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== 1.6#h(0.6em)Add and Subtract Fractions

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


=== Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

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

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

#math.equation(block: true, alt: "the fraction a over c plus the fraction b over c equals the fraction a plus b over c and the fraction a over c minus the fraction b over c equals the fraction a minus b over c")[$frac(a, c) + frac(b, c) = frac(a + b, c) #h(1em) "and" #h(1em) frac(a, c) − #h(0.2em) frac(b, c) = frac(a − b, c)$]To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
]

#examplebox("Example 1")[][
Find the sum: #math.equation(block: false, alt: "the fraction x over 3 plus the fraction 2 over 3 .")[$frac(x, 3) + frac(2, 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction x over 3 plus the fraction 2 over 3")[$frac(x, 3) + frac(2, 3)$]]),
  [Add the numerators and place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction x plus 2 over 3")[$frac(x + 2, 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "the fraction x over 4 plus the fraction 3 over 4 .")[$frac(x, 4) + frac(3, 4) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "the fraction y over 8 plus the fraction 5 over 8 .")[$frac(y, 8) + frac(5, 8) .$]

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

#examplebox("Example 2")[][
Find the difference: #math.equation(block: false, alt: "minus the fraction 23 over 24 minus the fraction 13 over 24 .")[$− #h(0.2em) frac(23, 24) − #h(0.2em) frac(13, 24) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 23 over 24 minus the fraction 13 over 24")[$− #h(0.2em) frac(23, 24) − #h(0.2em) frac(13, 24)$]]),
  [Subtract the numerators and place the difference over the common denominator.], [#math.equation(block: false, alt: "the fraction −23 minus 13 over 24")[$frac(−23 − 13, 24)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction −36 over 24")[$frac(−36, 24)$]],
  [Simplify. Remember, #math.equation(block: false, alt: "minus the fraction a over b equals the fraction − a over b")[$− #h(0.2em) frac(a, b) = frac("−" a, b)$].], [#math.equation(block: false, alt: "minus the fraction 3 over 2")[$− #h(0.2em) frac(3, 2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "minus the fraction 19 over 28 minus the fraction 7 over 28 .")[$− #h(0.2em) frac(19, 28) − #h(0.2em) frac(7, 28) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 13 over 14")[$− #h(0.2em) frac(13, 14)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "minus the fraction 27 over 32 minus the fraction 1 over 32 .")[$− #h(0.2em) frac(27, 32) − #h(0.2em) frac(1, 32) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 7 over 8")[$− #h(0.2em) frac(7, 8)$]
]
]

#examplebox("Example 3")[][
Simplify: #math.equation(block: false, alt: "minus the fraction 10 over x minus the fraction 4 over x .")[$− #h(0.2em) frac(10, x) − #h(0.2em) frac(4, x) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 10 over x minus the fraction 4 over x")[$− #h(0.2em) frac(10, x) − #h(0.2em) frac(4, x)$]]),
  [Subtract the numerators and place the difference over the common denominator.], [#math.equation(block: false, alt: "the fraction −14 over x")[$frac(−14, x)$]],
  [Rewrite with the sign in front of the fraction.], [#math.equation(block: false, alt: "minus the fraction 14 over x")[$− #h(0.2em) frac(14, x)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "minus the fraction 9 over x minus the fraction 7 over x .")[$− #h(0.2em) frac(9, x) − #h(0.2em) frac(7, x) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "minus the fraction 17 over a minus the fraction 5 over a .")[$− #h(0.2em) frac(17, a) − #h(0.2em) frac(5, a) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 22 over a")[$− #h(0.2em) frac(22, a)$]
]
]

Now we will do an example that has both addition and subtraction.

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "the fraction 3 over 8 plus open parenthesis minus the fraction 5 over 8 close parenthesis minus the fraction 1 over 8 .")[$frac(3, 8) + ( − #h(0.2em) frac(5, 8) ) − #h(0.2em) frac(1, 8) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Add and subtract fractions—do they have a common denominator? Yes.], [#math.equation(block: false, alt: "the fraction 3 over 8 plus open parenthesis minus the fraction 5 over 8 close parenthesis minus the fraction 1 over 8")[$frac(3, 8) + ( − #h(0.2em) frac(5, 8) ) − #h(0.2em) frac(1, 8)$]]),
  [Add and subtract the numerators and place the difference over the common denominator.], [#math.equation(block: false, alt: "the fraction 3 plus open parenthesis −5 close parenthesis minus 1 over 8")[$frac(3 + ( −5 ) − 1, 8)$]],
  [Simplify left to right.], [#math.equation(block: false, alt: "the fraction −2 minus 1 over 8")[$frac(−2 − 1, 8)$]],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 3 over 8")[$− #h(0.2em) frac(3, 8)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "minus the fraction 2 over 9 plus open parenthesis minus the fraction 4 over 9 close parenthesis minus the fraction 7 over 9 .")[$− frac(2, 9) + ( − #h(0.2em) frac(4, 9) ) − #h(0.2em) frac(7, 9) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 5 over 9 plus open parenthesis minus the fraction 4 over 9 close parenthesis minus the fraction 7 over 9 .")[$frac(5, 9) + ( − #h(0.2em) frac(4, 9) ) − #h(0.2em) frac(7, 9) .$]

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


=== Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The #strong[least common denominator] (LCD) of two fractions is the smallest number that can be used as a common #strong[denominator] of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Least Common Denominator]

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.
]

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

#examplebox("Example 5")[How to Add or Subtract Fractions][
Add: #math.equation(block: false, alt: "the fraction 7 over 12 plus the fraction 5 over 18 .")[$frac(7, 12) + frac(5, 18) .$]

#solutionbox[
#figure(figph[In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have “Step 1. Do they have a common denominator? No – rewrite each fraction with the LCD (least common denominator).” To the right of this, we have the statement “No. Find the LCD 12, 18.” To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have “Change into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!” To the right of this, we have 7/12 plus 5/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21/36 plus 10/36.], alt: "In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have “Step 1. Do they have a common denominator? No – rewrite each fraction with the LCD (least common denominator).” To the right of this, we have the statement “No. Find the LCD 12, 18.” To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have “Change into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!” To the right of this, we have 7/12 plus 5/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21/36 plus 10/36.", caption: none)

#figure(figph[The next step reads “Step 2. Add or subtract the fractions.” The hint reads “Add.” And we have 31/36.], alt: "The next step reads “Step 2. Add or subtract the fractions.” The hint reads “Add.” And we have 31/36.", caption: none)

#figure(figph[The final step reads “Step 3. Simplify, if possible.” The explanation reads “Because 31 is a prime number, it has no factors in common with 36. The answer is simplified.”], alt: "The final step reads “Step 3. Simplify, if possible.” The explanation reads “Because 31 is a prime number, it has no factors in common with 36. The answer is simplified.”", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction 7 over 12 plus the fraction 11 over 15 .")[$frac(7, 12) + frac(11, 15) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 79 over 60")[$frac(79, 60)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction 13 over 15 plus the fraction 17 over 20 .")[$frac(13, 15) + frac(17, 20) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 103 over 60")[$frac(103, 60)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Add or Subtract Fractions.]

+ Do they have a common denominator?
- Yes—go to step 2.
- No—rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
+ Add or subtract the fractions.
+ Simplify, if possible.
]

When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

#figure(figph[The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked “missing factors.” The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).], alt: "The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked “missing factors.” The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).", caption: none)

In , the LCD, 36, has two factors of 2 and two factors of #math.equation(block: false, alt: "3 .")[$3 .$]

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.

We will apply this method as we subtract the fractions in .

#examplebox("Example 6")[][
Subtract: #math.equation(block: false, alt: "the fraction 7 over 15 minus the fraction 19 over 24 .")[$frac(7, 15) − #h(0.2em) frac(19, 24) .$]

#solutionbox[
Do the fractions have a common denominator? No, so we need to find the LCD. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the LCD.$#h(5em)$#figure(figph[The image shows the calculation of the Least Common Denominator (LCD) for the fractions 7/15 and 19/24. It details the prime factorization of 15 (3\*5) and 24 (2\*2\*2\*3), leading to an LCD of 120.], alt: "The image shows the calculation of the Least Common Denominator (LCD) for the fractions 7/15 and 19/24. It details the prime factorization of 15 (3*5) and 24 (2*2*2*3), leading to an LCD of 120.", caption: none)], []),
  [Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.], [],
  [Rewrite as equivalent fractions with the LCD.], [#figure(figph[A mathematical expression showing the subtraction of two fractions: (7 multiplied by 8) divided by (15 multiplied by 8), minus (19 multiplied by 5) divided by (24 multiplied by 5). The numbers 8 and 5 are highlighted in red.], alt: "A mathematical expression showing the subtraction of two fractions: (7 multiplied by 8) divided by (15 multiplied by 8), minus (19 multiplied by 5) divided by (24 multiplied by 5). The numbers 8 and 5 are highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the subtraction of two fractions with a common denominator: 56/120 - 95/120.], alt: "A mathematical expression showing the subtraction of two fractions with a common denominator: 56/120 - 95/120.", caption: none)],
  [Subtract.], [#math.equation(block: false, alt: "− the fraction 39 over 120")[$#h(2em) − #h(0.2em) frac(39, 120)$]],
  [Check to see if the answer can be simplified.], [#math.equation(block: false, alt: "− the fraction 13 times 3 over 40 times 3")[$#h(2em) − #h(0.2em) frac(13 ⋅ 3, 40 ⋅ 3)$]],
  [Both 39 and 120 have a factor of 3.], [],
  [Simplify.], [#math.equation(block: false, alt: "− the fraction 13 over 40")[$#h(2em) − #h(0.2em) frac(13, 40)$]],
))

Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction 13 over 24 minus the fraction 17 over 32 .")[$frac(13, 24) − #h(0.2em) frac(17, 32) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 96")[$frac(1, 96)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction 21 over 32 minus the fraction 9 over 28 .")[$frac(21, 32) − #h(0.2em) frac(9, 28) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 75 over 224")[$frac(75, 224)$]
]
]

In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.

#examplebox("Example 7")[][
Add: #math.equation(block: false, alt: "the fraction 3 over 5 plus the fraction x over 8 .")[$frac(3, 5) + frac(x, 8) .$]

#solutionbox[
The fractions have different denominators. #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [], [$#h(2em)$#figure(figph[A mathematical expression showing the sum of two fractions: 3/5 plus x/8. The numbers and variable are in black text on a white background.], alt: "A mathematical expression showing the sum of two fractions: 3/5 plus x/8. The numbers and variable are in black text on a white background.", caption: none)]),
  [Find the LCD.$#h(4em)$#figure(figph[Calculation of the Least Common Denominator (LCD) for 5 and 8 using prime factorization, showing 5 = 5, 8 = 2x2x2, and LCD = 2x2x2x5 = 40.], alt: "Calculation of the Least Common Denominator (LCD) for 5 and 8 using prime factorization, showing 5 = 5, 8 = 2x2x2, and LCD = 2x2x2x5 = 40.", caption: none)], [], [],
  [Rewrite as equivalent fractions with the LCD.], [], [$#h(2em)$#figure(figph[A mathematical expression shows the addition of two fractions: (3 times 8) divided by (5 times 8) plus (x times 5) divided by (8 times 5). The numbers 8 and 5 in red highlight common factors.], alt: "A mathematical expression shows the addition of two fractions: (3 times 8) divided by (5 times 8) plus (x times 5) divided by (8 times 5). The numbers 8 and 5 in red highlight common factors.", caption: none)],
  [Simplify.], [], [$#h(2em)$#figure(figph[A mathematical expression shows the sum of two fractions, 24/40 and 5x/40. Both fractions share a common denominator of 40.], alt: "A mathematical expression shows the sum of two fractions, 24/40 and 5x/40. Both fractions share a common denominator of 40.", caption: none)],
  [Add.], [], [$#h(2em)$#figure(figph[A mathematical expression showing the fraction (24 + 5x) / 40.], alt: "A mathematical expression showing the fraction (24 + 5x) / 40.", caption: none)],
))

Remember, we can only add like terms: 24 and 5#emph[x] are not like terms.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction y over 6 plus the fraction 7 over 9 .")[$frac(y, 6) + frac(7, 9) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 y plus 14 over 18")[$frac(3 y + 14, 18)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction x over 6 plus the fraction 7 over 15 .")[$frac(x, 6) + frac(7, 15) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 x plus 14 over 30")[$frac(5 x + 14, 30)$]
]
]

We now have all four operations for fractions.  summarizes #strong[fraction operations].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#strong[Fraction Multiplication]], [#strong[Fraction Division]],
  [#math.equation(block: false, alt: "the fraction a over b times the fraction c over d equals the fraction a c over b d")[$frac(a, b) · frac(c, d) = frac(a c, b d)$] #linebreak()  #linebreak() Multiply the numerators and multiply the denominators], [#math.equation(block: false, alt: "the fraction a over b divided by the fraction c over d equals the fraction a over b times the fraction d over c")[$frac(a, b) ÷ frac(c, d) = frac(a, b) · frac(d, c)$] #linebreak()  #linebreak() Multiply the first fraction by the reciprocal of the second.],
  [#strong[Fraction Addition]], [#strong[Fraction Subtraction]],
  [#math.equation(block: false, alt: "the fraction a over c plus the fraction b over c equals the fraction a plus b over c")[$frac(a, c) + frac(b, c) = frac(a + b, c)$] #linebreak()  #linebreak() Add the numerators and place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction a over c minus the fraction b over c equals the fraction a minus b over c")[$frac(a, c) − #h(0.2em) frac(b, c) = frac(a − b, c)$] #linebreak()  #linebreak() Subtract the numerators and place the difference over the common denominator.],
  [To multiply or divide fractions, an LCD is NOT needed. #linebreak() 
To add or subtract fractions, an LCD is needed.], [],
))

#examplebox("Example 8")[][
Simplify: ⓐ #math.equation(block: false, alt: "the fraction 5 x over 6 minus the fraction 3 over 10")[$frac(5 x, 6) − #h(0.2em) frac(3, 10)$] ⓑ #math.equation(block: false, alt: "the fraction 5 x over 6 times the fraction 3 over 10 .")[$frac(5 x, 6) · frac(3, 10) .$]

#solutionbox[
First ask, “What is the operation?” Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ What is the operation? The operation is subtraction.
     #linebreak() Do the fractions have a common denominator? No.
     #linebreak() Rewrite each fraction as an equivalent fraction with the LCD.
     #linebreak() Subtract the numerators and place the difference over the common denominators.
     #linebreak() Simplify, if possible.
     #linebreak() There are no common factors. The fraction is simplified.], [#math.equation(block: false, alt: "the fraction 5 x over 6 minus the fraction 3 over 10; the fraction 5 x times 5 over 6 times 5 minus the fraction 3 times 3 over 10 times 3; the fraction 25 x over 30 minus the fraction 9 over 30; the fraction 25 x minus 9 over 30")[$frac(5 x, 6) − #h(0.2em) frac(3, 10) \ frac(5 x · 5, 6 · 5) − #h(0.2em) frac(3 · 3, 10 · 3) \ frac(25 x, 30) − #h(0.2em) frac(9, 30) \ frac(25 x − 9, 30)$]]),
  [ⓑ What is the operation? Multiplication.
     #linebreak() To multiply fractions, multiply the numerators and multiply the denominators.
     #linebreak() Rewrite, showing common factors. Remove common factors.
     #linebreak() Simplify.], [#math.equation(block: false, alt: "the fraction 5 x over 6 times the fraction 3 over 10; the fraction 5 x times 3 over 6 times 10; the fraction 5 x times 3 over 2 times 3 times 2 times 5; the fraction x over 4")[$frac(5 x, 6) · frac(3, 10) \ frac(5 x · 3, 6 · 10) \ frac(cancel(5) x · cancel(3), 2 · cancel(3) · 2 · cancel(5)) \ frac(x, 4)$]],
))

Notice we needed an LCD to add #math.equation(block: false, alt: "the fraction 5 x over 6 minus the fraction 3 over 10 ,")[$frac(5 x, 6) − #h(0.2em) frac(3, 10) ,$] but not to multiply #math.equation(block: false, alt: "the fraction 5 x over 6 times the fraction 3 over 10 .")[$frac(5 x, 6) · frac(3, 10) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the fraction 3 a over 4 minus the fraction 8 over 9")[$frac(3 a, 4) − #h(0.2em) frac(8, 9)$] ⓑ #math.equation(block: false, alt: "the fraction 3 a over 4 times the fraction 8 over 9 .")[$frac(3 a, 4) · frac(8, 9) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 27 a minus 32 over 36")[$frac(27 a − 32, 36)$] ⓑ #math.equation(block: false, alt: "the fraction 2 a over 3")[$frac(2 a, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the fraction 4 k over 5 minus the fraction 1 over 6")[$frac(4 k, 5) − #h(0.2em) frac(1, 6)$] ⓑ #math.equation(block: false, alt: "the fraction 4 k over 5 times the fraction 1 over 6 .")[$frac(4 k, 5) · frac(1, 6) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 24 k minus 5 over 30")[$frac(24 k − 5, 30)$] ⓑ #math.equation(block: false, alt: "the fraction 2 k over 15")[$frac(2 k, 15)$]
]
]


=== Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates #strong[division]. We simplified the complex fraction #math.equation(block: false, alt: "the fraction the fraction 3 over 4 over the fraction 5 over 8")[$frac(frac(3, 4), frac(5, 8))$] by dividing #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] by #math.equation(block: false, alt: "the fraction 5 over 8 .")[$frac(5, 8) .$]

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.

#examplebox("Example 9")[How to Simplify Complex Fractions][
Simplify: #math.equation(block: false, alt: "the fraction open parenthesis the fraction 1 over 2 close parenthesis squared over 4 plus 3 squared .")[$frac(attach(( frac(1, 2) ), t: 2), 4 + 3^(2)) .$]

#solutionbox[
#figure(figph[In this figure, we have a table with directions on the left and mathematical statements on the right. On the first line, we have “Step 1. Simplify the numerator. Remember one half squared means one half times one half.” To the right of this, we have the quantity (1/2) squared all over the quantity (4 plus 3 squared). Then, we have 1/4 over the quantity (4 plus 3 squared).], alt: "In this figure, we have a table with directions on the left and mathematical statements on the right. On the first line, we have “Step 1. Simplify the numerator. Remember one half squared means one half times one half.” To the right of this, we have the quantity (1/2) squared all over the quantity (4 plus 3 squared). Then, we have 1/4 over the quantity (4 plus 3 squared).", caption: none)

#figure(figph[The next line’s direction reads “Step 2. Simplify the denominator.” To the right of this, we have 1/4 over the quantity (4 plus 9), under which we have 1/4 over 13.], alt: "The next line’s direction reads “Step 2. Simplify the denominator.” To the right of this, we have 1/4 over the quantity (4 plus 9), under which we have 1/4 over 13.", caption: none)

#figure(figph[The final step is “Step 3. Divide the numerator by the denominator. Simplify if possible. Remember, thirteen equals thirteen over 1.” To the right we have 1/4 divided by 13. Then we have 1/4 times 1/13, which equals 1/52.], alt: "The final step is “Step 3. Divide the numerator by the denominator. Simplify if possible. Remember, thirteen equals thirteen over 1.” To the right we have 1/4 divided by 13. Then we have 1/4 times 1/13, which equals 1/52.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction open parenthesis the fraction 1 over 3 close parenthesis squared over 2 cubed plus 2 .")[$frac(attach(( frac(1, 3) ), t: 2), 2^(3) + 2) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 90")[$frac(1, 90)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 1 plus 4 squared over open parenthesis the fraction 1 over 4 close parenthesis squared .")[$frac(1 + 4^(2), attach(( frac(1, 4) ), t: 2)) .$]

#solutionbox[
#math.equation(block: true, alt: "272")[$272$]
]
]

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

+ Simplify the numerator.
+ Simplify the denominator.
+ Divide the numerator by the denominator. Simplify if possible.
]

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "the fraction the fraction 1 over 2 plus the fraction 2 over 3 over the fraction 3 over 4 minus the fraction 1 over 6 .")[$frac(frac(1, 2) + frac(2, 3), frac(3, 4) − #h(0.2em) frac(1, 6)) .$]

#solutionbox[
It may help to put parentheses around the numerator and the denominator.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction open parenthesis the fraction 1 over 2 plus the fraction 2 over 3 close parenthesis over open parenthesis the fraction 3 over 4 minus the fraction 1 over 6 close parenthesis")[$frac(( frac(1, 2) + frac(2, 3) ), ( frac(3, 4) − #h(0.2em) frac(1, 6) ))$]]),
  [Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12).], [#math.equation(block: false, alt: "the fraction open parenthesis the fraction 3 over 6 plus the fraction 4 over 6 close parenthesis over open parenthesis the fraction 9 over 12 minus the fraction 2 over 12 close parenthesis")[$frac(( frac(3, 6) + frac(4, 6) ), ( frac(9, 12) − #h(0.2em) frac(2, 12) ))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction open parenthesis the fraction 7 over 6 close parenthesis over open parenthesis the fraction 7 over 12 close parenthesis")[$frac(( frac(7, 6) ), ( frac(7, 12) ))$]],
  [Divide the numerator by the denominator.], [#math.equation(block: false, alt: "the fraction 7 over 6 divided by the fraction 7 over 12")[$frac(7, 6) ÷ frac(7, 12)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 7 over 6 times the fraction 12 over 7")[$frac(7, 6) · frac(12, 7)$]],
  [Divide out common factors.], [#math.equation(block: false, alt: "the fraction 7 times 6 times 2 over 6 times 7")[$frac(7 · 6 · 2, 6 · 7)$]],
  [Simplify.], [#math.equation(block: false, alt: "2")[$2$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the fraction 1 over 3 plus the fraction 1 over 2 over the fraction 3 over 4 minus the fraction 1 over 3 .")[$frac(frac(1, 3) + frac(1, 2), frac(3, 4) − #h(0.2em) frac(1, 3)) .$]

#solutionbox[
2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction the fraction 2 over 3 minus the fraction 1 over 2 over the fraction 1 over 4 plus the fraction 1 over 3 .")[$frac(frac(2, 3) − #h(0.2em) frac(1, 2), frac(1, 4) + frac(1, 3)) .$]

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


=== Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

#examplebox("Example 11")[][
Evaluate #math.equation(block: false, alt: "x plus the fraction 1 over 3")[$x + frac(1, 3)$] when ⓐ #math.equation(block: false, alt: "x equals minus the fraction 1 over 3")[$x = − #h(0.2em) frac(1, 3)$] ⓑ #math.equation(block: false, alt: "x equals minus the fraction 3 over 4 .")[$x = − #h(0.2em) frac(3, 4) .$]

#solutionbox[
+ ⓐ To evaluate #math.equation(block: false, alt: "x plus the fraction 1 over 3")[$x + frac(1, 3)$] when #math.equation(block: false, alt: "x equals minus the fraction 1 over 3 ,")[$x = − #h(0.2em) frac(1, 3) ,$] substitute #math.equation(block: false, alt: "minus the fraction 1 over 3")[$− #h(0.2em) frac(1, 3)$] for #math.equation(block: false, alt: "x")[$x$] in the expression. #linebreak() 
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing 'x' plus the fraction '1/3'.], alt: "A mathematical expression showing 'x' plus the fraction '1/3'.", caption: none)]),
  [#figure(figph[The text 'Substitute -1/3 for x.' is shown in the image, with '-1/3' highlighted in red.], alt: "The text 'Substitute -1/3 for x.' is shown in the image, with '-1/3' highlighted in red.", caption: none)], [#figure(figph[The mathematical expression -1/3 + 1/3, illustrating the concept of additive inverses where a number summed with its opposite results in zero.], alt: "The mathematical expression -1/3 + 1/3, illustrating the concept of additive inverses where a number summed with its opposite results in zero.", caption: none)],
  [Simplify.$#h(18em)$], [0],
))

 #linebreak()  #linebreak()
+ ⓑ To evaluate #math.equation(block: false, alt: "x plus the fraction 1 over 3")[$x + frac(1, 3)$] when #math.equation(block: false, alt: "x equals minus the fraction 3 over 4 ,")[$x = − #h(0.2em) frac(3, 4) ,$] we substitute #math.equation(block: false, alt: "minus the fraction 3 over 4")[$− #h(0.2em) frac(3, 4)$] for #emph[x] in the expression. #linebreak() 
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displaying 'x + 1/3' in black text on a white background.], alt: "A mathematical expression displaying 'x + 1/3' in black text on a white background.", caption: none)]),
  [#figure(figph[Text: Substitute -3/4 for x. The fraction -3/4 is highlighted in red.], alt: "Text: Substitute -3/4 for x. The fraction -3/4 is highlighted in red.", caption: none)], [#figure(figph[A mathematical expression showing the addition of two fractions: negative three-fourths plus one-third. The number three and four are in red color, while the negative sign and fractions for one-third are in black.], alt: "A mathematical expression showing the addition of two fractions: negative three-fourths plus one-third. The number three and four are in red color, while the negative sign and fractions for one-third are in black.", caption: none)],
  [Rewrite as equivalent fractions with the LCD, 12.], [#figure(figph[An equation for adding -3/4 and 1/3, showing the fractions multiplied by factors to achieve a common denominator of 12: -(3\*3)/(4\*3) + (1\*4)/(3\*4).], alt: "An equation for adding -3/4 and 1/3, showing the fractions multiplied by factors to achieve a common denominator of 12: -(3*3)/(4*3) + (1*4)/(3*4).", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the addition of two fractions with the same denominator: -9/12 + 4/12.], alt: "A mathematical expression showing the addition of two fractions with the same denominator: -9/12 + 4/12.", caption: none)],
  [Add.], [#math.equation(block: false, alt: "minus the fraction 5 over 12")[$− #h(0.2em) frac(5, 12)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "x plus the fraction 3 over 4")[$x + frac(3, 4)$] when ⓐ #math.equation(block: false, alt: "x equals minus the fraction 7 over 4")[$x = − #h(0.2em) frac(7, 4)$] ⓑ #math.equation(block: false, alt: "x equals minus the fraction 5 over 4 .")[$x = − #h(0.2em) frac(5, 4) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "y plus the fraction 1 over 2")[$y + frac(1, 2)$] when ⓐ #math.equation(block: false, alt: "y equals the fraction 2 over 3")[$y = frac(2, 3)$] ⓑ #math.equation(block: false, alt: "y equals minus the fraction 3 over 4 .")[$y = − #h(0.2em) frac(3, 4) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 7 over 6")[$frac(7, 6)$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 4")[$− #h(0.2em) frac(1, 4)$]
]
]

#examplebox("Example 12")[][
Evaluate #math.equation(block: false, alt: "minus the fraction 5 over 6 minus y")[$− #h(0.2em) frac(5, 6) − y$] when #math.equation(block: false, alt: "y equals minus the fraction 2 over 3 .")[$y = − #h(0.2em) frac(2, 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing negative five-sixths minus y, presented as a black text on a white background.], alt: "A mathematical expression showing negative five-sixths minus y, presented as a black text on a white background.", caption: none)]),
  [#figure(figph[The image displays the mathematical instruction 'Substitute -2/3 for y.' in a gray sans-serif font, with the fraction '-2/3' highlighted in red.], alt: "The image displays the mathematical instruction 'Substitute -2/3 for y.' in a gray sans-serif font, with the fraction '-2/3' highlighted in red.", caption: none)], [#figure(figph[A mathematical expression showing the subtraction of two negative fractions: -5/6 - (-2/3), where the second fraction -2/3 is enclosed in parentheses and highlighted in red.], alt: "A mathematical expression showing the subtraction of two negative fractions: -5/6 - (-2/3), where the second fraction -2/3 is enclosed in parentheses and highlighted in red.", caption: none)],
  [Rewrite as equivalent fractions with the LCD, 6.], [#figure(figph[A mathematical expression showing the subtraction of negative five-sixths and negative four-sixths, written as -5/6 - (-4/6), where the second fraction is enclosed in parentheses and in red.], alt: "A mathematical expression showing the subtraction of negative five-sixths and negative four-sixths, written as -5/6 - (-4/6), where the second fraction is enclosed in parentheses and in red.", caption: none)],
  [Subtract.], [#figure(figph[A mathematical expression showing the fraction: negative 5 minus negative 4, all divided by 6.], alt: "A mathematical expression showing the fraction: negative 5 minus negative 4, all divided by 6.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 1 over 6")[$− #h(0.2em) frac(1, 6)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "minus the fraction 1 over 2 minus y")[$− #h(0.2em) frac(1, 2) − y$] when #math.equation(block: false, alt: "y equals minus the fraction 1 over 4 .")[$y = − #h(0.2em) frac(1, 4) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 4")[$− #h(0.2em) frac(1, 4)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "minus the fraction 3 over 8 minus y")[$− #h(0.2em) frac(3, 8) − y$] when #math.equation(block: false, alt: "y equals minus the fraction 5 over 2 .")[$y = − #h(0.2em) frac(5, 2) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 17 over 8")[$− #h(0.2em) frac(17, 8)$]
]
]

#examplebox("Example 13")[][
Evaluate #math.equation(block: false, alt: "2 x squared y")[$2 x^(2) y$] when #math.equation(block: false, alt: "x equals the fraction 1 over 4")[$x = frac(1, 4)$] and #math.equation(block: false, alt: "y equals minus the fraction 2 over 3 .")[$y = − #h(0.2em) frac(2, 3) .$]

#solutionbox[
Substitute the values into the expression. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 x squared y")[$2 x^(2) y$]]),
  [#figure(figph[The image shows the instruction: 'Substitute 1/4 for x and -2/3 for y.' The fraction 1/4 is colored red, and -2/3 is colored light blue, making them visually distinct from the black text.], alt: "The image shows the instruction: 'Substitute 1/4 for x and -2/3 for y.' The fraction 1/4 is colored red, and -2/3 is colored light blue, making them visually distinct from the black text.", caption: none)], [#figure(figph[A mathematical expression showing the product of 2, the square of 1/4, and -2/3. The fraction 1/4 is colored red and -2/3 is colored blue, highlighting different parts of the expression.], alt: "A mathematical expression showing the product of 2, the square of 1/4, and -2/3. The fraction 1/4 is colored red and -2/3 is colored blue, highlighting different parts of the expression.", caption: none)],
  [Simplify exponents first.], [#math.equation(block: false, alt: "2 open parenthesis the fraction 1 over 16 close parenthesis open parenthesis minus the fraction 2 over 3 close parenthesis")[$2 ( frac(1, 16) ) ( − #h(0.2em) frac(2, 3) )$]],
  [Multiply. Divide out the common factors. Notice we write 16 as #math.equation(block: false, alt: "2 times 2 times 4")[$2 ⋅ 2 ⋅ 4$] to make it easy to remove common factors.], [#math.equation(block: false, alt: "minus the fraction 2 times 1 times 2 over 2 times 2 times 4 times 3")[$− #h(0.2em) frac(cancel(2) ⋅ 1 ⋅ cancel(2), cancel(2) ⋅ cancel(2) ⋅ 4 ⋅ 3)$]],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 1 over 12")[$− #h(0.2em) frac(1, 12)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "3 a b squared")[$3 a b^(2)$] when #math.equation(block: false, alt: "a equals minus the fraction 2 over 3")[$a = − #h(0.2em) frac(2, 3)$] and #math.equation(block: false, alt: "b equals minus the fraction 1 over 2 .")[$b = − #h(0.2em) frac(1, 2) .$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 2")[$− #h(0.2em) frac(1, 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "4 c cubed d")[$4 c^(3) d$] when #math.equation(block: false, alt: "c equals minus the fraction 1 over 2")[$c = − #h(0.2em) frac(1, 2)$] and #math.equation(block: false, alt: "d equals minus the fraction 4 over 3 .")[$d = − #h(0.2em) frac(4, 3) .$]

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

The next example will have only variables, no constants.

#examplebox("Example 14")[][
Evaluate #math.equation(block: false, alt: "the fraction p plus q over r")[$frac(p + q, r)$] when #math.equation(block: false, alt: "p equals −4 , q equals −2 , and r equals 8 .")[$p = −4 , q = −2 , #h(0.2em) "and" #h(0.2em) r = 8 .$]

#solutionbox[
To evaluate #math.equation(block: false, alt: "the fraction p plus q over r")[$frac(p + q, r)$] when #math.equation(block: false, alt: "p equals −4 , q equals −2 , and r equals 8 ,")[$p = −4 , q = −2 , #h(0.2em) "and" #h(0.2em) r = 8 ,$] we substitute the values into the expression. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction p plus q over r")[$frac(p + q, r)$]]),
  [#figure(figph[The image shows the instruction: 'Substitute -4 for p, -2 for q and 8 for r.' The numbers -4, -2, and 8 are highlighted in red, light blue, and yellow respectively.], alt: "The image shows the instruction: 'Substitute -4 for p, -2 for q and 8 for r.' The numbers -4, -2, and 8 are highlighted in red, light blue, and yellow respectively.", caption: none)], [#figure(figph[A mathematical fraction is displayed, with the numerator being '-4 + (-2)' and the denominator being '8'.], alt: "A mathematical fraction is displayed, with the numerator being '-4 + (-2)' and the denominator being '8'.", caption: none)],
  [Add in the numerator first.], [#math.equation(block: false, alt: "the fraction −6 over 8")[$frac(−6, 8)$]],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 3 over 4")[$− #h(0.2em) frac(3, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "the fraction a plus b over c")[$frac(a + b, c)$] when #math.equation(block: false, alt: "a equals −8 , b equals −7 , and c equals 6 .")[$a = −8 , b = −7 , #h(0.2em) "and" #h(0.2em) c = 6 .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "the fraction x plus y over z")[$frac(x + y, z)$] when #math.equation(block: false, alt: "x equals 9 , y equals −18 , and z equals −6 .")[$x = 9 , y = −18 , #h(0.2em) "and" #h(0.2em) z = −6 .$]

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


=== Key Concepts

- #strong[Fraction Addition and Subtraction:] If #math.equation(block: false, alt: "a , b , and c")[$a , b , #h(0.2em) "and" #h(0.2em) c$] are numbers where #math.equation(block: false, alt: "c not equal to 0 ,")[$c ≠ 0 ,$] then #linebreak() 
#math.equation(block: false, alt: "the fraction a over c plus the fraction b over c equals the fraction a plus b over c")[$frac(a, c) + frac(b, c) = frac(a + b, c)$] and #math.equation(block: false, alt: "the fraction a over c minus the fraction b over c equals the fraction a minus b over c .")[$frac(a, c) − #h(0.2em) frac(b, c) = frac(a − b, c) .$] #linebreak() 
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
- #strong[Strategy for Adding or Subtracting Fractions]
+ Do they have a common denominator? #linebreak() 
Yes—go to step 2. #linebreak() 
No—Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
+ Add or subtract the fractions.
+ Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
- #strong[Simplify Complex Fractions]
+ Simplify the numerator.
+ Simplify the denominator.
+ Divide the numerator by the denominator. Simplify if possible.


==== Practice Makes Perfect

#strong[Add and Subtract Fractions with a Common Denominator]

In the following exercises, add.

#math.equation(block: true, alt: "the fraction 6 over 13 plus the fraction 5 over 13")[$frac(6, 13) + frac(5, 13)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 13")[$frac(11, 13)$]
]

#math.equation(block: true, alt: "the fraction 4 over 15 plus the fraction 7 over 15")[$frac(4, 15) + frac(7, 15)$]

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

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

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

#math.equation(block: true, alt: "minus the fraction 3 over 16 plus open parenthesis minus the fraction 7 over 16 close parenthesis")[$− #h(0.2em) frac(3, 16) + ( − #h(0.2em) frac(7, 16) )$]

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

#math.equation(block: true, alt: "minus the fraction 5 over 16 plus open parenthesis minus the fraction 9 over 16 close parenthesis")[$− #h(0.2em) frac(5, 16) + ( − #h(0.2em) frac(9, 16) )$]

#math.equation(block: true, alt: "minus the fraction 8 over 17 plus the fraction 15 over 17")[$− #h(0.2em) frac(8, 17) + frac(15, 17)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 17")[$frac(7, 17)$]
]

#math.equation(block: true, alt: "minus the fraction 9 over 19 plus the fraction 17 over 19")[$− #h(0.2em) frac(9, 19) + frac(17, 19)$]

#math.equation(block: true, alt: "the fraction 6 over 13 plus open parenthesis minus the fraction 10 over 13 close parenthesis plus open parenthesis minus the fraction 12 over 13 close parenthesis")[$frac(6, 13) + ( − #h(0.2em) frac(10, 13) ) + ( − #h(0.2em) frac(12, 13) )$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 16 over 13")[$− #h(0.2em) frac(16, 13)$]
]

#math.equation(block: true, alt: "the fraction 5 over 12 plus open parenthesis minus the fraction 7 over 12 close parenthesis plus open parenthesis minus the fraction 11 over 12 close parenthesis")[$frac(5, 12) + ( − #h(0.2em) frac(7, 12) ) + ( − #h(0.2em) frac(11, 12) )$]

In the following exercises, subtract.

#math.equation(block: true, alt: "the fraction 11 over 15 minus the fraction 7 over 15")[$frac(11, 15) − #h(0.2em) frac(7, 15)$]

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

#math.equation(block: true, alt: "the fraction 9 over 13 minus the fraction 4 over 13")[$frac(9, 13) − #h(0.2em) frac(4, 13)$]

#math.equation(block: true, alt: "the fraction 11 over 12 minus the fraction 5 over 12")[$frac(11, 12) − #h(0.2em) frac(5, 12)$]

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

#math.equation(block: true, alt: "the fraction 7 over 12 minus the fraction 5 over 12")[$frac(7, 12) − #h(0.2em) frac(5, 12)$]

#math.equation(block: true, alt: "the fraction 19 over 21 minus the fraction 4 over 21")[$frac(19, 21) − #h(0.2em) frac(4, 21)$]

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

#math.equation(block: true, alt: "the fraction 17 over 21 minus the fraction 8 over 21")[$frac(17, 21) − #h(0.2em) frac(8, 21)$]

#math.equation(block: true, alt: "the fraction 5 y over 8 minus the fraction 7 over 8")[$frac(5 y, 8) − #h(0.2em) frac(7, 8)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 y minus 7 over 8")[$frac(5 y − 7, 8)$]
]

#math.equation(block: true, alt: "the fraction 11 z over 13 minus the fraction 8 over 13")[$frac(11 z, 13) − #h(0.2em) frac(8, 13)$]

#math.equation(block: true, alt: "minus the fraction 23 over u minus the fraction 15 over u")[$− #h(0.2em) frac(23, u) − #h(0.2em) frac(15, u)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 38 over u")[$− #h(0.2em) frac(38, u)$]
]

#math.equation(block: true, alt: "minus the fraction 29 over v minus the fraction 26 over v")[$− #h(0.2em) frac(29, v) − #h(0.2em) frac(26, v)$]

#math.equation(block: true, alt: "minus the fraction 3 over 5 minus open parenthesis minus the fraction 4 over 5 close parenthesis")[$− #h(0.2em) frac(3, 5) − ( − #h(0.2em) frac(4, 5) )$]

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

#math.equation(block: true, alt: "minus the fraction 3 over 7 minus open parenthesis minus the fraction 5 over 7 close parenthesis")[$− #h(0.2em) frac(3, 7) − ( − #h(0.2em) frac(5, 7) )$]

#math.equation(block: true, alt: "minus the fraction 7 over 9 minus open parenthesis minus the fraction 5 over 9 close parenthesis")[$− #h(0.2em) frac(7, 9) − ( − #h(0.2em) frac(5, 9) )$]

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

#math.equation(block: true, alt: "minus the fraction 8 over 11 minus open parenthesis minus the fraction 5 over 11 close parenthesis")[$− #h(0.2em) frac(8, 11) − ( − #h(0.2em) frac(5, 11) )$]

#strong[Mixed Practice]

In the following exercises, simplify.

#math.equation(block: true, alt: "minus the fraction 5 over 18 times the fraction 9 over 10")[$− #h(0.2em) frac(5, 18) · frac(9, 10)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 4")[$− #h(0.2em) frac(1, 4)$]
]

#math.equation(block: true, alt: "minus the fraction 3 over 14 times the fraction 7 over 12")[$− #h(0.2em) frac(3, 14) · frac(7, 12)$]

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

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

#math.equation(block: true, alt: "the fraction 6 over 11 minus the fraction s over 11")[$frac(6, 11) − #h(0.2em) frac(s, 11)$]

#math.equation(block: true, alt: "minus the fraction 7 over 24 plus the fraction 2 over 24")[$− #h(0.2em) frac(7, 24) + frac(2, 24)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 5 over 24")[$− #h(0.2em) frac(5, 24)$]
]

#math.equation(block: true, alt: "minus the fraction 5 over 18 plus the fraction 1 over 18")[$− #h(0.2em) frac(5, 18) + frac(1, 18)$]

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

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

#math.equation(block: true, alt: "the fraction 7 over 12 divided by the fraction 9 over 28")[$frac(7, 12) ÷ frac(9, 28)$]

#strong[Add or Subtract Fractions with Different Denominators]

In the following exercises, add or subtract.

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

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

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

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

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

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

#math.equation(block: true, alt: "the fraction 7 over 12 plus the fraction 5 over 8")[$frac(7, 12) + frac(5, 8)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 29 over 24")[$frac(29, 24)$]
]

#math.equation(block: true, alt: "the fraction 5 over 12 plus the fraction 3 over 8")[$frac(5, 12) + frac(3, 8)$]

#math.equation(block: true, alt: "the fraction 7 over 12 minus the fraction 9 over 16")[$frac(7, 12) − #h(0.2em) frac(9, 16)$]

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

#math.equation(block: true, alt: "the fraction 7 over 16 minus the fraction 5 over 12")[$frac(7, 16) − #h(0.2em) frac(5, 12)$]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 24")[$frac(7, 24)$]
]

#math.equation(block: true, alt: "the fraction 5 over 6 minus the fraction 3 over 4")[$frac(5, 6) − #h(0.2em) frac(3, 4)$]

#math.equation(block: true, alt: "minus the fraction 11 over 30 plus the fraction 27 over 40")[$− #h(0.2em) frac(11, 30) + frac(27, 40)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 37 over 120")[$frac(37, 120)$]
]

#math.equation(block: true, alt: "minus the fraction 9 over 20 plus the fraction 17 over 30")[$− #h(0.2em) frac(9, 20) + frac(17, 30)$]

#math.equation(block: true, alt: "minus the fraction 13 over 30 plus the fraction 25 over 42")[$− #h(0.2em) frac(13, 30) + frac(25, 42)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 17 over 105")[$frac(17, 105)$]
]

#math.equation(block: true, alt: "minus the fraction 23 over 30 plus the fraction 5 over 48")[$− #h(0.2em) frac(23, 30) + frac(5, 48)$]

#math.equation(block: true, alt: "minus the fraction 39 over 56 minus the fraction 22 over 35")[$− #h(0.2em) frac(39, 56) − #h(0.2em) frac(22, 35)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 53 over 40")[$− #h(0.2em) frac(53, 40)$]
]

#math.equation(block: true, alt: "minus the fraction 33 over 49 minus the fraction 18 over 35")[$− #h(0.2em) frac(33, 49) − #h(0.2em) frac(18, 35)$]

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

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

#math.equation(block: true, alt: "minus the fraction 3 over 4 minus open parenthesis minus the fraction 4 over 5 close parenthesis")[$− #h(0.2em) frac(3, 4) − ( − #h(0.2em) frac(4, 5) )$]

#math.equation(block: true, alt: "1 plus the fraction 7 over 8")[$1 + frac(7, 8)$]

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

#math.equation(block: true, alt: "1 minus the fraction 3 over 10")[$1 − #h(0.2em) frac(3, 10)$]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 x plus 3 over 12")[$frac(4 x + 3, 12)$]
]

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

#math.equation(block: true, alt: "the fraction y over 4 minus the fraction 3 over 5")[$frac(y, 4) − #h(0.2em) frac(3, 5)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 y minus 12 over 20")[$frac(5 y − 12, 20)$]
]

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

#strong[Mixed Practice]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "the fraction 2 over 3 plus the fraction 1 over 6")[$frac(2, 3) + frac(1, 6)$] ⓑ #math.equation(block: false, alt: "the fraction 2 over 3 divided by the fraction 1 over 6")[$frac(2, 3) ÷ frac(1, 6)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$] ⓑ 4
]

ⓐ #math.equation(block: false, alt: "minus the fraction 2 over 5 minus the fraction 1 over 8")[$− #h(0.2em) frac(2, 5) − #h(0.2em) frac(1, 8)$] ⓑ #math.equation(block: false, alt: "minus the fraction 2 over 5 times the fraction 1 over 8")[$− #h(0.2em) frac(2, 5) · frac(1, 8)$]

ⓐ #math.equation(block: false, alt: "the fraction 5 n over 6 divided by the fraction 8 over 15")[$frac(5 n, 6) ÷ frac(8, 15)$] ⓑ #math.equation(block: false, alt: "the fraction 5 n over 6 minus the fraction 8 over 15")[$frac(5 n, 6) − #h(0.2em) frac(8, 15)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 25 n over 16")[$frac(25 n, 16)$] ⓑ #math.equation(block: false, alt: "the fraction 25 n minus 16 over 30")[$frac(25 n − 16, 30)$]
]

ⓐ #math.equation(block: false, alt: "the fraction 3 a over 8 divided by the fraction 7 over 12")[$frac(3 a, 8) ÷ frac(7, 12)$] ⓑ #math.equation(block: false, alt: "the fraction 3 a over 8 minus the fraction 7 over 12")[$frac(3 a, 8) − #h(0.2em) frac(7, 12)$]

#math.equation(block: true, alt: "minus the fraction 3 over 8 divided by open parenthesis minus the fraction 3 over 10 close parenthesis")[$− #h(0.2em) frac(3, 8) ÷ ( − #h(0.2em) frac(3, 10) )$]

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

#math.equation(block: true, alt: "minus the fraction 5 over 12 divided by open parenthesis minus the fraction 5 over 9 close parenthesis")[$− #h(0.2em) frac(5, 12) ÷ ( − #h(0.2em) frac(5, 9) )$]

#math.equation(block: true, alt: "minus the fraction 3 over 8 plus the fraction 5 over 12")[$− #h(0.2em) frac(3, 8) + frac(5, 12)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 24")[$frac(1, 24)$]
]

#math.equation(block: true, alt: "minus the fraction 1 over 8 plus the fraction 7 over 12")[$− #h(0.2em) frac(1, 8) + frac(7, 12)$]

#math.equation(block: true, alt: "the fraction 5 over 6 minus the fraction 1 over 9")[$frac(5, 6) − #h(0.2em) frac(1, 9)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over 18")[$frac(13, 18)$]
]

#math.equation(block: true, alt: "the fraction 5 over 9 minus the fraction 1 over 6")[$frac(5, 9) − #h(0.2em) frac(1, 6)$]

#math.equation(block: true, alt: "minus the fraction 7 over 15 minus the fraction y over 4")[$− #h(0.2em) frac(7, 15) − #h(0.2em) frac(y, 4)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction −28 minus 15 y over 60")[$frac(−28 − 15 y, 60)$]
]

#math.equation(block: true, alt: "minus the fraction 3 over 8 minus the fraction x over 11")[$− #h(0.2em) frac(3, 8) − #h(0.2em) frac(x, 11)$]

#math.equation(block: true, alt: "the fraction 11 over 12 a times the fraction 9 a over 16")[$frac(11, 12 a) · frac(9 a, 16)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 33 over 64")[$frac(33, 64)$]
]

#math.equation(block: true, alt: "the fraction 10 y over 13 times the fraction 8 over 15 y")[$frac(10 y, 13) · frac(8, 15 y)$]

#strong[Use the Order of Operations to Simplify Complex Fractions]

In the following exercises, simplify.

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

#solutionbox[
54
]

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

#math.equation(block: true, alt: "the fraction open parenthesis the fraction 3 over 5 close parenthesis squared over open parenthesis the fraction 3 over 7 close parenthesis squared")[$frac(attach(( frac(3, 5) ), t: 2), attach(( frac(3, 7) ), t: 2))$]

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

#math.equation(block: true, alt: "the fraction open parenthesis the fraction 3 over 4 close parenthesis squared over open parenthesis the fraction 5 over 8 close parenthesis squared")[$frac(attach(( frac(3, 4) ), t: 2), attach(( frac(5, 8) ), t: 2))$]

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

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

#math.equation(block: true, alt: "the fraction 5 over the fraction 1 over 4 plus the fraction 1 over 3")[$frac(5, frac(1, 4) + frac(1, 3))$]

#math.equation(block: true, alt: "the fraction the fraction 7 over 8 minus the fraction 2 over 3 over the fraction 1 over 2 plus the fraction 3 over 8")[$frac(frac(7, 8) − #h(0.2em) frac(2, 3), frac(1, 2) + frac(3, 8))$]

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

#math.equation(block: true, alt: "the fraction the fraction 3 over 4 minus the fraction 3 over 5 over the fraction 1 over 4 plus the fraction 2 over 5")[$frac(frac(3, 4) − #h(0.2em) frac(3, 5), frac(1, 4) + frac(2, 5))$]

#math.equation(block: true, alt: "the fraction 1 over 2 plus the fraction 2 over 3 times the fraction 5 over 12")[$frac(1, 2) + frac(2, 3) · frac(5, 12)$]

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

#math.equation(block: true, alt: "the fraction 1 over 3 plus the fraction 2 over 5 times the fraction 3 over 4")[$frac(1, 3) + frac(2, 5) · frac(3, 4)$]

#math.equation(block: true, alt: "1 minus the fraction 3 over 5 divided by the fraction 1 over 10")[$1 − #h(0.2em) frac(3, 5) ÷ frac(1, 10)$]

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

#math.equation(block: true, alt: "1 minus the fraction 5 over 6 divided by the fraction 1 over 12")[$1 − #h(0.2em) frac(5, 6) ÷ frac(1, 12)$]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 19 over 12")[$frac(19, 12)$]
]

#math.equation(block: true, alt: "the fraction 2 over 3 plus the fraction 1 over 4 plus the fraction 3 over 5")[$frac(2, 3) + frac(1, 4) + frac(3, 5)$]

#math.equation(block: true, alt: "the fraction 3 over 8 minus the fraction 1 over 6 plus the fraction 3 over 4")[$frac(3, 8) − #h(0.2em) frac(1, 6) + frac(3, 4)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 23 over 24")[$frac(23, 24)$]
]

#math.equation(block: true, alt: "the fraction 2 over 5 plus the fraction 5 over 8 minus the fraction 3 over 4")[$frac(2, 5) + frac(5, 8) − #h(0.2em) frac(3, 4)$]

#math.equation(block: true, alt: "12 open parenthesis the fraction 9 over 20 minus the fraction 4 over 15 close parenthesis")[$12 ( frac(9, 20) − #h(0.2em) frac(4, 15) )$]

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

#math.equation(block: true, alt: "8 open parenthesis the fraction 15 over 16 minus the fraction 5 over 6 close parenthesis")[$8 ( frac(15, 16) − #h(0.2em) frac(5, 6) )$]

#math.equation(block: true, alt: "the fraction the fraction 5 over 8 plus the fraction 1 over 6 over the fraction 19 over 24")[$frac(frac(5, 8) + frac(1, 6), frac(19, 24))$]

#solutionbox[
1
]

#math.equation(block: true, alt: "the fraction the fraction 1 over 6 plus the fraction 3 over 10 over the fraction 14 over 30")[$frac(frac(1, 6) + frac(3, 10), frac(14, 30))$]

#math.equation(block: true, alt: "open parenthesis the fraction 5 over 9 plus the fraction 1 over 6 close parenthesis divided by open parenthesis the fraction 2 over 3 minus the fraction 1 over 2 close parenthesis")[$( frac(5, 9) + frac(1, 6) ) ÷ ( frac(2, 3) − #h(0.2em) frac(1, 2) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over 3")[$frac(13, 3)$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 3 over 4 plus the fraction 1 over 6 close parenthesis divided by open parenthesis the fraction 5 over 8 minus the fraction 1 over 3 close parenthesis")[$( frac(3, 4) + frac(1, 6) ) ÷ ( frac(5, 8) − #h(0.2em) frac(1, 3) )$]

#strong[Evaluate Variable Expressions with Fractions]

In the following exercises, evaluate.

#math.equation(block: false, alt: "x plus open parenthesis minus the fraction 5 over 6 close parenthesis")[$x + ( − #h(0.2em) frac(5, 6) )$] when #linebreak() ⓐ #math.equation(block: false, alt: "x equals the fraction 1 over 3")[$x = frac(1, 3)$] #linebreak() ⓑ #math.equation(block: false, alt: "x equals minus the fraction 1 over 6")[$x = − #h(0.2em) frac(1, 6)$]

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

#math.equation(block: false, alt: "x plus open parenthesis minus the fraction 11 over 12 close parenthesis")[$x + ( − #h(0.2em) frac(11, 12) )$] when #linebreak() ⓐ #math.equation(block: false, alt: "x equals the fraction 11 over 12")[$x = frac(11, 12)$]ⓑ #math.equation(block: false, alt: "x equals the fraction 3 over 4")[$x = frac(3, 4)$]

#math.equation(block: false, alt: "x minus the fraction 2 over 5")[$x − #h(0.2em) frac(2, 5)$] when ⓐ #math.equation(block: false, alt: "x equals the fraction 3 over 5")[$x = frac(3, 5)$] ⓑ #math.equation(block: false, alt: "x equals minus the fraction 3 over 5")[$x = − #h(0.2em) frac(3, 5)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] ⓑ #math.equation(block: false, alt: "−1")[$−1$]
]

#math.equation(block: false, alt: "x minus the fraction 1 over 3")[$x − #h(0.2em) frac(1, 3)$] when ⓐ #math.equation(block: false, alt: "x equals the fraction 2 over 3")[$x = frac(2, 3)$] ⓑ #math.equation(block: false, alt: "x equals minus the fraction 2 over 3")[$x = − #h(0.2em) frac(2, 3)$]

#math.equation(block: false, alt: "the fraction 7 over 10 minus w")[$frac(7, 10) − w$] when ⓐ #math.equation(block: false, alt: "w equals the fraction 1 over 2")[$w = frac(1, 2)$] ⓑ #math.equation(block: false, alt: "w equals minus the fraction 1 over 2")[$w = − #h(0.2em) frac(1, 2)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] ⓑ #math.equation(block: false, alt: "the fraction 6 over 5")[$frac(6, 5)$]
]

#math.equation(block: false, alt: "the fraction 5 over 12 minus w")[$frac(5, 12) − w$] when ⓐ #math.equation(block: false, alt: "w equals the fraction 1 over 4")[$w = frac(1, 4)$] ⓑ #math.equation(block: false, alt: "w equals minus the fraction 1 over 4")[$w = − #h(0.2em) frac(1, 4)$]

#math.equation(block: false, alt: "2 x squared y cubed")[$2 x^(2) y^(3)$] when #math.equation(block: false, alt: "x equals minus the fraction 2 over 3")[$x = − #h(0.2em) frac(2, 3)$] and #math.equation(block: false, alt: "y equals minus the fraction 1 over 2")[$y = − #h(0.2em) frac(1, 2)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 9")[$− #h(0.2em) frac(1, 9)$]
]

#math.equation(block: false, alt: "8 u squared v cubed")[$8 u^(2) v^(3)$] when #math.equation(block: false, alt: "u equals minus the fraction 3 over 4")[$u = − #h(0.2em) frac(3, 4)$] and #math.equation(block: false, alt: "v equals minus the fraction 1 over 2")[$v = − #h(0.2em) frac(1, 2)$]

#math.equation(block: false, alt: "the fraction a plus b over a minus b")[$frac(a + b, a − b)$] when #math.equation(block: false, alt: "a equals −3 , b equals 8")[$a = −3 , b = 8$]

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

#math.equation(block: false, alt: "the fraction r minus s over r plus s")[$frac(r − s, r + s)$] when #math.equation(block: false, alt: "r equals 10 , s equals −5")[$r = 10 , s = −5$]


==== Everyday Math

#strong[Decorating] Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] yard of print fabric and #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

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

#strong[Baking] Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] cup of sugar for the chocolate chip cookies and #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] of sugar for the oatmeal cookies. How much sugar does she need altogether?


==== Writing Exercises

Why do you need a common denominator to add or subtract fractions? Explain.

#solutionbox[
Answers may vary
]

How do you find the LCD of 2 fractions?


==== Self Check

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

#figure(figph[This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “add and subtract fractions with different denominators,” “identify and use fraction operations,” “use the order of operations to simplify complex fractions,” and “evaluate variable expressions with fractions.” The rest of the cells are blank.], alt: "This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “add and subtract fractions with different denominators,” “identify and use fraction operations,” “use the order of operations to simplify complex fractions,” and “evaluate variable expressions with fractions.” The rest of the cells are blank.", caption: none)

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