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== 4.4#h(0.6em)Add and Subtract Fractions with Common Denominators

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

Simplify: #math.equation(block: false, alt: "2 x plus 9 plus 3 x minus 4 .")[$2 x + 9 + 3 x − 4 .$] #linebreak() 
If you missed this problem, review .

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

#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"))[
Simplify: #math.equation(block: false, alt: "the fraction 3 plus 2 over 6 .")[$frac(3 + 2, 6) .$] #linebreak() 
If you missed this problem, review .

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


=== Model Fraction Addition

How many quarters are pictured? One quarter plus #math.equation(block: false, alt: "2")[$2$] quarters equals #math.equation(block: false, alt: "3")[$3$] quarters.

#figure(figph[Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.], alt: "Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.", caption: none)

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

#math.equation(block: true, alt: "the fraction 1 over 4, the fraction 2 over 4, the fraction 3 over 4; one quarter, plus, two quarters, equals, three quarters")[$frac(1, 4) &  & frac(2, 4) &  & frac(3, 4) \ "one quarter" & + & "two quarters" & = & "three quarters"$]Let’s use fraction circles to model the same example, #math.equation(block: false, alt: "the fraction 1 over 4 plus the fraction 2 over 4 .")[$frac(1, 4) + frac(2, 4) .$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Start with one #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] piece.], [#figure(figph[A light salmon-colored quarter circle with a red outline is positioned on the left side of a white background, suggesting a geometric shape or part of a larger circle.], alt: "A light salmon-colored quarter circle with a red outline is positioned on the left side of a white background, suggesting a geometric shape or part of a larger circle.", caption: none)], [#figure(figph[The mathematical fraction 1/4, representing one-fourth, is clearly displayed on a white background.], alt: "The mathematical fraction 1/4, representing one-fourth, is clearly displayed on a white background.", caption: none)]),
  [Add two more #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]pieces.], [#figure(figph[A light orange semi-circle, divided in half vertically, with a small black plus sign to its left, rests above a thin horizontal grey line against a white background.], alt: "A light orange semi-circle, divided in half vertically, with a small black plus sign to its left, rests above a thin horizontal grey line against a white background.", caption: none)], [#figure(figph[A mathematical expression featuring a plus sign followed by the fraction two over four, with a blank line or underscore beneath it, indicating an operation or a problem to be solved.], alt: "A mathematical expression featuring a plus sign followed by the fraction two over four, with a blank line or underscore beneath it, indicating an operation or a problem to be solved.", caption: none)],
  [The result is #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$].], [#figure(figph[A circle is divided into four equal sections. Three sections are shaded in a light orange-red color, illustrating three-quarters of the circle being filled, with one section remaining white.], alt: "A circle is divided into four equal sections. Three sections are shaded in a light orange-red color, illustrating three-quarters of the circle being filled, with one section remaining white.", caption: none)], [#figure(figph[A white background with the fraction 3/4 written in black text on the right side.], alt: "A white background with the fraction 3/4 written in black text on the right side.", caption: none)],
))

So again, we see that

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

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Start with three #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] pieces.], [#figure(figph[A quarter-circle shape, filled with a pale orange color and outlined in red, is sectioned into three distinct segments by two lines extending from the central point of the arc.], alt: "A quarter-circle shape, filled with a pale orange color and outlined in red, is sectioned into three distinct segments by two lines extending from the central point of the arc.", caption: none)], [#figure(figph[The image displays the fraction 3/8, written vertically with the numeral 3 above a horizontal line and the numeral 8 below it, centered on a white background.], alt: "The image displays the fraction 3/8, written vertically with the numeral 3 above a horizontal line and the numeral 8 below it, centered on a white background.", caption: none)]),
  [Add two #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]pieces.], [#figure(figph[An abstract image featuring a light orange, fan-like shape divided vertically, with a plus sign to its left and a horizontal line underneath, suggesting a mathematical or symbolic representation.], alt: "An abstract image featuring a light orange, fan-like shape divided vertically, with a plus sign to its left and a horizontal line underneath, suggesting a mathematical or symbolic representation.", caption: none)], [#figure(figph[A mathematical expression shows a plus sign followed by the fraction two-eighths (+ 2/8) with a horizontal line underneath, likely indicating part of a larger sum or equation.], alt: "A mathematical expression shows a plus sign followed by the fraction two-eighths (+ 2/8) with a horizontal line underneath, likely indicating part of a larger sum or equation.", caption: none)],
  [How many #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]pieces are there?], [#figure(figph[A circle is divided into eight equal segments, with five of these segments shaded in a light orange color and three segments unshaded.], alt: "A circle is divided into eight equal segments, with five of these segments shaded in a light orange color and three segments unshaded.", caption: none)], [#figure(figph[The fraction 5/8 is displayed against a plain white background.], alt: "The fraction 5/8 is displayed against a plain white background.", caption: none)],
))

There are five #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] pieces, or five-eighths. The model shows that #math.equation(block: false, alt: "the fraction 3 over 8 plus the fraction 2 over 8 equals the fraction 5 over 8 .")[$frac(3, 8) + frac(2, 8) = frac(5, 8) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use a model to find each sum. Show a diagram to illustrate your model.

#math.equation(block: true, alt: "the fraction 1 over 8 plus the fraction 4 over 8")[$frac(1, 8) + frac(4, 8)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 8")[$frac(5, 8)$] #linebreak() #figure(figph[A circle divided into 8 sections, 5 of which are shaded.], alt: "A circle divided into 8 sections, 5 of which are shaded.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use a model to find each sum. Show a diagram to illustrate your model.

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 6")[$frac(5, 6)$] #linebreak() #figure(figph[A circle divided into 6 sections, 5 of which are shaded.], alt: "A circle divided into 6 sections, 5 of which are shaded.", caption: none)
]
]


=== Add Fractions with a Common Denominator

shows that to add the same-size pieces—meaning that the fractions have the same #strong[denominator]—we just add the number of pieces.

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

If #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c")[$c$] 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")[$frac(a, c) + frac(b, c) = frac(a + b, c)$]To add fractions with a common denominator, add the numerators and place the sum over the common denominator.
]

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 3 over 5 plus the fraction 1 over 5")[$frac(3, 5) + frac(1, 5)$]]),
  [Add the numerators and place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction 3 plus 1 over 5")[$frac(3 + 1, 5)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 4 over 5")[$frac(4, 5)$]],
))
]
]

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

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find each sum: #math.equation(block: false, alt: "the fraction 3 over 10 plus the fraction 7 over 10 .")[$frac(3, 10) + frac(7, 10) .$]

#solutionbox[
1
]
]

#examplebox("Example 3")[][
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")[$frac(x + 2, 3)$]],
))

Note that we cannot simplify this fraction any more. Since #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "2")[$2$] are not like terms, we cannot combine them.
]
]

#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 4")[][
Find the sum: #math.equation(block: false, alt: "minus the fraction 9 over d plus the fraction 3 over d .")[$− frac(9, d) + frac(3, d) .$]

#solutionbox[
We will begin by rewriting the first fraction with the negative sign in the numerator.

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 9 over d plus the fraction 3 over d")[$− frac(9, d) + frac(3, d)$]]),
  [Rewrite the first fraction with the negative in the numerator.], [#math.equation(block: false, alt: "the fraction −9 over d plus the fraction 3 over d")[$frac(−9, d) + frac(3, d)$]],
  [Add the numerators and place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction −9 plus 3 over d")[$frac(−9 + 3, d)$]],
  [Simplify the numerator.], [#math.equation(block: false, alt: "the fraction −6 over d")[$frac(−6, d)$]],
  [Rewrite with negative sign in front of the fraction.], [#math.equation(block: false, alt: "minus the fraction 6 over d")[$− frac(6, d)$]],
))
]
]

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

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "minus the fraction 6 over m plus the fraction 9 over m .")[$− frac(6, m) + frac(9, m) .$]

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

#examplebox("Example 5")[][
Find the sum: #math.equation(block: false, alt: "the fraction 2 n over 11 plus the fraction 5 n over 11 .")[$frac(2 n, 11) + frac(5 n, 11) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 2 n over 11 plus the fraction 5 n over 11")[$frac(2 n, 11) + frac(5 n, 11)$]]),
  [Add the numerators and place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction 2 n plus 5 n over 11")[$frac(2 n + 5 n, 11)$]],
  [Combine like terms.], [#math.equation(block: false, alt: "the fraction 7 n over 11")[$frac(7 n, 11)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "the fraction 3 p over 8 plus the fraction 6 p over 8 .")[$frac(3 p, 8) + frac(6 p, 8) .$]

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

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

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

#examplebox("Example 6")[][
Find the sum: #math.equation(block: false, alt: "minus the fraction 3 over 12 plus open parenthesis minus the fraction 5 over 12 close parenthesis .")[$− frac(3, 12) + ( − frac(5, 12) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus the fraction 3 over 12 plus open parenthesis minus the fraction 5 over 12 close parenthesis")[$− frac(3, 12) + ( − frac(5, 12) )$]]),
  [Add the numerators and place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction −3 plus open parenthesis −5 close parenthesis over 12")[$frac(−3 + ( −5 ), 12)$]],
  [Add.], [#math.equation(block: false, alt: "the fraction −8 over 12")[$frac(−8, 12)$]],
  [Simplify the fraction.], [#math.equation(block: false, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find each sum: #math.equation(block: false, alt: "minus the fraction 4 over 15 plus open parenthesis minus the fraction 6 over 15 close parenthesis .")[$− frac(4, 15) + ( − frac(6, 15) ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find each sum: #math.equation(block: false, alt: "minus the fraction 5 over 21 plus open parenthesis minus the fraction 9 over 21 close parenthesis .")[$− frac(5, 21) + ( − frac(9, 21) ) .$]

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


=== Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into #math.equation(block: false, alt: "12")[$12$] slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] of the pizza) left in the box. If Leonardo eats #math.equation(block: false, alt: "2")[$2$] of these remaining pieces (or #math.equation(block: false, alt: "the fraction 2 over 12")[$frac(2, 12)$] of the pizza), how much is left? There would be #math.equation(block: false, alt: "5")[$5$] pieces left (or #math.equation(block: false, alt: "the fraction 5 over 12")[$frac(5, 12)$] of the pizza).

#math.equation(block: true, alt: "the fraction 7 over 12 minus the fraction 2 over 12 equals the fraction 5 over 12")[$frac(7, 12) − frac(2, 12) = frac(5, 12)$]Let’s use fraction circles to model the same example, #math.equation(block: false, alt: "the fraction 7 over 12 minus the fraction 2 over 12 .")[$frac(7, 12) − frac(2, 12) .$]

Start with seven #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$] pieces. Take away two #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$] pieces. How many twelfths are left?

#figure(figph[The bottom reads 7 twelfths minus 2 twelfths equals 5 twelfths. Above 7 twelfths, there is a circle divided into 12 equal pieces, with 7 pieces shaded in orange. Above 2 twelfths, the same circle is shown, but 2 of the 7 pieces are shaded in grey. Above 5 twelfths, the 2 grey pieces are no longer shaded, so there is a circle divided into 12 pieces with 5 of the pieces shaded in orange.], alt: "The bottom reads 7 twelfths minus 2 twelfths equals 5 twelfths. Above 7 twelfths, there is a circle divided into 12 equal pieces, with 7 pieces shaded in orange. Above 2 twelfths, the same circle is shown, but 2 of the 7 pieces are shaded in grey. Above 5 twelfths, the 2 grey pieces are no longer shaded, so there is a circle divided into 12 pieces with 5 of the pieces shaded in orange.", caption: none)

Again, we have five twelfths, #math.equation(block: false, alt: "the fraction 5 over 12 .")[$frac(5, 12) .$]

#examplebox("Example 7")[][
Use fraction circles to find the difference: #math.equation(block: false, alt: "the fraction 4 over 5 minus the fraction 1 over 5 .")[$frac(4, 5) − frac(1, 5) .$]

#solutionbox[
Start with four #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] pieces. Take away one #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] piece. Count how many fifths are left. There are three #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] pieces left.

#figure(figph[The bottom reads 4 fifths minus 1 fifth equals 3 fifths. Above 4 fifths, there is a circle divided into 5 equal pieces, with 4 pieces shaded in orange. Above 1 fifth, the same circle is shown, but 1 of the 4 shaded pieces is shaded in grey. Above 3 fifths, the 1 grey piece is no longer shaded, so there is a circle divided into 5 pieces with 3 of the pieces shaded in orange.], alt: "The bottom reads 4 fifths minus 1 fifth equals 3 fifths. Above 4 fifths, there is a circle divided into 5 equal pieces, with 4 pieces shaded in orange. Above 1 fifth, the same circle is shown, but 1 of the 4 shaded pieces is shaded in grey. Above 3 fifths, the 1 grey piece is no longer shaded, so there is a circle divided into 5 pieces with 3 of the pieces shaded in orange.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use a model to find each difference. Show a diagram to illustrate your model.

#math.equation(block: true, alt: "the fraction 7 over 8 minus the fraction 4 over 8")[$frac(7, 8) − frac(4, 8)$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use a model to find each difference. Show a diagram to illustrate your model.

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

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


=== Subtract Fractions with a Common Denominator

We subtract fractions with a common #strong[denominator] in much the same way as we add fractions with a common denominator.

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

If #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c")[$c$] 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 minus the fraction b over c equals the fraction a minus b over c")[$frac(a, c) − frac(b, c) = frac(a − b, c)$]To subtract fractions with a common denominator, we subtract the numerators and place the difference over the common denominator.
]

#examplebox("Example 8")[][
Find the difference: #math.equation(block: false, alt: "the fraction 23 over 24 minus the fraction 14 over 24 .")[$frac(23, 24) − frac(14, 24) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 23 over 24 minus the fraction 14 over 24")[$frac(23, 24) − frac(14, 24)$]]),
  [Subtract the numerators and place the difference over the common denominator.], [#math.equation(block: false, alt: "the fraction 23 minus 14 over 24")[$frac(23 − 14, 24)$]],
  [Simplify the numerator.], [#math.equation(block: false, alt: "the fraction 9 over 24")[$frac(9, 24)$]],
  [Simplify the fraction by removing common factors.], [#math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$]],
))
]
]

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

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "the fraction 27 over 32 minus the fraction 11 over 32 .")[$frac(27, 32) − frac(11, 32) .$]

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

#examplebox("Example 9")[][
Find the difference: #math.equation(block: false, alt: "the fraction y over 6 minus the fraction 1 over 6 .")[$frac(y, 6) − frac(1, 6) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction y over 6 minus the fraction 1 over 6")[$frac(y, 6) − frac(1, 6)$]]),
  [Subtract the numerators and place the difference over the common denominator.], [#math.equation(block: false, alt: "the fraction y minus 1 over 6")[$frac(y − 1, 6)$]],
))

The fraction is simplified because we cannot combine the terms in the numerator.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "the fraction x over 7 minus the fraction 2 over 7 .")[$frac(x, 7) − frac(2, 7) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "the fraction y over 14 minus the fraction 13 over 14 .")[$frac(y, 14) − frac(13, 14) .$]

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

#examplebox("Example 10")[][
Find the difference: #math.equation(block: false, alt: "minus the fraction 10 over x minus the fraction 4 over x .")[$− frac(10, x) − frac(4, x) .$]

#solutionbox[
Remember, the fraction #math.equation(block: false, alt: "minus the fraction 10 over x")[$− frac(10, x)$] can be written as #math.equation(block: false, alt: "the fraction −10 over x .")[$frac(−10, x) .$]

#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")[$− frac(10, x) − frac(4, x)$]]),
  [Subtract the numerators.], [#math.equation(block: false, alt: "the fraction −10 minus 4 over x")[$frac(−10 − 4, x)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction −14 over x")[$frac(−14, x)$]],
  [Rewrite with the negative sign in front of the fraction.], [#math.equation(block: false, alt: "minus the fraction 14 over x")[$− frac(14, x)$]],
))
]
]

#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 .")[$− frac(9, x) − frac(7, x) .$]

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

#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 .")[$− frac(17, a) − frac(5, a) .$]

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

Now lets do an example that involves both addition and subtraction.

#examplebox("Example 11")[][
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 .")[$frac(3, 8) + ( − frac(5, 8) ) − frac(1, 8) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#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")[$frac(3, 8) + ( − frac(5, 8) ) − frac(1, 8)$]]),
  [Combine the numerators 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 the numerator, working left to right.], [#math.equation(block: false, alt: "the fraction −2 minus 1 over 8")[$frac(−2 − 1, 8)$]],
  [Subtract the terms in the numerator.], [#math.equation(block: false, alt: "the fraction −3 over 8")[$frac(−3, 8)$]],
  [Rewrite with the negative sign in front of the fraction.], [#math.equation(block: false, alt: "minus the fraction 3 over 8")[$− frac(3, 8)$]],
))
]
]

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

#solutionbox[
−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 .")[$frac(5, 9) + ( − frac(4, 9) ) − frac(7, 9) .$]

#solutionbox[
#math.equation(block: true, alt: "minus 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/24AddFraction")[Adding Fractions With Pattern Blocks]
- #link("https://www.openstax.org/l/24AddLikeDenom")[Adding Fractions With Like Denominators]
- #link("https://www.openstax.org/l/24SubtrLikeDeno")[Subtracting Fractions With Like Denominators]
]


=== Key Concepts

- #emph[Fraction Addition]- If #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c")[$c$] are numbers where #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$], then #math.equation(block: false, alt: "the fraction a over c plus the fraction b over c equals the fraction a plus b over c")[$frac(a, c) + frac(b, c) = frac(a + b, c)$].
- To add fractions, add the numerators and place the sum over the common denominator.
- #emph[Fraction Subtraction]- If #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c")[$c$] are numbers where #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$], then #math.equation(block: false, alt: "the fraction a over c minus the fraction b over c equals the fraction a minus b over c")[$frac(a, c) − frac(b, c) = frac(a − b, c)$].
- To subtract fractions, subtract the numerators and place the difference over the common denominator.


==== Practice Makes Perfect

#strong[Model Fraction Addition]

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

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

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

#solutionbox[
#figure(figph[A circle is divided into 10 equal pieces. 7 of the pieces are shaded.], alt: "A circle is divided into 10 equal pieces. 7 of the pieces are shaded.", caption: none)

 #linebreak() 
#math.equation(block: true, alt: "the fraction 7 over 10")[$frac(7, 10)$]
]

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

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

#solutionbox[
#figure(figph[A circle is divided into 8 equal pieces. 6 of the pieces are shaded.], alt: "A circle is divided into 8 equal pieces. 6 of the pieces are shaded.", caption: none)

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

#strong[Add Fractions with a Common Denominator]

In the following exercises, find each sum.

#math.equation(block: true, alt: "the fraction 4 over 9 plus the fraction 1 over 9")[$frac(4, 9) + frac(1, 9)$]

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

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

#math.equation(block: true, alt: "the fraction 6 over 13 plus the fraction 7 over 13")[$frac(6, 13) + frac(7, 13)$]

#math.equation(block: true, alt: "the fraction 9 over 15 plus the fraction 7 over 15")[$frac(9, 15) + frac(7, 15)$]

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

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

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

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

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

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

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

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

#math.equation(block: true, alt: "the fraction 5 a over 7 plus the fraction 4 a over 7")[$frac(5 a, 7) + frac(4 a, 7)$]

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

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

#math.equation(block: true, alt: "the fraction −11 x over 5 plus the fraction 7 x over 5")[$frac(−11 x, 5) + frac(7 x, 5)$]

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

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

#math.equation(block: true, alt: "minus the fraction 1 over 8 plus open parenthesis minus the fraction 5 over 8 close parenthesis")[$− frac(1, 8) + ( − frac(5, 8) )$]

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

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

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 over 19")[$frac(8, 19)$]
]

#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")[$frac(6, 13) + ( − frac(10, 13) ) + ( − frac(12, 13) )$]

#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")[$frac(5, 12) + ( − frac(7, 12) ) + ( − frac(11, 12) )$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 13 over 12")[$− frac(13, 12)$]
]

#strong[Model Fraction Subtraction]

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

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

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

#solutionbox[
#figure(figph[A circle is divided into eight sections, three of which are shaded.], alt: "A circle is divided into eight sections, three of which are shaded.", caption: none)

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

#strong[Subtract Fractions with a Common Denominator]

In the following exercises, find the difference.

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

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

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

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

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

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

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

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

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

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

#math.equation(block: true, alt: "minus the fraction 8 over 9 minus the fraction 16 over 9")[$− frac(8, 9) − frac(16, 9)$]

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

#math.equation(block: true, alt: "the fraction y over 17 minus the fraction 9 over 17")[$frac(y, 17) − frac(9, 17)$]

#math.equation(block: true, alt: "the fraction x over 19 minus the fraction 8 over 19")[$frac(x, 19) − frac(8, 19)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x minus 8 over 19")[$frac(x − 8, 19)$]
]

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

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

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

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

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 55 over v")[$− frac(55, v)$]
]

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

#math.equation(block: true, alt: "the fraction 12 d over 11 minus the fraction 9 d over 11")[$frac(12 d, 11) − frac(9 d, 11)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 d over 11")[$frac(3 d, 11)$]
]

#math.equation(block: true, alt: "the fraction −4 r over 13 minus the fraction 5 r over 13")[$frac(−4 r, 13) − frac(5 r, 13)$]

#math.equation(block: true, alt: "the fraction −7 s over 3 minus the fraction 7 s over 3")[$frac(−7 s, 3) − frac(7 s, 3)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 14 s over 3")[$− frac(14 s, 3)$]
]

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

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 3 over 11")[$− frac(3, 11)$]
]

#strong[Mixed Practice]

In the following exercises, perform the indicated operation and write your answers in simplified form.

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

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

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

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

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

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

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

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 49 over 27")[$frac(49, 27)$]
]


==== Everyday Math

#strong[Trail Mix] Jacob is mixing together nuts and raisins to make trail mix. He has #math.equation(block: false, alt: "the fraction 6 over 10")[$frac(6, 10)$] of a pound of nuts and #math.equation(block: false, alt: "the fraction 3 over 10")[$frac(3, 10)$] of a pound of raisins. How much trail mix can he make?

#strong[Baking] Janet needs #math.equation(block: false, alt: "the fraction 5 over 8")[$frac(5, 8)$] of a cup of flour for a recipe she is making. She only has #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] of a cup of flour and will ask to borrow the rest from her next-door neighbor. How much flour does she have to borrow?

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


==== Writing Exercises

Greg dropped his case of drill bits and three of the bits fell out. The case has slots for the drill bits, and the slots are arranged in order from smallest to largest. Greg needs to put the bits that fell out back in the case in the empty slots. Where do the three bits go? Explain how you know. #linebreak()  Bits in case: #math.equation(block: false, alt: "the fraction 1 over 16")[$frac(1, 16)$], #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$], \_\_\_, \_\_\_, #math.equation(block: false, alt: "the fraction 5 over 16")[$frac(5, 16)$], #math.equation(block: false, alt: "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 16")[$frac(9, 16)$], #math.equation(block: false, alt: "the fraction 5 over 8")[$frac(5, 8)$].  #linebreak() Bits that fell out: #math.equation(block: false, alt: "the fraction 7 over 16")[$frac(7, 16)$], #math.equation(block: false, alt: "the fraction 3 over 16")[$frac(3, 16)$], #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$].

After a party, Lupe has #math.equation(block: false, alt: "the fraction 5 over 12")[$frac(5, 12)$] of a cheese pizza, #math.equation(block: false, alt: "the fraction 4 over 12")[$frac(4, 12)$] of a pepperoni pizza, and #math.equation(block: false, alt: "the fraction 4 over 12")[$frac(4, 12)$] of a veggie pizza left. Will all the slices fit into #math.equation(block: false, alt: "1")[$1$] pizza box? Explain your reasoning.

#solutionbox[
No, adding up the number of pieces gives #math.equation(block: false, alt: "the fraction 13 over 12")[$frac(13, 12)$], which is greater than 1. (Answers may vary.)
]


==== Self Check

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

#figure(figph[Fraction skills self-assessment table. Users rate their ability to model and add/subtract fractions with common denominators: 'Confidently', 'With some help', or 'No - I don't get it!'], alt: "Fraction skills self-assessment table. Users rate their ability to model and add/subtract fractions with common denominators: 'Confidently', 'With some help', or 'No - I don't get it!'", caption: none)

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?