#set document(title: "4.5 Add and Subtract Fractions with Different Denominators", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 4.5#h(0.6em)Add and Subtract Fractions with Different Denominators #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Find two fractions equivalent to #math.equation(block: false, alt: "the fraction 5 over 6 .")[$frac(5, 6) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "the fraction 10 over 12 , the fraction 15 over 18")[$frac(10, 12) , frac(15, 18)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 1 plus 5 times 3 over 2 squared plus 4 .")[$frac(1 + 5 · 3, 2^(2) + 4) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "2")[$2$] ] ] === Find the Least Common Denominator In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators? Let’s think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals #math.equation(block: false, alt: "25")[$25$] cents and one dime equals #math.equation(block: false, alt: "10")[$10$] cents, so the sum is #math.equation(block: false, alt: "35")[$35$] cents. #figure(figph[A quarter and a dime are shown. Below them, it reads 25 cents plus 10 cents. Below that, it reads 35 cents.], alt: "A quarter and a dime are shown. Below them, it reads 25 cents plus 10 cents. Below that, it reads 35 cents.", caption: [Together, a quarter and a dime are worth #math.equation(block: false, alt: "35")[$35$] cents, or #math.equation(block: false, alt: "the fraction 35 over 100")[$frac(35, 100)$] of a dollar.]) Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is #math.equation(block: false, alt: "100 .")[$100 .$] Since there are #math.equation(block: false, alt: "100")[$100$] cents in one dollar, #math.equation(block: false, alt: "25")[$25$] cents is #math.equation(block: false, alt: "the fraction 25 over 100")[$frac(25, 100)$] and #math.equation(block: false, alt: "10")[$10$] cents is #math.equation(block: false, alt: "the fraction 10 over 100 .")[$frac(10, 100) .$] So we add #math.equation(block: false, alt: "the fraction 25 over 100 plus the fraction 10 over 100")[$frac(25, 100) + frac(10, 100)$] to get #math.equation(block: false, alt: "the fraction 35 over 100 ,")[$frac(35, 100) ,$] which is #math.equation(block: false, alt: "35")[$35$] cents. You have practiced adding and subtracting fractions with common denominators. Now let’s see what you need to do with fractions that have different denominators. First, we will use fraction tiles to model finding the common denominator of #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] and #math.equation(block: false, alt: "the fraction 1 over 3 .")[$frac(1, 3) .$] We’ll start with one #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] tile and #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] tile. We want to find a common fraction tile that we can use to match #emph[both] #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] and #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] exactly. If we try the #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] pieces, #math.equation(block: false, alt: "2")[$2$] of them exactly match the #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] piece, but they do not exactly match the #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] piece. #figure(figph[Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into two pieces, each labeled 1 fourth. Underneath the second rectangle are two pieces, each labeled 1 fourth. These rectangles together are longer than the rectangle labeled as 1 third.], alt: "Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into two pieces, each labeled 1 fourth. Underneath the second rectangle are two pieces, each labeled 1 fourth. These rectangles together are longer than the rectangle labeled as 1 third.", caption: none) If we try the #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] pieces, they do not exactly cover the #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] piece or the #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] piece. #figure(figph[Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into three pieces, each labeled 1 sixth. Underneath the second rectangle is an equally sized rectangle split vertically into 2 pieces, each labeled 1 sixth.], alt: "Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into three pieces, each labeled 1 sixth. Underneath the second rectangle is an equally sized rectangle split vertically into 2 pieces, each labeled 1 sixth.", caption: none) If we try the #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] pieces, we see that exactly #math.equation(block: false, alt: "3")[$3$] of them cover the #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] piece, and exactly #math.equation(block: false, alt: "2")[$2$] of them cover the #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] piece. #figure(figph[Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle are three smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 half rectangle. Below the 1 third rectangle are two smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 third rectangle.], alt: "Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle are three smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 half rectangle. Below the 1 third rectangle are two smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 third rectangle.", caption: none) If we were to try the #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$] pieces, they would also work. #figure(figph[Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into 6 pieces, each labeled 1 twelfth. Underneath the second rectangle is an equally sized rectangle split vertically into 4 pieces, each labeled 1 twelfth.], alt: "Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into 6 pieces, each labeled 1 twelfth. Underneath the second rectangle is an equally sized rectangle split vertically into 4 pieces, each labeled 1 twelfth.", caption: none) Even smaller tiles, such as #math.equation(block: false, alt: "the fraction 1 over 24")[$frac(1, 24)$] and #math.equation(block: false, alt: "the fraction 1 over 48 ,")[$frac(1, 48) ,$] would also exactly cover the #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] piece and the #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] piece. The denominator of the largest piece that covers both fractions is the #strong[least common denominator (LCD)] of the two fractions. So, the least common denominator of #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] and #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] is #math.equation(block: false, alt: "6 .")[$6 .$] Notice that all of the tiles that cover #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] and #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] have something in common: Their denominators are common multiples of #math.equation(block: false, alt: "2")[$2$] and #math.equation(block: false, alt: "3 ,")[$3 ,$] the denominators of #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] and #math.equation(block: false, alt: "the fraction 1 over 3 .")[$frac(1, 3) .$] The least common multiple (LCM) of the denominators is #math.equation(block: false, alt: "6 ,")[$6 ,$] and so we say that #math.equation(block: false, alt: "6")[$6$] is the least common denominator (LCD) of the fractions #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] and #math.equation(block: false, alt: "the fraction 1 over 3 .")[$frac(1, 3) .$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Least Common Denominator] ] To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD. #examplebox("Example 1")[][ Find the LCD for the fractions #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] and #math.equation(block: false, alt: "the fraction 5 over 18 .")[$frac(5, 18) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Factor each denominator into its primes.], [#figure(figph[Two factor trees illustrate prime factorization. One tree shows 12 factoring into 3, 2, 2. The other shows 18 factoring into 3, 2, 3. The circled numbers are prime factors.], alt: "Two factor trees illustrate prime factorization. One tree shows 12 factoring into 3, 2, 2. The other shows 18 factoring into 3, 2, 3. The circled numbers are prime factors.", caption: none)]), [List the primes of 12 and the primes of 18 lining them up in columns when possible.], [#figure(figph[Prime factorization of 12 and 18, showing 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3, a setup for finding the greatest common factor or least common multiple.], alt: "Prime factorization of 12 and 18, showing 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3, a setup for finding the greatest common factor or least common multiple.", caption: none)], [Bring down the columns.], [#figure(figph[This image illustrates how to calculate the Least Common Multiple (LCM) of 12 and 18 using their prime factorization. It shows 12 = 2 as 2 \* 2 \* 3 and 18 = 2 \* 3 \* 3. The LCM is derived by taking all prime factors with their highest powers from either number, resulting in LCM = 2 \* 2 \* 3 \* 3.], alt: "This image illustrates how to calculate the Least Common Multiple (LCM) of 12 and 18 using their prime factorization. It shows 12 = 2 as 2 * 2 * 3 and 18 = 2 * 3 * 3. The LCM is derived by taking all prime factors with their highest powers from either number, resulting in LCM = 2 * 2 * 3 * 3.", caption: none)], [Multiply the factors. The product is the LCM.], [LCM#math.equation(block: false, alt: "equals 36")[$= 36$]], [The LCM of 12 and 18 is 36, so the LCD of #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] and #math.equation(block: false, alt: "the fraction 5 over 18")[$frac(5, 18)$] is 36.], [LCD of #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] and #math.equation(block: false, alt: "the fraction 5 over 18")[$frac(5, 18)$] is 36.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the least common denominator for the fractions: #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] and #math.equation(block: false, alt: "the fraction 11 over 15 .")[$frac(11, 15) .$] #solutionbox[ 60 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the least common denominator for the fractions: #math.equation(block: false, alt: "the fraction 13 over 15")[$frac(13, 15)$] and #math.equation(block: false, alt: "the fraction 17 over 5 .")[$frac(17, 5) .$] #solutionbox[ 15 ] ] To find the LCD of two fractions, find the LCM of their denominators. Notice how the steps shown below are similar to the steps we took to find the LCM. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find the least common denominator (LCD) of two fractions.] + Factor each denominator into its primes. + List the primes, matching primes in columns when possible. + Bring down the columns. + Multiply the factors. The product is the LCM of the denominators. + The LCM of the denominators is the LCD of the fractions. ] #examplebox("Example 2")[][ Find the least common denominator for the fractions #math.equation(block: false, alt: "the fraction 8 over 15")[$frac(8, 15)$] and #math.equation(block: false, alt: "the fraction 11 over 24 .")[$frac(11, 24) .$] #solutionbox[ To find the LCD, we find the LCM of the denominators. Find the LCM of #math.equation(block: false, alt: "15")[$15$] and #math.equation(block: false, alt: "24 .")[$24 .$] #figure(figph[The top line shows 15 equals 3 times 5. The next line shows 24 equals 2 times 2 times 2 times 3. The 3s are lined up vertically. The next line shows LCM equals 2 times 2 times 2 times 3 times 5. The last line shows LCM equals 120.], alt: "The top line shows 15 equals 3 times 5. The next line shows 24 equals 2 times 2 times 2 times 3. The 3s are lined up vertically. The next line shows LCM equals 2 times 2 times 2 times 3 times 5. The last line shows LCM equals 120.", caption: none) The LCM of #math.equation(block: false, alt: "15")[$15$] and #math.equation(block: false, alt: "24")[$24$] is #math.equation(block: false, alt: "120 .")[$120 .$] So, the LCD of #math.equation(block: false, alt: "the fraction 8 over 15")[$frac(8, 15)$] and #math.equation(block: false, alt: "the fraction 11 over 24")[$frac(11, 24)$] is #math.equation(block: false, alt: "120 .")[$120 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the least common denominator for the fractions: #math.equation(block: false, alt: "the fraction 13 over 24")[$frac(13, 24)$] and #math.equation(block: false, alt: "the fraction 17 over 32 .")[$frac(17, 32) .$] #solutionbox[ 96 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the least common denominator for the fractions: #math.equation(block: false, alt: "the fraction 9 over 28")[$frac(9, 28)$] and #math.equation(block: false, alt: "the fraction 21 over 32 .")[$frac(21, 32) .$] #solutionbox[ 224 ] ] === Convert Fractions to Equivalent Fractions with the LCD Earlier, we used fraction tiles to see that the LCD of #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] when #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] is #math.equation(block: false, alt: "12 .")[$12 .$] We saw that three #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$] pieces exactly covered #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] and two #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$] pieces exactly covered #math.equation(block: false, alt: "the fraction 1 over 6 ,")[$frac(1, 6) ,$] so #math.equation(block: true, alt: "the fraction 1 over 4 equals the fraction 3 over 12 and the fraction 1 over 6 equals the fraction 2 over 12 .")[$frac(1, 4) = frac(3, 12) " and " frac(1, 6) = frac(2, 12) .$]#figure(figph[On the left is a rectangle labeled 1 fourth. Below it is an identical rectangle split vertically into 3 equal pieces, each labeled 1 twelfth. On the right is a rectangle labeled 1 sixth. Below it is an identical rectangle split vertically into 2 equal pieces, each labeled 1 twelfth.], alt: "On the left is a rectangle labeled 1 fourth. Below it is an identical rectangle split vertically into 3 equal pieces, each labeled 1 twelfth. On the right is a rectangle labeled 1 sixth. Below it is an identical rectangle split vertically into 2 equal pieces, each labeled 1 twelfth.", caption: none) We say that #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] and #math.equation(block: false, alt: "the fraction 3 over 12")[$frac(3, 12)$] are equivalent fractions and also that #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] and #math.equation(block: false, alt: "the fraction 2 over 12")[$frac(2, 12)$] are equivalent fractions. We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Equivalent Fractions Property] If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are whole numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0 ,")[$b ≠ 0 , c ≠ 0 ,$] then #math.equation(block: true, alt: "the fraction a over b equals the fraction a times c over b times c and the fraction a times c over b times c equals the fraction a over b")[$frac(a, b) = frac(a · c, b · c) #h(1.0em) "and" #h(1.0em) frac(a · c, b · c) = frac(a, b)$] ] To add or subtract fractions with different denominators, we will first have to convert each fraction to an #strong[equivalent] fraction with the LCD. Let’s see how to change #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] and #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] to equivalent fractions with denominator #math.equation(block: false, alt: "12")[$12$] without using models. #examplebox("Example 3")[][ Convert #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] and #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] to equivalent fractions with denominator #math.equation(block: false, alt: "12 ,")[$12 ,$] their LCD. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find the LCD.], [The LCD of #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] and #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] is 12.]), [Find the number to multiply 4 to get 12.], [#figure(figph[A multiplication equation displays 4 multiplied by 3 (highlighted in red), equaling 12. This is a basic arithmetic problem, demonstrating a fundamental concept in mathematics.], alt: "A multiplication equation displays 4 multiplied by 3 (highlighted in red), equaling 12. This is a basic arithmetic problem, demonstrating a fundamental concept in mathematics.", caption: none)], [Find the number to multiply 6 to get 12.], [#figure(figph[A mathematical equation shows '6 2 = 12' on a white background, with the number 2 highlighted in red. The dot symbol signifies multiplication, indicating that six multiplied by two equals twelve.], alt: "A mathematical equation shows '6 2 = 12' on a white background, with the number 2 highlighted in red. The dot symbol signifies multiplication, indicating that six multiplied by two equals twelve.", caption: none)], [Use the Equivalent Fractions Property to convert each fraction to an equivalent fraction with the LCD, multiplying both the numerator and denominator of each fraction by the same number.], [#figure(figph[Fractions 1/4 and 1/6 are shown being converted to equivalent fractions by multiplying their numerators and denominators by 3 and 2, respectively, to find a common denominator.], alt: "Fractions 1/4 and 1/6 are shown being converted to equivalent fractions by multiplying their numerators and denominators by 3 and 2, respectively, to find a common denominator.", caption: none)], [Simplify the numerators and denominators.], [#figure(figph[Two fractions, 3/12 and 2/12, are displayed on a white background.], alt: "Two fractions, 3/12 and 2/12, are displayed on a white background.", caption: none)], )) We do not reduce the resulting fractions. If we did, we would get back to our original fractions and lose the common denominator. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Change to equivalent fractions with the LCD: #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] and #math.equation(block: false, alt: "the fraction 5 over 6 ,")[$frac(5, 6) ,$] LCD #math.equation(block: false, alt: "equals 12")[$= 12$] #solutionbox[ #math.equation(block: true, alt: "the fraction 9 over 12 , the fraction 10 over 12")[$frac(9, 12) , frac(10, 12)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Change to equivalent fractions with the LCD: #math.equation(block: false, alt: "minus the fraction 7 over 12")[$− frac(7, 12)$] and #math.equation(block: false, alt: "the fraction 11 over 15 ,")[$frac(11, 15) ,$] LCD #math.equation(block: false, alt: "equals 60")[$= 60$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 35 over 60 , the fraction 44 over 60")[$− frac(35, 60) , frac(44, 60)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Convert two fractions to equivalent fractions with their LCD as the common denominator.] + Find the LCD. + For each fraction, determine the number needed to multiply the denominator to get the LCD. + Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2. + Simplify the numerator and denominator. ] #examplebox("Example 4")[][ Convert #math.equation(block: false, alt: "the fraction 8 over 15")[$frac(8, 15)$] and #math.equation(block: false, alt: "the fraction 11 over 24")[$frac(11, 24)$] to equivalent fractions with denominator #math.equation(block: false, alt: "120 ,")[$120 ,$] their LCD. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [The LCD is 120. We will start at Step 2.]), [Find the number that must multiply 15 to get 120.], [#figure(figph[A mathematical equation showing 15 multiplied by 8 equals 120. The number '8' is highlighted in red, indicating it might be a specific element or variable in a larger problem.], alt: "A mathematical equation showing 15 multiplied by 8 equals 120. The number '8' is highlighted in red, indicating it might be a specific element or variable in a larger problem.", caption: none)], [Find the number that must multiply 24 to get 120.], [#figure(figph[The mathematical equation '24 \* 5 = 120' is displayed on a white background, with the number '5' highlighted in red.], alt: "The mathematical equation '24 * 5 = 120' is displayed on a white background, with the number '5' highlighted in red.", caption: none)], [Use the Equivalent Fractions Property.], [#figure(figph[Fractions illustrating that multiplying both the numerator and denominator by the same non-zero number (highlighted in red) results in an equivalent fraction.], alt: "Fractions illustrating that multiplying both the numerator and denominator by the same non-zero number (highlighted in red) results in an equivalent fraction.", caption: none)], [Simplify the numerators and denominators.], [#figure(figph[Two fractions, 64 over 120 and 55 over 120, are displayed on a white background.], alt: "Two fractions, 64 over 120 and 55 over 120, are displayed on a white background.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Change to equivalent fractions with the LCD: #math.equation(block: false, alt: "the fraction 13 over 24")[$frac(13, 24)$] and #math.equation(block: false, alt: "the fraction 17 over 32 ,")[$frac(17, 32) ,$] LCD #math.equation(block: false, alt: "96")[$96$] #solutionbox[ #math.equation(block: true, alt: "the fraction 52 over 96 , the fraction 51 over 96")[$frac(52, 96) , frac(51, 96)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Change to equivalent fractions with the LCD: #math.equation(block: false, alt: "the fraction 9 over 28")[$frac(9, 28)$] and #math.equation(block: false, alt: "the fraction 27 over 32 ,")[$frac(27, 32) ,$] LCD #math.equation(block: false, alt: "224")[$224$] #solutionbox[ #math.equation(block: true, alt: "the fraction 72 over 224 , the fraction 189 over 224")[$frac(72, 224) , frac(189, 224)$] ] ] === Add and Subtract Fractions with Different Denominators Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Add or subtract fractions with different denominators.] + Find the LCD. + Convert each fraction to an equivalent form with the LCD as the denominator. + Add or subtract the fractions. + Write the result in simplified form. ] #examplebox("Example 5")[][ Add: #math.equation(block: false, alt: "the fraction 1 over 2 plus the fraction 1 over 3 .")[$frac(1, 2) + frac(1, 3) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 1 over 2 plus the fraction 1 over 3")[$frac(1, 2) + frac(1, 3)$]]), [Find the LCD of 2, 3. #linebreak() #figure(figph[Calculating the Least Common Denominator (LCD) for 2 and 3, showing the product 2x3 equals 6.], alt: "Calculating the Least Common Denominator (LCD) for 2 and 3, showing the product 2x3 equals 6.", caption: none)], [], [Change into equivalent fractions with the LCD 6.], [#figure(figph[A math problem showing the addition of 1/2 and 1/3 by finding a common denominator, represented as (1\*3)/(2\*3) + (1\*2)/(3\*2).], alt: "A math problem showing the addition of 1/2 and 1/3 by finding a common denominator, represented as (1*3)/(2*3) + (1*2)/(3*2).", caption: none)], [Simplify the numerators and denominators.], [#math.equation(block: false, alt: "the fraction 3 over 6 plus the fraction 2 over 6")[$frac(3, 6) + frac(2, 6)$]], [Add.], [#math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$]], )) Remember, always check to see if the answer can be simplified. Since #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "6")[$6$] have no common factors, the fraction #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$] cannot be reduced. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 1 over 4 plus the fraction 1 over 3 .")[$frac(1, 4) + frac(1, 3) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 12")[$frac(7, 12)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "the fraction 1 over 2 plus the fraction 1 over 5 .")[$frac(1, 2) + frac(1, 5) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 10")[$frac(7, 10)$] ] ] #examplebox("Example 6")[][ Subtract: #math.equation(block: false, alt: "the fraction 1 over 2 minus open parenthesis minus the fraction 1 over 4 close parenthesis .")[$frac(1, 2) − ( − frac(1, 4) ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 1 over 2 minus open parenthesis minus the fraction 1 over 4 close parenthesis")[$frac(1, 2) − ( − frac(1, 4) )$]]), [Find the LCD of 2 and 4. #linebreak() #figure(figph[The image demonstrates finding the Least Common Denominator (LCD) for 2 and 4, detailing their prime factorizations (2 and 2\*2) and concluding that the LCD is 4.], alt: "The image demonstrates finding the Least Common Denominator (LCD) for 2 and 4, detailing their prime factorizations (2 and 2*2) and concluding that the LCD is 4.", caption: none)], [], [Rewrite as equivalent fractions using the LCD 4.], [#figure(figph[A mathematical expression featuring a fraction (1 times 2, over 2 times 2, with the '2's highlighted in red) subtracted by a negative fraction (-1/4).], alt: "A mathematical expression featuring a fraction (1 times 2, over 2 times 2, with the '2's highlighted in red) subtracted by a negative fraction (-1/4).", caption: none)], [Simplify the first fraction.], [#math.equation(block: false, alt: "the fraction 2 over 4 minus open parenthesis minus the fraction 1 over 4 close parenthesis")[$frac(2, 4) − ( − frac(1, 4) )$]], [Subtract.], [#math.equation(block: false, alt: "the fraction 2 minus open parenthesis −1 close parenthesis over 4")[$frac(2 #h(0.2em) − #h(0.2em) ( −1 ), 4)$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$]], )) One of the fractions already had the least common denominator, so we only had to convert the other fraction. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 1 over 2 minus open parenthesis minus the fraction 1 over 8 close parenthesis .")[$frac(1, 2) − ( − frac(1, 8) ) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 8")[$frac(5, 8)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 1 over 3 minus open parenthesis minus the fraction 1 over 6 close parenthesis .")[$frac(1, 3) − ( − frac(1, 6) ) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 2")[$frac(1, 2)$] ] ] #examplebox("Example 7")[][ 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(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 7 over 12 plus the fraction 5 over 18")[$frac(7, 12) + frac(5, 18)$]]), [Find the LCD of 12 and 18. #linebreak() #figure(figph[Calculating the Least Common Denominator (LCD) of 12 and 18 using prime factorization, resulting in 36.], alt: "Calculating the Least Common Denominator (LCD) of 12 and 18 using prime factorization, resulting in 36.", caption: none)], [], [Rewrite as equivalent fractions with the LCD.], [#figure(figph[An arithmetic expression showing the addition of two fractions, 7/12 and 5/18, where each fraction is multiplied by a factor (3/3 and 2/2 respectively) to achieve a common denominator for summation.], alt: "An arithmetic expression showing the addition of two fractions, 7/12 and 5/18, where each fraction is multiplied by a factor (3/3 and 2/2 respectively) to achieve a common denominator for summation.", caption: none)], [Simplify the numerators and denominators.], [#math.equation(block: false, alt: "the fraction 21 over 36 plus the fraction 10 over 36")[$frac(21, 36) + frac(10, 36)$]], [Add.], [#math.equation(block: false, alt: "the fraction 31 over 36")[$frac(31, 36)$]], )) Because #math.equation(block: false, alt: "31")[$31$] is a prime number, it has no factors in common with #math.equation(block: false, alt: "36 .")[$36 .$] The answer is simplified. ] ] #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")[$frac(103, 60)$] ] ] When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The “missing” factors of each denominator are the numbers you need. #figure(figph[The first line says 12 equals 2 times 2 times 3. There is a blank space next to the 3. The next line says 18 equals 2 times 3 times 3. There is a blank space between the 2 and the first 3. There are red lines drawn from the blank spaces. This is labeled as missing factors. There is a horizontal line. Below the line, it says LCD equals 2 times 2 times 3 times 3. Below this, it says LCD equals 36.], alt: "The first line says 12 equals 2 times 2 times 3. There is a blank space next to the 3. The next line says 18 equals 2 times 3 times 3. There is a blank space between the 2 and the first 3. There are red lines drawn from the blank spaces. This is labeled as missing factors. There is a horizontal line. Below the line, it says LCD equals 2 times 2 times 3 times 3. Below this, it says LCD equals 36.", caption: none) The LCD, #math.equation(block: false, alt: "36 ,")[$36 ,$] has #math.equation(block: false, alt: "2")[$2$] factors of #math.equation(block: false, alt: "2")[$2$] and #math.equation(block: false, alt: "2")[$2$] factors of #math.equation(block: false, alt: "3 .")[$3 .$] Twelve has two factors of #math.equation(block: false, alt: "2 ,")[$2 ,$] but only one of #math.equation(block: false, alt: "3")[$3$]—so it is ‘missing‘ one #math.equation(block: false, alt: "3 .")[$3 .$] We multiplied the numerator and denominator of #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] by #math.equation(block: false, alt: "3")[$3$] to get an equivalent fraction with denominator #math.equation(block: false, alt: "36 .")[$36 .$] Eighteen is missing one factor of #math.equation(block: false, alt: "2")[$2$]—so you multiply the numerator and denominator #math.equation(block: false, alt: "the fraction 5 over 18")[$frac(5, 18)$] by #math.equation(block: false, alt: "2")[$2$] to get an equivalent fraction with denominator #math.equation(block: false, alt: "36 .")[$36 .$] We will apply this method as we subtract the fractions in the next example. #examplebox("Example 8")[][ Subtract: #math.equation(block: false, alt: "the fraction 7 over 15 minus the fraction 19 over 24 .")[$frac(7, 15) − frac(19, 24) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 7 over 15 minus the fraction 19 over 24")[$frac(7, 15) − frac(19, 24)$]]), [Find the LCD. #linebreak() #figure(figph[Calculation of the Least Common Denominator (LCD) for 15 and 24 using prime factorization, demonstrating how 15 = 3x5 and 24 = 2x2x2x3 lead to an LCD of 2x2x2x3x5 = 120.], alt: "Calculation of the Least Common Denominator (LCD) for 15 and 24 using prime factorization, demonstrating how 15 = 3x5 and 24 = 2x2x2x3 lead to an LCD of 2x2x2x3x5 = 120.", caption: none) #linebreak() 15 is 'missing' three factors of 2 #linebreak() 24 is 'missing' a factor of 5], [], [Rewrite as equivalent fractions with the LCD.], [#figure(figph[A math problem displaying the subtraction of two fractions, (7 \* 8) / (15 \* 8) and (19 \* 5) / (24 \* 5). Common factors 8 and 5 in the numerators and denominators are highlighted in red, suggesting simplification.], alt: "A math problem displaying the subtraction of two fractions, (7 * 8) / (15 * 8) and (19 * 5) / (24 * 5). Common factors 8 and 5 in the numerators and denominators are highlighted in red, suggesting simplification.", caption: none)], [Simplify each numerator and denominator.], [#math.equation(block: false, alt: "the fraction 56 over 120 minus the fraction 95 over 120")[$frac(56, 120) − frac(95, 120)$]], [Subtract.], [#math.equation(block: false, alt: "minus the fraction 39 over 120")[$− frac(39, 120)$]], [Rewrite showing the common factor of 3.], [#math.equation(block: false, alt: "minus the fraction 13 times 3 over 40 times 3")[$− frac(13 · 3, 40 · 3)$]], [Remove the common factor to simplify.], [#math.equation(block: false, alt: "minus the fraction 13 over 40")[$− frac(13, 40)$]], )) ] ] #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 .")[$frac(13, 24) − frac(17, 32) .$] #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 .")[$frac(21, 32) − frac(9, 28) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 75 over 224")[$frac(75, 224)$] ] ] #examplebox("Example 9")[][ Add: #math.equation(block: false, alt: "minus the fraction 11 over 30 plus the fraction 23 over 42 .")[$− #h(0.2em) frac(11, 30) + frac(23, 42) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "minus the fraction 11 over 30 plus the fraction 23 over 42")[$− frac(11, 30) + frac(23, 42)$]]), [Find the LCD. #linebreak() #figure(figph[The image illustrates the calculation of the Least Common Denominator (LCD) for 30 and 42 using prime factorization, showing 30 = 2\*3\*5, 42 = 2\*3\*7, and the resulting LCD = 2\*3\*5\*7 = 210.], alt: "The image illustrates the calculation of the Least Common Denominator (LCD) for 30 and 42 using prime factorization, showing 30 = 2*3*5, 42 = 2*3*7, and the resulting LCD = 2*3*5*7 = 210.", caption: none)], [], [Rewrite as equivalent fractions with the LCD.], [#figure(figph[Math expression for adding fractions: -11/30 and 23/42. Each fraction's numerator and denominator are multiplied by a distinct factor (7 or 5, in red) to find a common denominator.], alt: "Math expression for adding fractions: -11/30 and 23/42. Each fraction's numerator and denominator are multiplied by a distinct factor (7 or 5, in red) to find a common denominator.", caption: none)], [Simplify each numerator and denominator.], [#math.equation(block: false, alt: "minus the fraction 77 over 210 plus the fraction 115 over 210")[$− frac(77, 210) + frac(115, 210)$]], [Add.], [#math.equation(block: false, alt: "the fraction 38 over 210")[$frac(38, 210)$]], [Rewrite showing the common factor of 2.], [#math.equation(block: false, alt: "the fraction 19 times 2 over 105 times 2")[$frac(19 · 2, 105 · 2)$]], [Remove the common factor to simplify.], [#math.equation(block: false, alt: "the fraction 19 over 105")[$frac(19, 105)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "minus the fraction 13 over 42 plus the fraction 17 over 35 .")[$− frac(13, 42) + frac(17, 35) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 37 over 210")[$frac(37, 210)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "minus the fraction 19 over 24 plus the fraction 17 over 32 .")[$− frac(19, 24) + frac(17, 32) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 25 over 96")[$− frac(25, 96)$] ] ] In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers. #examplebox("Example 10")[][ 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. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 3 over 5 plus the fraction x over 8")[$frac(3, 5) + frac(x, 8)$]]), [Find the LCD. #linebreak() #figure(figph[A step-by-step calculation showing how to find the Least Common Denominator (LCD) of 5 and 8. The prime factorization of each number is displayed, leading to an LCD of 2 x 2 x 2 x 5, which equals 40.], alt: "A step-by-step calculation showing how to find the Least Common Denominator (LCD) of 5 and 8. The prime factorization of each number is displayed, leading to an LCD of 2 x 2 x 2 x 5, which equals 40.", caption: none)], [], [Rewrite as equivalent fractions with the LCD.], [#figure(figph[A mathematical expression featuring two fractions being added. The first fraction is (3 times 8) divided by (5 times 8). The second fraction is (x times 5) divided by (8 times 5).], alt: "A mathematical expression featuring two fractions being added. The first fraction is (3 times 8) divided by (5 times 8). The second fraction is (x times 5) divided by (8 times 5).", caption: none)], [Simplify the numerators and denominators.], [#math.equation(block: false, alt: "the fraction 24 over 40 plus the fraction 5 x over 40")[$frac(24, 40) + frac(5 x, 40)$]], [Add.], [#math.equation(block: false, alt: "the fraction 24 plus 5 x over 40")[$frac(24 #h(0.2em) + #h(0.2em) 5 x, 40)$]], )) We cannot add #math.equation(block: false, alt: "24")[$24$] and #math.equation(block: false, alt: "5 x")[$5 x$] since they are not like terms, so we cannot simplify the expression any further. ] ] #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")[$frac(5 x + 14, 30)$] ] ] === Identify and Use Fraction Operations By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Summary of Fraction Operations] #emph[Fraction multiplication:] Multiply the numerators and multiply the denominators. #math.equation(block: true, alt: "the fraction a over b times the fraction c over d equals the fraction a c over b d")[$frac(a, b) · frac(c, d) = frac(a c, b d)$]#emph[Fraction division:] Multiply the first fraction by the reciprocal of the second. #math.equation(block: true, alt: "the fraction a over b divided by the fraction c over d equals the fraction a over b times the fraction d over c")[$frac(a, b) ÷ frac(c, d) = frac(a, b) · frac(d, c)$]#emph[Fraction addition:] Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD. #math.equation(block: true, alt: "the fraction a over c plus the fraction b over c equals the fraction a plus b over c")[$frac(a, c) + frac(b, c) = frac(a + b, c)$]#emph[Fraction subtraction:] Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD. #math.equation(block: true, alt: "the fraction a over c minus the fraction b over c equals the fraction a minus b over c")[$frac(a, c) − frac(b, c) = frac(a − b, c)$] ] #examplebox("Example 11")[][ Simplify: #linebreak() + ⓐ #math.equation(block: false, alt: "minus the fraction 1 over 4 plus the fraction 1 over 6")[$− frac(1, 4) + frac(1, 6)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 4 divided by the fraction 1 over 6")[$− frac(1, 4) ÷ frac(1, 6)$] #solutionbox[ First we ask ourselves, “What is the operation?” ⓐ The operation is addition. Do the fractions have a common denominator? No. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "minus the fraction 1 over 4 plus the fraction 1 over 6")[$− frac(1, 4) + frac(1, 6)$]]), [Find the LCD. #linebreak() #figure(figph[Calculation of the Least Common Denominator (LCD) for numbers 4 and 6, showing prime factorization to arrive at an LCD of 12.], alt: "Calculation of the Least Common Denominator (LCD) for numbers 4 and 6, showing prime factorization to arrive at an LCD of 12.", caption: none)], [], [Rewrite each fraction as an equivalent fraction with the LCD.], [#figure(figph[A mathematical expression showing the addition of two fractions: - (1 \* 3) / (4 \* 3) + (1 \* 2) / (6 \* 2). The numbers 3 and 2, used to adjust fractions to a common denominator, are highlighted in red.], alt: "A mathematical expression showing the addition of two fractions: - (1 * 3) / (4 * 3) + (1 * 2) / (6 * 2). The numbers 3 and 2, used to adjust fractions to a common denominator, are highlighted in red.", caption: none)], [Simplify the numerators and denominators.], [#math.equation(block: false, alt: "minus the fraction 3 over 12 plus the fraction 2 over 12")[$− frac(3, 12) + frac(2, 12)$]], [Add the numerators and place the sum over the common denominator.], [#math.equation(block: false, alt: "minus the fraction 1 over 12")[$− frac(1, 12)$]], [Check to see if the answer can be simplified. It cannot.], [], )) ⓑ The operation is division. We do not need a common denominator. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "minus the fraction 1 over 4 divided by the fraction 1 over 6")[$− frac(1, 4) ÷ frac(1, 6)$]]), [To divide fractions, multiply the first fraction by the reciprocal of the second.], [#math.equation(block: false, alt: "minus the fraction 1 over 4 times the fraction 6 over 1")[$− frac(1, 4) · frac(6, 1)$]], [Multiply.], [#math.equation(block: false, alt: "minus the fraction 6 over 4")[$− frac(6, 4)$]], [Simplify.], [#math.equation(block: false, alt: "minus the fraction 3 over 2")[$− frac(3, 2)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify each expression: + ⓐ #math.equation(block: false, alt: "minus the fraction 3 over 4 minus the fraction 1 over 6")[$− frac(3, 4) − frac(1, 6)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 3 over 4 times the fraction 1 over 6")[$− frac(3, 4) · frac(1, 6)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "minus the fraction 11 over 12")[$− frac(11, 12)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 8")[$− frac(1, 8)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify each expression: + ⓐ #math.equation(block: false, alt: "the fraction 5 over 6 divided by open parenthesis minus the fraction 1 over 4 close parenthesis")[$frac(5, 6) ÷ ( − frac(1, 4) )$] + ⓑ #math.equation(block: false, alt: "the fraction 5 over 6 minus open parenthesis minus the fraction 1 over 4 close parenthesis")[$frac(5, 6) − ( − frac(1, 4) )$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "minus the fraction 10 over 3")[$− frac(10, 3)$] + ⓑ #math.equation(block: false, alt: "the fraction 13 over 12")[$frac(13, 12)$] ] ] #examplebox("Example 12")[][ Simplify: #linebreak() + ⓐ #math.equation(block: false, alt: "the fraction 5 x over 6 minus the fraction 3 over 10")[$frac(5 x, 6) − frac(3, 10)$] + ⓑ #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[ ⓐ The operation is subtraction. The fractions do not have a common denominator. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 5 x over 6 minus the fraction 3 over 10")[$frac(5 x, 6) − frac(3, 10)$]]), [Rewrite each fraction as an equivalent fraction with the LCD, 30.], [#math.equation(block: false, alt: "the fraction 5 x times 5 over 6 times 5 minus the fraction 3 times 3 over 10 times 3")[$frac(5 x · 5, 6 · 5) − frac(3 · 3, 10 · 3)$]], [], [#math.equation(block: false, alt: "the fraction 25 x over 30 minus the fraction 9 over 30")[$frac(25 x, 30) − frac(9, 30)$]], [Subtract the numerators and place the difference over the common denominator.], [#math.equation(block: false, alt: "the fraction 25 x minus 9 over 30")[$frac(25 x − 9, 30)$]], )) ⓑ The operation is multiplication; no need for a common denominator. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 5 x over 6 times the fraction 3 over 10")[$frac(5 x, 6) · frac(3, 10)$]]), [To multiply fractions, multiply the numerators and multiply the denominators.], [#math.equation(block: false, alt: "the fraction 5 x times 3 over 6 times 10")[$frac(5 x · 3, 6 · 10)$]], [Rewrite, showing common factors.], [#math.equation(block: false, alt: "the fraction 5 times x times 3 over 2 times 3 times 2 times 5")[$frac(cancel(5) · x · cancel(3), 2 · cancel(3) · 2 · cancel(5))$]], [Remove common factors to simplify.], [#math.equation(block: false, alt: "the fraction x over 4")[$frac(x, 4)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #linebreak() + ⓐ #math.equation(block: false, alt: "the fraction open parenthesis 27 a minus 32 close parenthesis over 36")[$frac(( 27 a − 32 ), 36)$] + ⓑ #math.equation(block: false, alt: "the fraction 2 a over 3")[$frac(2 a, 3)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction open parenthesis 27 a minus 32 close parenthesis 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: #linebreak() + ⓐ #math.equation(block: false, alt: "the fraction open parenthesis 24 k plus 25 close parenthesis over 30")[$frac(( 24 k + 25 ), 30)$] + ⓑ #math.equation(block: false, alt: "the fraction 24 k over 5")[$frac(24 k, 5)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction open parenthesis 24 k plus 25 close parenthesis over 30")[$frac(( 24 k + 25 ), 30)$] + ⓑ #math.equation(block: false, alt: "the fraction 24 k over 5")[$frac(24 k, 5)$] ] ] === Use the Order of Operations to Simplify Complex Fractions In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example, #math.equation(block: true, alt: "the fraction the fraction 3 over 4 over the fraction 5 over 8 equals the fraction 3 over 4 divided by the fraction 5 over 8")[$frac(#h(0.2em) frac(3, 4) #h(0.2em), #h(0.2em) frac(5, 8) #h(0.2em)) = frac(3, 4) ÷ frac(5, 8)$]Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator. #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 13")[][ 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(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction open parenthesis the fraction 1 over 2 close parenthesis squared over 4 plus 3 squared")[$frac(attach(( frac(1, 2) ), t: 2), 4 + 3^(2))$]]), [Simplify the numerator.], [#math.equation(block: false, alt: "the fraction the fraction 1 over 4 over 4 plus 3 squared")[$frac(frac(1, 4), 4 + 3^(2))$]], [Simplify the term with the exponent in the denominator.], [#math.equation(block: false, alt: "the fraction the fraction 1 over 4 over 4 plus 9")[$frac(frac(1, 4), 4 + 9)$]], [Add the terms in the denominator.], [#math.equation(block: false, alt: "the fraction the fraction 1 over 4 over 13")[$frac(frac(1, 4), 13)$]], [Divide the numerator by the denominator.], [#math.equation(block: false, alt: "the fraction 1 over 4 divided by 13")[$frac(1, 4) ÷ 13$]], [Rewrite as multiplication by the reciprocal.], [#math.equation(block: false, alt: "the fraction 1 over 4 times the fraction 1 over 13")[$frac(1, 4) · frac(1, 13)$]], [Multiply.], [#math.equation(block: false, alt: "the fraction 1 over 52")[$frac(1, 52)$]], )) ] ] #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[ 272 ] ] #examplebox("Example 14")[][ 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 .")[$frac(frac(1, 2) + frac(2, 3), frac(3, 4) − frac(1, 6)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#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")[$frac(frac(1, 2) + frac(2, 3), frac(3, 4) − frac(1, 6))$]]), [Rewrite numerator with the LCD of 6 and denominator with LCD of 12.], [#math.equation(block: false, alt: "the fraction the fraction 3 over 6 plus the fraction 4 over 6 over the fraction 9 over 12 minus the fraction 2 over 12")[$frac(frac(3, 6) + frac(4, 6), frac(9, 12) − frac(2, 12))$]], [Add in the numerator. Subtract in the denominator.], [#math.equation(block: false, alt: "the fraction the fraction 7 over 6 over the fraction 7 over 12")[$frac(#h(0.2em) frac(7, 6) #h(0.2em), #h(0.2em) frac(7, 12) #h(0.2em))$]], [Divide the numerator by the denominator.], [#math.equation(block: false, alt: "the fraction 7 over 6 divided by the fraction 7 over 12")[$frac(7, 6) ÷ frac(7, 12)$]], [Rewrite as multiplication by the reciprocal.], [#math.equation(block: false, alt: "the fraction 7 over 6 times the fraction 12 over 7")[$frac(7, 6) · frac(12, 7)$]], [Rewrite, showing common factors.], [#math.equation(block: false, alt: "the fraction 7 times 6 times 2 over 6 times 7 times 1")[$frac(cancel(7) · cancel(6) · 2, cancel(6) · cancel(7) · 1)$]], [Simplify.], [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")[$frac(frac(1, 3) + frac(1, 2), frac(3, 4) − frac(1, 3))$]. #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")[$frac(frac(2, 3) − frac(1, 2), frac(1, 4) + frac(1, 3))$]. #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 also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify. #examplebox("Example 15")[][ Evaluate #math.equation(block: false, alt: "x plus the fraction 1 over 3")[$x + frac(1, 3)$] when #linebreak() + ⓐ #math.equation(block: false, alt: "x equals minus the fraction 1 over 3")[$x = − frac(1, 3)$] + ⓑ #math.equation(block: false, alt: "x equals minus the fraction 3 over 4 .")[$x = − frac(3, 4) .$] #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 ,")[$x = − frac(1, 3) ,$] substitute #math.equation(block: false, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$] for #math.equation(block: false, alt: "x")[$x$] in the expression. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "x plus the fraction 1 over 3")[$x + frac(1, 3)$]]), [#figure(figph[The image displays the instruction 'Substitute -1/3 for x.' in a dark teal font, with the fraction '-1/3' highlighted in red, against a white background.], alt: "The image displays the instruction 'Substitute -1/3 for x.' in a dark teal font, with the fraction '-1/3' highlighted in red, against a white background.", caption: none)], [#figure(figph[The mathematical expression for negative one-third plus one-third, illustrating additive inverses that sum to zero.], alt: "The mathematical expression for negative one-third plus one-third, illustrating additive inverses that sum to zero.", caption: none)], [Simplify.], [#math.equation(block: false, alt: "0")[$0$]], )) ⓑ 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 ,")[$x = − frac(3, 4) ,$] we substitute #math.equation(block: false, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] for #math.equation(block: false, alt: "x")[$x$] in the expression. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "x plus the fraction 1 over 3")[$x + frac(1, 3)$]]), [#figure(figph[The text 'Substitute -3/4 for x.' is shown in a math problem, indicating an instruction to replace the variable 'x' with the fractional value -3/4.], alt: "The text 'Substitute -3/4 for x.' is shown in a math problem, indicating an instruction to replace the variable 'x' with the fractional value -3/4.", caption: none)], [#figure(figph[A mathematical expression showing the sum of negative three-fourths and one-third.], alt: "A mathematical expression showing the sum of negative three-fourths and one-third.", caption: none)], [Rewrite as equivalent fractions with the LCD, 12.], [#math.equation(block: false, alt: "minus the fraction 3 times 3 over 4 times 3 plus the fraction 1 times 4 over 3 times 4")[$− frac(3 · 3, 4 · 3) + frac(1 · 4, 3 · 4)$]], [Simplify the numerators and denominators.], [#math.equation(block: false, alt: "minus the fraction 9 over 12 plus the fraction 4 over 12")[$− frac(9, 12) + frac(4, 12)$]], [Add.], [#math.equation(block: false, alt: "minus the fraction 5 over 12")[$− frac(5, 12)$]], )) ] ] #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 #linebreak() + ⓐ #math.equation(block: false, alt: "x equals minus the fraction 7 over 4")[$x = − frac(7, 4)$] + ⓑ #math.equation(block: false, alt: "x equals minus the fraction 5 over 4")[$x = − frac(5, 4)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "−1")[$−1$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] ] ] #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 #linebreak() + ⓐ #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")[$y = − frac(3, 4)$] #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")[$− frac(1, 4)$] ] ] #examplebox("Example 16")[][ Evaluate #math.equation(block: false, alt: "y minus the fraction 5 over 6")[$y − frac(5, 6)$] when #math.equation(block: false, alt: "y equals minus the fraction 2 over 3 .")[$y = − frac(2, 3) .$] #solutionbox[ We substitute #math.equation(block: false, alt: "minus the fraction 2 over 3")[$− frac(2, 3)$] for #math.equation(block: false, alt: "y")[$y$] in the expression. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "y minus the fraction 5 over 6")[$y − frac(5, 6)$]]), [#figure(figph[The text 'Substitute -2/3 for y.' is displayed, instructing to replace the variable y with the fraction -2/3.], alt: "The text 'Substitute -2/3 for y.' is displayed, instructing to replace the variable y with the fraction -2/3.", caption: none)], [#figure(figph[A mathematical expression showing the subtraction of two fractions: negative two-thirds minus five-sixths. The first fraction's numerator and denominator are colored red.], alt: "A mathematical expression showing the subtraction of two fractions: negative two-thirds minus five-sixths. The first fraction's numerator and denominator are colored red.", caption: none)], [Rewrite as equivalent fractions with the LCD, 6.], [#math.equation(block: false, alt: "minus the fraction 4 over 6 minus the fraction 5 over 6")[$− frac(4, 6) − frac(5, 6)$]], [Subtract.], [#math.equation(block: false, alt: "minus the fraction 9 over 6")[$− frac(9, 6)$]], [Simplify.], [#math.equation(block: false, alt: "minus the fraction 3 over 2")[$− frac(3, 2)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: false, alt: "y minus the fraction 1 over 2")[$y − frac(1, 2)$] when #math.equation(block: false, alt: "y equals minus the fraction 1 over 4 .")[$y = − frac(1, 4) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: false, alt: "x minus the fraction 3 over 8")[$x − frac(3, 8)$] when #math.equation(block: false, alt: "x equals minus the fraction 5 over 2 .")[$x = − frac(5, 2) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 23 over 8")[$− frac(23, 8)$] ] ] #examplebox("Example 17")[][ 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 .")[$y = − frac(2, 3) .$] #solutionbox[ Substitute the values into the expression. In #math.equation(block: false, alt: "2 x squared y ,")[$2 x^(2) y ,$] the exponent applies only to #math.equation(block: false, alt: "x .")[$x .$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The mathematical expression 2x^2y is shown in black text on a white background.], alt: "The mathematical expression 2x^2y is shown in black text on a white background.", caption: none)]), [#figure(figph[Substitute 1/4 for x and -2/3 for y.], alt: "Substitute 1/4 for x and -2/3 for y.", caption: none)], [#figure(figph[The image displays the mathematical expression 2(1/4)^2(-2/3), showing a multiplication of a whole number, a fraction squared, and a negative fraction. The numbers 1/4 are red, and -2/3 are blue.], alt: "The image displays the mathematical expression 2(1/4)^2(-2/3), showing a multiplication of a whole number, a fraction squared, and a negative fraction. The numbers 1/4 are red, and -2/3 are blue.", caption: none)], [Simplify exponents first.], [#figure(figph[A mathematical expression showing the product of three terms: 2, the fraction 1/16, and the negative fraction -2/3. The terms are enclosed in parentheses, indicating multiplication.], alt: "A mathematical expression showing the product of three terms: 2, the fraction 1/16, and the negative fraction -2/3. The terms are enclosed in parentheses, indicating multiplication.", caption: none)], [Multiply. The product will be negative.], [#figure(figph[A mathematical expression showing the multiplication of three fractions: negative two over one, one over sixteen, and two over three.], alt: "A mathematical expression showing the multiplication of three fractions: negative two over one, one over sixteen, and two over three.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression showing the fraction -4/48, presented in black text on a white background. The negative sign precedes the fraction bar, with '4' as the numerator and '48' as the denominator.], alt: "A mathematical expression showing the fraction -4/48, presented in black text on a white background. The negative sign precedes the fraction bar, with '4' as the numerator and '48' as the denominator.", caption: none)], [Remove the common factors.], [#figure(figph[A mathematical expression showing a fraction with a negative sign, where the common factor '4' in the numerator and denominator has been crossed out as part of simplification, resulting in -(1\*4)/(4\*12).], alt: "A mathematical expression showing a fraction with a negative sign, where the common factor '4' in the numerator and denominator has been crossed out as part of simplification, resulting in -(1*4)/(4*12).", caption: none)], [Simplify.], [#figure(figph[A mathematical expression displaying the fraction minus 1 over 12, written as -1/12.], alt: "A mathematical expression displaying the fraction minus 1 over 12, written as -1/12.", caption: none)], )) ] ] #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")[$a = − frac(2, 3)$] and #math.equation(block: false, alt: "b equals minus the fraction 1 over 2 .")[$b = − frac(1, 2) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] ] ] #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")[$c = − frac(1, 2)$] and #math.equation(block: false, alt: "d equals minus the fraction 4 over 3 .")[$d = − frac(4, 3) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 3")[$frac(2, 3)$] ] ] #examplebox("Example 18")[][ 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 ,")[$p = −4 , q = −2 ,$] and #math.equation(block: false, alt: "r equals 8 .")[$r = 8 .$] #solutionbox[ We substitute the values into the expression and simplify. #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 displays the text 'Substitute -4 for p, -2 for q and 8 for r.' against a white background.], alt: "The image displays the text 'Substitute -4 for p, -2 for q and 8 for r.' against a white background.", caption: none)], [#figure(figph[A mathematical expression showing the sum of negative four and negative two, all divided by eight.], alt: "A mathematical expression showing the sum of negative four and negative two, all divided by eight.", caption: none)], [Add in the numerator first.], [#math.equation(block: false, alt: "minus the fraction 6 over 8")[$− frac(6, 8)$]], [Simplify.], [#math.equation(block: false, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$]], )) ] ] #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 ,")[$a = −8 , b = −7 ,$] and #math.equation(block: false, alt: "c equals 6 .")[$#h(0.2em) c = 6 .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 2")[$− frac(5, 2)$] ] ] #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 ,")[$x = 9 , y = −18 ,$] and #math.equation(block: false, alt: "z equals −6 .")[$z = −6 .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 2")[$frac(3, 2)$] ] ] === Key Concepts - #emph[Find the least common denominator (LCD) of two fractions.]+ Factor each denominator into its primes. + List the primes, matching primes in columns when possible. + Bring down the columns. + Multiply the factors. The product is the LCM of the denominators. + The LCM of the denominators is the LCD of the fractions. - #emph[Equivalent Fractions Property]- If #math.equation(block: false, alt: "a , b")[$a , b$], and #math.equation(block: false, alt: "c")[$c$] are whole numbers where #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$] then #linebreak() #math.equation(block: false, alt: "the fraction a over b equals the fraction a times c over b times c")[$frac(a, b) = frac(a ⋅ c, b ⋅ c)$] and #math.equation(block: false, alt: "the fraction a times c over b times c equals the fraction a over b")[$frac(a ⋅ c, b ⋅ c) = frac(a, b)$] - #emph[Convert two fractions to equivalent fractions with their LCD as the common denominator.]+ Find the LCD. + For each fraction, determine the number needed to multiply the denominator to get the LCD. + Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2. + Simplify the numerator and denominator. - #emph[Add or subtract fractions with different denominators.]+ Find the LCD. + Convert each fraction to an equivalent form with the LCD as the denominator. + Add or subtract the fractions. + Write the result in simplified form. - #emph[Summary of Fraction Operations]- #emph[Fraction multiplication:] Multiply the numerators and multiply the denominators. #math.equation(block: true, alt: "the fraction a over b times the fraction c over d equals the fraction a c over b d")[$frac(a, b) ⋅ frac(c, d) = frac(a c, b d)$] - #emph[Fraction division:] Multiply the first fraction by the reciprocal of the second.#math.equation(block: true, alt: "the fraction a over b plus the fraction c over d equals the fraction a over b times the fraction d over c")[$frac(a, b) + frac(c, d) = frac(a, b) ⋅ frac(d, c)$] - #emph[Fraction addition:] Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.#math.equation(block: true, alt: "the fraction a over c plus the fraction b over c equals the fraction a plus b over c")[$frac(a, c) + frac(b, c) = frac(a + b, c)$] - #emph[Fraction subtraction:] Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.#math.equation(block: true, alt: "the fraction a over c minus the fraction b over c equals the fraction a minus b over c")[$frac(a, c) − frac(b, c) = frac(a − b, c)$] - #emph[Simplify complex fractions.]+ Simplify the numerator. + Simplify the denominator. + Divide the numerator by the denominator. + Simplify if possible. ==== Practice Makes Perfect #strong[Find the Least Common Denominator (LCD)] In the following exercises, find the least common denominator (LCD) for each set of fractions. #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] and #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] and #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$] #solutionbox[ 20 ] #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] and #math.equation(block: false, alt: "the fraction 5 over 8")[$frac(5, 8)$] #math.equation(block: false, alt: "the fraction 9 over 16")[$frac(9, 16)$] and #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] #solutionbox[ 48 ] #math.equation(block: false, alt: "the fraction 13 over 30")[$frac(13, 30)$] and #math.equation(block: false, alt: "the fraction 25 over 42")[$frac(25, 42)$] #math.equation(block: false, alt: "the fraction 23 over 30")[$frac(23, 30)$] and #math.equation(block: false, alt: "the fraction 5 over 48")[$frac(5, 48)$] #solutionbox[ 240 ] #math.equation(block: false, alt: "the fraction 21 over 35")[$frac(21, 35)$] and #math.equation(block: false, alt: "the fraction 39 over 56")[$frac(39, 56)$] #math.equation(block: false, alt: "the fraction 18 over 35")[$frac(18, 35)$] and #math.equation(block: false, alt: "the fraction 33 over 49")[$frac(33, 49)$] #solutionbox[ 245 ] #math.equation(block: false, alt: "the fraction 2 over 3 , the fraction 1 over 6 ,")[$frac(2, 3) , frac(1, 6) ,$] and #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] #math.equation(block: false, alt: "the fraction 2 over 3 , the fraction 1 over 4 ,")[$frac(2, 3) , frac(1, 4) ,$] and #math.equation(block: false, alt: "the fraction 3 over 5")[$frac(3, 5)$] #solutionbox[ 60 ] #strong[Convert Fractions to Equivalent Fractions with the LCD] In the following exercises, convert to equivalent fractions using the LCD. #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] and #math.equation(block: false, alt: "the fraction 1 over 4 ,")[$frac(1, 4) ,$] LCD #math.equation(block: false, alt: "equals 12")[$= 12$] #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] and #math.equation(block: false, alt: "the fraction 1 over 5 ,")[$frac(1, 5) ,$] LCD #math.equation(block: false, alt: "equals 20")[$= 20$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 20 , the fraction 4 over 20")[$frac(5, 20) , frac(4, 20)$] ] #math.equation(block: false, alt: "the fraction 5 over 12")[$frac(5, 12)$] and #math.equation(block: false, alt: "the fraction 7 over 8 ,")[$frac(7, 8) ,$] LCD #math.equation(block: false, alt: "equals 24")[$= 24$] #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] and #math.equation(block: false, alt: "the fraction 5 over 8 ,")[$frac(5, 8) ,$] LCD #math.equation(block: false, alt: "equals 24")[$= 24$] #solutionbox[ #math.equation(block: true, alt: "the fraction 14 over 24 , the fraction 15 over 24")[$frac(14, 24) , frac(15, 24)$] ] #math.equation(block: false, alt: "the fraction 13 over 16")[$frac(13, 16)$] and #math.equation(block: false, alt: "minus the fraction 11 over 12 ,")[$− frac(11, 12) ,$] LCD #math.equation(block: false, alt: "equals 48")[$= 48$] #math.equation(block: false, alt: "the fraction 11 over 16")[$frac(11, 16)$] and #math.equation(block: false, alt: "minus the fraction 5 over 12 ,")[$− frac(5, 12) ,$] LCD #math.equation(block: false, alt: "equals 48")[$= 48$] #solutionbox[ #math.equation(block: true, alt: "the fraction 33 over 48 , minus the fraction 20 over 48")[$frac(33, 48) , − frac(20, 48)$] ] #math.equation(block: false, alt: "the fraction 1 over 3 , the fraction 5 over 6 ,")[$frac(1, 3) , frac(5, 6) ,$] and #math.equation(block: false, alt: "the fraction 3 over 4 ,")[$frac(3, 4) ,$] LCD #math.equation(block: false, alt: "equals 12")[$= 12$] #math.equation(block: false, alt: "the fraction 1 over 3 , the fraction 3 over 4 ,")[$frac(1, 3) , frac(3, 4) ,$] and #math.equation(block: false, alt: "the fraction 3 over 5 ,")[$frac(3, 5) ,$] LCD #math.equation(block: false, alt: "equals 60")[$= 60$] #solutionbox[ #math.equation(block: true, alt: "the fraction 20 over 60 , the fraction 45 over 60 , the fraction 36 over 60")[$frac(20, 60) , frac(45, 60) , frac(36, 60)$] ] #strong[Add and Subtract Fractions with Different Denominators] In the following exercises, add or subtract. Write the result in simplified form. #math.equation(block: true, alt: "the fraction 1 over 3 plus the fraction 1 over 5")[$frac(1, 3) + frac(1, 5)$] #math.equation(block: true, alt: "the fraction 1 over 4 plus the fraction 1 over 5")[$frac(1, 4) + frac(1, 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 9 over 20")[$frac(9, 20)$] ] #math.equation(block: true, alt: "the fraction 1 over 2 plus the fraction 1 over 7")[$frac(1, 2) + frac(1, 7)$] #math.equation(block: true, alt: "the fraction 1 over 3 plus the fraction 1 over 8")[$frac(1, 3) + frac(1, 8)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 11 over 24")[$frac(11, 24)$] ] #math.equation(block: true, alt: "the fraction 1 over 3 minus open parenthesis minus the fraction 1 over 9 close parenthesis")[$frac(1, 3) − ( − frac(1, 9) )$] #math.equation(block: true, alt: "the fraction 1 over 4 minus open parenthesis minus the fraction 1 over 8 close parenthesis")[$frac(1, 4) − ( − frac(1, 8) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 8")[$frac(3, 8)$] ] #math.equation(block: true, alt: "the fraction 1 over 5 minus open parenthesis minus the fraction 1 over 10 close parenthesis")[$frac(1, 5) − ( − frac(1, 10) )$] #math.equation(block: true, alt: "the fraction 1 over 2 minus open parenthesis minus the fraction 1 over 6 close parenthesis")[$frac(1, 2) − ( − frac(1, 6) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 3")[$frac(2, 3)$] ] #math.equation(block: true, alt: "the fraction 2 over 3 plus the fraction 3 over 4")[$frac(2, 3) + frac(3, 4)$] #math.equation(block: true, alt: "the fraction 3 over 4 plus the fraction 2 over 5")[$frac(3, 4) + frac(2, 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 23 over 20")[$frac(23, 20)$] ] #math.equation(block: true, alt: "the fraction 7 over 12 plus the fraction 5 over 8")[$frac(7, 12) + frac(5, 8)$] #math.equation(block: true, alt: "the fraction 5 over 12 plus the fraction 3 over 8")[$frac(5, 12) + frac(3, 8)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 19 over 24")[$frac(19, 24)$] ] #math.equation(block: true, alt: "the fraction 7 over 12 minus the fraction 9 over 16")[$frac(7, 12) − frac(9, 16)$] #math.equation(block: true, alt: "the fraction 7 over 16 minus the fraction 5 over 12")[$frac(7, 16) − frac(5, 12)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 48")[$frac(1, 48)$] ] #math.equation(block: true, alt: "the fraction 11 over 12 minus the fraction 3 over 8")[$frac(11, 12) − frac(3, 8)$] #math.equation(block: true, alt: "the fraction 5 over 8 minus the fraction 7 over 12")[$frac(5, 8) − frac(7, 12)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 24")[$frac(1, 24)$] ] #math.equation(block: true, alt: "the fraction 2 over 3 minus the fraction 3 over 8")[$frac(2, 3) − frac(3, 8)$] #math.equation(block: true, alt: "the fraction 5 over 6 minus the fraction 3 over 4")[$frac(5, 6) − frac(3, 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 12")[$frac(1, 12)$] ] #math.equation(block: true, alt: "minus the fraction 11 over 30 plus the fraction 27 over 40")[$− frac(11, 30) + frac(27, 40)$] #math.equation(block: true, alt: "minus the fraction 9 over 20 plus the fraction 17 over 30")[$− frac(9, 20) + frac(17, 30)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 60")[$frac(7, 60)$] ] #math.equation(block: true, alt: "minus the fraction 13 over 30 plus the fraction 25 over 42")[$− frac(13, 30) + frac(25, 42)$] #math.equation(block: true, alt: "minus the fraction 23 over 30 plus the fraction 5 over 48")[$− frac(23, 30) + frac(5, 48)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 53 over 80")[$− frac(53, 80)$] ] #math.equation(block: true, alt: "minus the fraction 39 over 56 minus the fraction 22 over 35")[$− frac(39, 56) − frac(22, 35)$] #math.equation(block: true, alt: "minus the fraction 33 over 49 minus the fraction 18 over 35")[$− frac(33, 49) − frac(18, 35)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 291 over 245")[$− frac(291, 245)$] ] #math.equation(block: true, alt: "minus the fraction 2 over 3 minus open parenthesis minus the fraction 3 over 4 close parenthesis")[$− frac(2, 3) − ( − frac(3, 4) )$] #math.equation(block: true, alt: "minus the fraction 3 over 4 minus open parenthesis minus the fraction 4 over 5 close parenthesis")[$− frac(3, 4) − ( − frac(4, 5) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 20")[$frac(1, 20)$] ] #math.equation(block: true, alt: "minus the fraction 9 over 16 minus open parenthesis minus the fraction 4 over 5 close parenthesis")[$− frac(9, 16) − ( − frac(4, 5) )$] #math.equation(block: true, alt: "minus the fraction 7 over 20 minus open parenthesis minus the fraction 5 over 8 close parenthesis")[$− frac(7, 20) − ( − frac(5, 8) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 11 over 40")[$frac(11, 40)$] ] #math.equation(block: true, alt: "1 plus the fraction 7 over 8")[$1 + frac(7, 8)$] #math.equation(block: true, alt: "1 plus the fraction 5 over 6")[$1 + frac(5, 6)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 11 over 6")[$frac(11, 6)$] ] #math.equation(block: true, alt: "1 minus the fraction 5 over 9")[$1 − frac(5, 9)$] #math.equation(block: true, alt: "1 minus the fraction 3 over 10")[$1 − frac(3, 10)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 10")[$frac(7, 10)$] ] #math.equation(block: true, alt: "the fraction x over 3 plus the fraction 1 over 4")[$frac(x, 3) + frac(1, 4)$] #math.equation(block: true, alt: "the fraction y over 2 plus the fraction 2 over 3")[$frac(y, 2) + frac(2, 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 y plus 4 over 6")[$frac(3 y + 4, 6)$] ] #math.equation(block: true, alt: "the fraction y over 4 minus the fraction 3 over 5")[$frac(y, 4) − frac(3, 5)$] #math.equation(block: true, alt: "the fraction x over 5 minus the fraction 1 over 4")[$frac(x, 5) − frac(1, 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 x minus 5 over 20")[$frac(4 x − 5, 20)$] ] #strong[Identify and Use Fraction Operations] In the following exercises, perform the indicated operations. Write your answers in simplified form. + ⓐ #math.equation(block: false, alt: "the fraction 3 over 4 plus the fraction 1 over 6")[$frac(3, 4) + frac(1, 6)$] + ⓑ #math.equation(block: false, alt: "the fraction 3 over 4 divided by the fraction 1 over 6")[$frac(3, 4) ÷ frac(1, 6)$] + ⓐ #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")[$frac(5, 6)$] + ⓑ #math.equation(block: false, alt: "4")[$4$] ] + ⓐ #math.equation(block: false, alt: "minus the fraction 2 over 5 minus the fraction 1 over 8")[$− frac(2, 5) − frac(1, 8)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 2 over 5 times the fraction 1 over 8")[$− frac(2, 5) · frac(1, 8)$] + ⓐ #math.equation(block: false, alt: "minus the fraction 4 over 5 minus the fraction 1 over 8")[$− frac(4, 5) − frac(1, 8)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 4 over 5 times the fraction 1 over 8")[$− frac(4, 5) · frac(1, 8)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "minus the fraction 37 over 40")[$− frac(37, 40)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 10")[$− frac(1, 10)$] ] + ⓐ #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")[$frac(5 n, 6) − frac(8, 15)$] + ⓐ #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")[$frac(3 a, 8) − frac(7, 12)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 9 a over 14")[$frac(9 a, 14)$] + ⓑ #math.equation(block: false, alt: "the fraction 9 a minus 14 over 24")[$frac(9 a − 14, 24)$] ] + ⓐ #math.equation(block: false, alt: "the fraction 9 over 10 times open parenthesis minus the fraction 11 d over 12 close parenthesis")[$frac(9, 10) · ( − frac(11 d, 12) )$] + ⓑ #math.equation(block: false, alt: "the fraction 9 over 10 plus open parenthesis minus the fraction 11 d over 12 close parenthesis")[$frac(9, 10) + ( − frac(11 d, 12) )$] + ⓐ #math.equation(block: false, alt: "the fraction 4 over 15 times open parenthesis minus the fraction 5 q over 9 close parenthesis")[$frac(4, 15) · ( − frac(5 q, 9) )$] + ⓑ #math.equation(block: false, alt: "the fraction 4 over 15 plus open parenthesis minus the fraction 5 q over 9 close parenthesis")[$frac(4, 15) + ( − frac(5 q, 9) )$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "minus the fraction 4 q over 27")[$− frac(4 q, 27)$] + ⓑ #math.equation(block: false, alt: "the fraction 12 minus 25 q over 45")[$frac(12 − 25 q, 45)$] ] #math.equation(block: true, alt: "minus the fraction 3 over 8 divided by open parenthesis minus the fraction 3 over 10 close parenthesis")[$− frac(3, 8) ÷ ( − frac(3, 10) )$] #math.equation(block: true, alt: "minus the fraction 5 over 12 divided by open parenthesis minus the fraction 5 over 9 close parenthesis")[$− frac(5, 12) ÷ ( − frac(5, 9) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] #math.equation(block: true, alt: "minus the fraction 3 over 8 plus the fraction 5 over 12")[$− frac(3, 8) + frac(5, 12)$] #math.equation(block: true, alt: "minus the fraction 1 over 8 plus the fraction 7 over 12")[$− frac(1, 8) + frac(7, 12)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 11 over 24")[$frac(11, 24)$] ] #math.equation(block: true, alt: "the fraction 5 over 6 minus the fraction 1 over 9")[$frac(5, 6) − frac(1, 9)$] #math.equation(block: true, alt: "the fraction 5 over 9 minus the fraction 1 over 6")[$frac(5, 9) − frac(1, 6)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 18")[$frac(7, 18)$] ] #math.equation(block: true, alt: "the fraction 3 over 8 times open parenthesis minus the fraction 10 over 21 close parenthesis")[$frac(3, 8) · ( − frac(10, 21) )$] #math.equation(block: true, alt: "the fraction 7 over 12 times open parenthesis minus the fraction 8 over 35 close parenthesis")[$frac(7, 12) · ( − frac(8, 35) )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 2 over 15")[$− frac(2, 15)$] ] #math.equation(block: true, alt: "minus the fraction 7 over 15 minus the fraction y over 4")[$− frac(7, 15) − frac(y, 4)$] #math.equation(block: true, alt: "minus the fraction 3 over 8 minus the fraction x over 11")[$− frac(3, 8) − frac(x, 11)$] #solutionbox[ #math.equation(block: true, alt: "the fraction −33 minus 8 x over 88")[$frac(−33 − 8 x, 88)$] ] #math.equation(block: true, alt: "the fraction 11 over 12 a times the fraction 9 a over 16")[$frac(11, 12 a) · frac(9 a, 16)$] #math.equation(block: true, alt: "the fraction 10 y over 13 times the fraction 8 over 15 y")[$frac(10 y, 13) · frac(8, 15 y)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 16 over 39")[$frac(16, 39)$] ] #strong[Use the Order of Operations to Simplify Complex Fractions] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction open parenthesis the fraction 1 over 5 close parenthesis squared over 2 plus 3 squared")[$frac(attach(( frac(1, 5) ), t: 2), 2 + 3^(2))$] #math.equation(block: true, alt: "the fraction open parenthesis the fraction 1 over 3 close parenthesis squared over 5 plus 2 squared")[$frac(attach(( frac(1, 3) ), t: 2), 5 + 2^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 81")[$frac(1, 81)$] ] #math.equation(block: true, alt: "the fraction 2 cubed plus 4 squared over open parenthesis the fraction 2 over 3 close parenthesis squared")[$frac(2^(3) + 4^(2), attach(( frac(2, 3) ), t: 2))$] #math.equation(block: true, alt: "the fraction 3 cubed minus 3 squared over open parenthesis the fraction 3 over 4 close parenthesis squared")[$frac(3^(3) − 3^(2), attach(( frac(3, 4) ), t: 2))$] #solutionbox[ 32 ] #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")[$frac(attach(( frac(3, 5) ), t: 2), attach(( frac(3, 7) ), t: 2))$] #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")[$frac(attach(( frac(3, 4) ), t: 2), attach(( frac(5, 8) ), t: 2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 36 over 25")[$frac(36, 25)$] ] #math.equation(block: true, alt: "the fraction 2 over the fraction 1 over 3 plus the fraction 1 over 5")[$frac(2, frac(1, 3) + frac(1, 5))$] #math.equation(block: true, alt: "the fraction 5 over the fraction 1 over 4 plus the fraction 1 over 3")[$frac(5, frac(1, 4) + frac(1, 3))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 60 over 7")[$frac(60, 7)$] ] #math.equation(block: true, alt: "the fraction the fraction 2 over 3 plus the fraction 1 over 2 over the fraction 3 over 4 minus the fraction 2 over 3")[$frac(frac(2, 3) + frac(1, 2), frac(3, 4) − frac(2, 3))$] #math.equation(block: true, alt: "the fraction the fraction 3 over 4 plus the fraction 1 over 2 over the fraction 5 over 6 minus the fraction 2 over 3")[$frac(frac(3, 4) + frac(1, 2), frac(5, 6) − frac(2, 3))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 15 over 2")[$frac(15, 2)$] ] #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")[$frac(frac(7, 8) − frac(2, 3), frac(1, 2) + frac(3, 8))$] #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")[$frac(frac(3, 4) − frac(3, 5), frac(1, 4) + frac(2, 5))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 13")[$frac(3, 13)$] ] #strong[Mixed Practice] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction 1 over 2 plus the fraction 2 over 3 times the fraction 5 over 12")[$frac(1, 2) + frac(2, 3) · frac(5, 12)$] #math.equation(block: true, alt: "the fraction 1 over 3 plus the fraction 2 over 5 times the fraction 3 over 4")[$frac(1, 3) + frac(2, 5) · frac(3, 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 19 over 30")[$frac(19, 30)$] ] #math.equation(block: true, alt: "1 minus the fraction 3 over 5 divided by the fraction 1 over 10")[$1 − frac(3, 5) ÷ frac(1, 10)$] #math.equation(block: true, alt: "1 minus the fraction 5 over 6 divided by the fraction 1 over 12")[$1 − frac(5, 6) ÷ frac(1, 12)$] #solutionbox[ −9 ] #math.equation(block: true, alt: "the fraction 2 over 3 plus the fraction 1 over 6 plus the fraction 3 over 4")[$frac(2, 3) + frac(1, 6) + frac(3, 4)$] #math.equation(block: true, alt: "the fraction 2 over 3 plus the fraction 1 over 4 plus the fraction 3 over 5")[$frac(2, 3) + frac(1, 4) + frac(3, 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 91 over 60")[$frac(91, 60)$] ] #math.equation(block: true, alt: "the fraction 3 over 8 minus the fraction 1 over 6 plus the fraction 3 over 4")[$frac(3, 8) − frac(1, 6) + frac(3, 4)$] #math.equation(block: true, alt: "the fraction 2 over 5 plus the fraction 5 over 8 minus the fraction 3 over 4")[$frac(2, 5) + frac(5, 8) − frac(3, 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 11 over 40")[$frac(11, 40)$] ] #math.equation(block: true, alt: "12 open parenthesis the fraction 9 over 20 minus the fraction 4 over 15 close parenthesis")[$12 ( frac(9, 20) − frac(4, 15) )$] #math.equation(block: true, alt: "8 open parenthesis the fraction 15 over 16 minus the fraction 5 over 6 close parenthesis")[$8 ( frac(15, 16) − frac(5, 6) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 6")[$frac(5, 6)$] ] #math.equation(block: true, alt: "the fraction the fraction 5 over 8 plus the fraction 1 over 6 over the fraction 19 over 24")[$frac(frac(5, 8) + frac(1, 6), frac(19, 24))$] #math.equation(block: true, alt: "the fraction the fraction 1 over 6 plus the fraction 3 over 10 over the fraction 14 over 30")[$frac(frac(1, 6) + frac(3, 10), frac(14, 30))$] #solutionbox[ 1 ] #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")[$( frac(5, 9) + frac(1, 6) ) ÷ ( frac(2, 3) − frac(1, 2) )$] #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")[$( frac(3, 4) + frac(1, 6) ) ÷ ( frac(5, 8) − frac(1, 3) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 22 over 7")[$frac(22, 7)$] ] In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. #math.equation(block: false, alt: "x plus the fraction 1 over 2")[$x + frac(1, 2)$] when #linebreak() + ⓐ #math.equation(block: false, alt: "x equals minus the fraction 1 over 8")[$x = − frac(1, 8)$] + ⓑ #math.equation(block: false, alt: "x equals minus the fraction 1 over 2")[$x = − frac(1, 2)$] #math.equation(block: false, alt: "x plus the fraction 2 over 3")[$x + frac(2, 3)$] when #linebreak() + ⓐ #math.equation(block: false, alt: "x equals minus the fraction 1 over 6")[$x = − frac(1, 6)$] + ⓑ #math.equation(block: false, alt: "x equals minus the fraction 5 over 3")[$x = − frac(5, 3)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] + ⓑ #math.equation(block: false, alt: "−1")[$−1$] ] #math.equation(block: false, alt: "x plus open parenthesis minus the fraction 5 over 6 close parenthesis")[$x + ( − frac(5, 6) )$] when #linebreak() + ⓐ #math.equation(block: false, alt: "x equals the fraction 1 over 3")[$x = frac(1, 3)$] + ⓑ #math.equation(block: false, alt: "x equals minus the fraction 1 over 6")[$x = − frac(1, 6)$] #math.equation(block: false, alt: "x plus open parenthesis minus the fraction 11 over 12 close parenthesis")[$x + ( − frac(11, 12) )$] 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")[$x = frac(3, 4)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "0")[$0$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 6")[$− frac(1, 6)$] ] #math.equation(block: false, alt: "x minus the fraction 2 over 5")[$x − frac(2, 5)$] 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")[$x = − frac(3, 5)$] #math.equation(block: false, alt: "x minus the fraction 1 over 3")[$x − frac(1, 3)$] 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")[$x = − frac(2, 3)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] + ⓑ #math.equation(block: false, alt: "−1")[$−1$] ] #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")[$w = − frac(1, 2)$] #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")[$w = − frac(1, 4)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] + ⓑ #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] ] #math.equation(block: false, alt: "4 p squared q")[$4 p^(2) q$] when #math.equation(block: false, alt: "p equals minus the fraction 1 over 2")[$p = − frac(1, 2)$] and #math.equation(block: false, alt: "q equals the fraction 5 over 9")[$q = frac(5, 9)$] #math.equation(block: false, alt: "5 m squared n")[$5 m^(2) n$] when #math.equation(block: false, alt: "m equals minus the fraction 2 over 5")[$m = − frac(2, 5)$] and #math.equation(block: false, alt: "n equals the fraction 1 over 3")[$n = frac(1, 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 15")[$frac(4, 15)$] ] #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")[$x = − frac(2, 3)$] and #math.equation(block: false, alt: "y equals minus the fraction 1 over 2")[$y = − frac(1, 2)$] #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")[$#h(0.2em) u = − frac(3, 4)$] and #math.equation(block: false, alt: "v equals minus the fraction 1 over 2")[$v = − frac(1, 2)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 9 over 16")[$− frac(9, 16)$] ] #math.equation(block: false, alt: "the fraction u plus v over w")[$frac(u + v, w)$] when #math.equation(block: false, alt: "u equals −4 , v equals −8 , w equals 2")[$u = −4 , v = −8 , w = 2$] #math.equation(block: false, alt: "the fraction m plus n over p")[$frac(m + n, p)$] when #math.equation(block: false, alt: "m equals −6 , n equals −2 , p equals 4")[$m = −6 , n = −2 , p = 4$] #solutionbox[ −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")[$a = −3 , b = 8$] #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")[$r = 10 , s = −5$] #solutionbox[ 3 ] ==== 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 3 over 16")[$frac(3, 16)$] 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? #strong[Baking] Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs #math.equation(block: false, alt: "1 the fraction 1 over 4")[$1 frac(1, 4)$] cups of sugar for the chocolate chip cookies, and #math.equation(block: false, alt: "1 the fraction 1 over 8")[$1 frac(1, 8)$] cups for the oatmeal cookies How much sugar does she need altogether? #solutionbox[ She needs #math.equation(block: false, alt: "2 the fraction 3 over 8")[$2 frac(3, 8)$] cups ] ==== Writing Exercises Explain why it is necessary to have a common denominator to add or subtract fractions. Explain how to find the LCD of two fractions. #solutionbox[ Answers will vary. ] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[An empty self-evaluation chart for math students to gauge their understanding of fraction concepts, from basic addition to complex expressions, marked as 'Confidently', 'With some help', or 'No-I don't get it!'.], alt: "An empty self-evaluation chart for math students to gauge their understanding of fraction concepts, from basic addition to complex expressions, marked as 'Confidently', 'With some help', or 'No-I don't get it!'.", caption: none) ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?