4 over 7 close parenthesis to the power 4")[$attach(( frac(4, 7) ), t: 4)$]

#strong[Find Reciprocals]

In the following exercises, find the reciprocal.

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

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

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

#math.equation(block: true, alt: "minus the fraction 5 over 17")[$− frac(5, 17)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 17 over 5")[$− frac(17, 5)$]
]

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

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

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

#math.equation(block: true, alt: "−13")[$−13$]

#math.equation(block: true, alt: "−19")[$−19$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 19")[$− frac(1, 19)$]
]

#math.equation(block: true, alt: "−1")[$−1$]

#math.equation(block: true, alt: "1")[$1$]

#solutionbox[
1
]

Fill in the chart.

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([], [Opposite], [Absolute Value], [Reciprocal]),
  [#math.equation(block: false, alt: "minus the fraction 7 over 11")[$− frac(7, 11)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 4 over 5")[$frac(4, 5)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 10 over 7")[$frac(10, 7)$]], [], [], [],
  [#math.equation(block: false, alt: "−8")[$−8$]], [], [], [],
))

#solutionbox[
#figure(figph[This table demonstrates how to find the opposite, absolute value, and reciprocal for various numbers, including fractions and integers, as fundamental mathematical concepts.], alt: "This table demonstrates how to find the opposite, absolute value, and reciprocal for various numbers, including fractions and integers, as fundamental mathematical concepts.", caption: none)
]

Fill in the chart.

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([], [Opposite], [Absolute Value], [Reciprocal]),
  [#math.equation(block: false, alt: "minus the fraction 3 over 13")[$− frac(3, 13)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 9 over 14")[$frac(9, 14)$]], [], [], [],
  [#math.equation(block: false, alt: "the fraction 15 over 7")[$frac(15, 7)$]], [], [], [],
  [#math.equation(block: false, alt: "−9")[$−9$]], [], [], [],
))

#solutionbox[
#figure(figph[A table is shown with four columns and five rows. The first row reads Number, Opposite, Absolute Value, and Reciprocal. The second row reads negative three thirteenths, three thirteenths, three thirteenths, negative thirteen thirds. The third row reads nine fourteenths, negative nine fourteenths, nine fourteenths, and fourteen ninths. The fourth row reads fifteen sevenths, negative fifteen sevenths, fifteen sevenths, and seven fifteenths. The last row reads negative nine, nine, nine, negative one ninth.], alt: "A table is shown with four columns and five rows. The first row reads Number, Opposite, Absolute Value, and Reciprocal. The second row reads negative three thirteenths, three thirteenths, three thirteenths, negative thirteen thirds. The third row reads nine fourteenths, negative nine fourteenths, nine fourteenths, and fourteen ninths. The fourth row reads fifteen sevenths, negative fifteen sevenths, fifteen sevenths, and seven fifteenths. The last row reads negative nine, nine, nine, negative one ninth.", caption: none)
]

#strong[Divide Fractions]

In the following exercises, model each fraction division.

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

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

#solutionbox[
4
 #linebreak() 
#figure(figph[The image shows blocks to demonstrate the fraction division problem given.], alt: "The image shows blocks to demonstrate the fraction division problem given.", caption: none)
]

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

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

#solutionbox[
12
 #linebreak() 
#figure(figph[The image shows blocks to demonstrate the fraction division problem given.], alt: "The image shows blocks to demonstrate the fraction division problem given.", caption: none)
]

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

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

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

#solutionbox[
4
]

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 16 over 15")[$frac(16, 15)$]
]

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

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

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

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

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

#solutionbox[
1
]

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

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

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

#math.equation(block: true, alt: "the fraction 5 over 8 divided by the fraction a over 10")[$frac(5, 8) ÷ frac(a, 10)$]

#math.equation(block: true, alt: "the fraction 5 over 6 divided by the fraction c over 15")[$frac(5, 6) ÷ frac(c, 15)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 25 over 2 c")[$frac(25, 2 c)$]
]

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

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

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

#math.equation(block: true, alt: "the fraction 7 p over 12 divided by the fraction 21 p over 8")[$frac(7 p, 12) ÷ frac(21 p, 8)$]

#math.equation(block: true, alt: "the fraction 5 q over 12 divided by the fraction 15 q over 8")[$frac(5 q, 12) ÷ frac(15 q, 8)$]

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

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 14 r over 15 s")[$frac(14 r, 15 s)$]
]

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

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

#solutionbox[
−12
]

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

#math.equation(block: true, alt: "the fraction 2 over 5 divided by open parenthesis −10 close parenthesis")[$frac(2, 5) ÷ ( −10 )$]

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

#math.equation(block: true, alt: "−18 divided by open parenthesis minus the fraction 9 over 2 close parenthesis")[$−18 ÷ ( − frac(9, 2) )$]

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

#solutionbox[
9
]

#math.equation(block: true, alt: "the fraction 1 over 2 divided by open parenthesis minus the fraction 3 over 4 close parenthesis divided by the fraction 7 over 8")[$frac(1, 2) ÷ ( − frac(3, 4) ) ÷ frac(7, 8)$]

#math.equation(block: true, alt: "the fraction 11 over 2 divided by the fraction 7 over 8 times the fraction 2 over 11")[$frac(11, 2) ÷ frac(7, 8) · frac(2, 11)$]

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


==== 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. Write your result as an improper fraction and as a mixed number.

ⓑ Measuring cups usually come in sets of #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 Imelda could measure the brown sugar needed to double the recipe.

#strong[Baking] Nina is making #math.equation(block: false, alt: "4")[$4$] 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. Write your result as an improper fraction and as a mixed number.
+ ⓑ Measuring cups usually come in sets of #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 Nina could measure the condensed milk she needs.

#solutionbox[
- ⓐ #math.equation(block: false, alt: "4 times the fraction 2 over 3")[$4 · frac(2, 3)$] cups#math.equation(block: false, alt: "equals the fraction 8 over 3")[$= frac(8, 3)$] cups#math.equation(block: false, alt: "equals 2 the fraction 2 over 3")[$= 2 frac(2, 3)$] cups$$
- ⓑ Answers will vary.
]

#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?

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

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


==== Writing Exercises

Explain how you find the reciprocal of a fraction.

Explain how you find the reciprocal of a negative fraction.

#solutionbox[
Answers will vary.
]

Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into #math.equation(block: false, alt: "6")[$6$] or #math.equation(block: false, alt: "8")[$8$] slices. Would he prefer #math.equation(block: false, alt: "3")[$3$] out of #math.equation(block: false, alt: "6")[$6$] slices or #math.equation(block: false, alt: "4")[$4$] out of #math.equation(block: false, alt: "8")[$8$] slices? Rafael replied that since he wasn’t very hungry, he would prefer #math.equation(block: false, alt: "3")[$3$] out of #math.equation(block: false, alt: "6")[$6$] slices. Explain what is wrong with Rafael’s reasoning.

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

#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 student self-evaluation chart for fractions skills: simplifying, multiplying, reciprocals, and dividing, rated as 'Confidently', 'With some help', or 'No-I don't get it!'.], alt: "A student self-evaluation chart for fractions skills: simplifying, multiplying, reciprocals, and dividing, rated as 'Confidently', 'With some help', or 'No-I don't get it!'.", caption: none)

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