#set document(title: "3.4 Rational Numbers", 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")) == 3.4#h(0.6em)Rational Numbers #figure(figph[A close up view of a newspaper showing stock price changes.], alt: "A close up view of a newspaper showing stock price changes.", caption: [Stock gains and losses are often represented as percentages.]) === Learning Objectives After completing this section, you should be able to: + Define and identify numbers that are rational. + Simplify rational numbers and express in lowest terms. + Add and subtract rational numbers. + Convert between improper fractions and mixed numbers. + Convert rational numbers between decimal and fraction form. + Multiply and divide rational numbers. + Apply the order of operations to rational numbers to simplify expressions. + Apply density property of rational numbers. + Solve problems involving rational numbers. + Use fractions to convert between units. + Define and apply percent. + Solve problems using percent. We are often presented with percentages or fractions to explain how much of a population has a certain feature. For example, the 6-year graduation rate of college students at public institutions is 57.6%, or 72/125. That fraction may be unsettling. But without the context, the percentage is hard to judge. So how does that compare to private institutions? There, the 6-year graduation rate is 65.4%, or 327/500. Comparing the percentages is straightforward, but the fractions are harder to interpret due to different denominators. For more context, historical data could be found. One study reported that the 6-year graduation rate in 1995 was 56.4%. Comparing that historical number to the recent 6-year graduation rate at public institutions of 57.6% shows that there hasn't been much change in that rate. === Defining and Identifying Numbers That Are Rational A #strong[rational number] (called rational since it is a ratio) is just a fraction where the numerator is an integer and the denominator is a non-zero integer. As simple as that is, they can be represented in many ways. It should be noted here that any integer is a rational number. An integer, #math.equation(block: false, alt: "n")[$n$], written as a fraction of two integers is #math.equation(block: false, alt: "the fraction n over 1")[$frac(n, 1)$]. In its most basic representation, a rational number is an integer divided by a non-zero integer, such as #math.equation(block: false, alt: "the fraction 3 over 12")[$frac(3, 12)$]. Fractions may be used to represent parts of a whole. The denominator is the total number of parts to the object, and the numerator is how many of those parts are being used or selected. So, if a pizza is cut into 8 equal pieces, each piece is #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] of the pizza. If you take three slices, you have #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] of the pizza . Similarly, if in a group of 20 people, 5 are wearing hats, then #math.equation(block: false, alt: "the fraction 5 over 20")[$frac(5, 20)$] of the people are wearing hats . #figure(figph[8 slices of pizza. Three slices are highlighted.], alt: "8 slices of pizza. Three slices are highlighted.", caption: [Pizza cut in 8 slices, with 3 slices highlighted]) #figure(figph[A group of 20 people. Five people are wearing hats.], alt: "A group of 20 people. Five people are wearing hats.", caption: [Group of 20 people, with 5 people wearing hats]) Another representation of rational numbers is as a mixed number, such as #math.equation(block: false, alt: "2 the fraction 5 over 8")[$2 frac(5, 8)$] . This represents a whole number (2 in this case), plus a fraction (the #math.equation(block: false, alt: "the fraction 5 over 8")[$frac(5, 8)$]). #figure(figph[Two whole pizzas and 5 slices of a third pizza. The whole pizzas have 8 slices each.], alt: "Two whole pizzas and 5 slices of a third pizza. The whole pizzas have 8 slices each.", caption: [Two whole pizzas and one partial pizza]) Rational numbers may also be expressed in decimal form; for instance, as 1.34. When 1.34 is written, the decimal part, 0.34, represents the fraction #math.equation(block: false, alt: "the fraction 34 over 100")[$frac(34, 100)$], and the number 1.34 is equal to #math.equation(block: false, alt: "1 the fraction 34 over 100")[$1 frac(34, 100)$]. However, not all decimal representations are rational numbers. A number written in decimal form where there is a last decimal digit (after a given decimal digit, all following decimal digits are 0) is a #strong[terminating decimal], as in 1.34 above. Alternately, any decimal numeral that, after a finite number of decimal digits, has digits equal to 0 for all digits following the last non-zero digit. All numbers that can be expressed as a terminating decimal are rational. This comes from what the decimal represents. The decimal part is the fraction of the decimal part divided by the appropriate power of 10. That power of 10 is the number of decimal digits present, as for 0.34, with two decimal digits, being equal to #math.equation(block: false, alt: "the fraction 34 over 100")[$frac(34, 100)$]. Another form that is a rational number is a decimal that repeats a pattern, such as 67.1313… When a rational number is expressed in decimal form and the decimal is a repeated pattern, we use special notation to designate the part that repeats. For example, if we have the repeating decimal 4.3636…, we write this as #math.equation(block: false, alt: "4. 36 ¯")[$4. overline(36)$]. The bar over the 36 indicates that the 36 repeats forever. If the decimal representation of a number does not terminate or form a repeating decimal, that number is not a rational number. One class of numbers that is not rational is the #strong[square roots] of integers or rational numbers that are not #strong[perfect squares], such as #math.equation(block: false, alt: "the square root of 10")[$sqrt(10)$] and #math.equation(block: false, alt: "the square root of the fraction 25 over 6")[$sqrt(frac(25, 6))$]. More generally, the number #math.equation(block: false, alt: "b")[$b$] is the square root of the number #math.equation(block: false, alt: "a")[$a$] if #math.equation(block: false, alt: "a equals b squared")[$a = b^(2)$]. The notation for this is #math.equation(block: false, alt: "b equals the square root of a")[$b = sqrt(a)$], where the symbol #math.equation(block: false, alt: "the square root of")[$sqrt(#h(0.28em))$] is the square root sign. An integer perfect square is any integer that can be written as the square of another integer. A rational perfect square is any rational number that can be written as a fraction of two integers that are perfect squares. Sometimes you may be able to identify a perfect square from memory. Another process that may be used is to factor the number into the product of an integer with itself. Or a calculator (such as Desmos) may be used to find the square root of the number. If the calculator yields an integer, the original number was a perfect square. #notebox("Tech Check", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[Using Desmos to Find the Square Root of a Number] When Desmos is used, there is a tab at the bottom of the screen that opens the keyboard for Desmos. The keyboard is shown below. On the keyboard is the square root symbol #math.equation(block: false, alt: "open parenthesis the square root of close parenthesis")[$( sqrt(" ") )$]. To find the square root of a number, click the square root key, and then type the number. Desmos will automatically display the value of the square root as you enter the number. #figure(figph[Desmos keyboard is displayed. Three sets of keys are displayed. The first set has 4 rows of 4 keys, each. Row 1: x, y, a squared, and a to the power b. Row 2: open parenthesis, close parenthesis, lesser than, and greater than. Row 3: modulus of a, comma, lesser than or equal to, and greater than or equal to. Row 4: A B C, sound, square root, and pi. The second set has 4 rows of 4 keys, each. Row 1: 7, 8, 9, and division symbol. Row 2: 4, 5, 6, and multiplication symbol. Row 3: 1, 2, 3, and minus symbol. Row 4: 0, dot, equals sign, and plus sign. The third set has the following keys: functions, left arrow, right arrow, close, and back.], alt: "Desmos keyboard is displayed. Three sets of keys are displayed. The first set has 4 rows of 4 keys, each. Row 1: x, y, a squared, and a to the power b. Row 2: open parenthesis, close parenthesis, lesser than, and greater than. Row 3: modulus of a, comma, lesser than or equal to, and greater than or equal to. Row 4: A B C, sound, square root, and pi. The second set has 4 rows of 4 keys, each. Row 1: 7, 8, 9, and division symbol. Row 2: 4, 5, 6, and multiplication symbol. Row 3: 1, 2, 3, and minus symbol. Row 4: 0, dot, equals sign, and plus sign. The third set has the following keys: functions, left arrow, right arrow, close, and back.", caption: [Desmos keyboard with square root key]) ] #examplebox("Example 1")[Identifying Perfect Squares][ Which of the following are perfect squares? + 45 + 144 #solutionbox[ + We could attempt to find the perfect square by factoring. Writing all the factor pairs of 45 results in #math.equation(block: false, alt: "1 times 45 , 3 times 15")[$1 × 45 , 3 × 15$], and #math.equation(block: false, alt: "5 times 9")[$5 × 9$]. None of the pairs is a square, so 45 is not a perfect square. Using a calculator to find the square root of 45, we obtain 6.708 (rounded to three decimal places). Since this was not an integer, the original number was not a perfect square. + We could attempt to find the perfect square by factoring. Writing all the factor pairs of 144 results in #math.equation(block: false, alt: "1 times 144 , 2 times 72 , 3 times 48 , 6 times 24 , 8 times 18")[$1 × 144 , 2 × 72 , 3 × 48 , 6 × 24 , 8 × 18$], and #math.equation(block: false, alt: "12 times 12")[$12 × 12$]. Since the last pair is an integer multiplied by itself, 144 is a perfect square. Alternately, using Desmos to find the square root of 144, we obtain 12. Since the square root of 144 is an integer, 144 is a perfect square. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Introduction_to_Fractions")[Introduction to Fractions] ] #examplebox("Example 2")[Identifying Rational Numbers][ Determine which of the following are rational numbers: + #math.equation(block: false, alt: "the square root of 73")[$sqrt(73)$] + #math.equation(block: false, alt: "4.556")[$4.556$] + #math.equation(block: false, alt: "3 the fraction 1 over 5")[$3 frac(1, 5)$] + #math.equation(block: false, alt: "the fraction 41 over 17")[$frac(41, 17)$] + #math.equation(block: false, alt: "5 . 64 ¯")[$5 . overline(64)$] #solutionbox[ + Since 73 is not a perfect square, its square root is not a rational number. This can also be seen when a calculator is used. Entering #math.equation(block: false, alt: "the square root of 73")[$sqrt(73)$] into a calculator results in 8.544003745317 (and then more decimal values after that). There is no repeated pattern, so this is not a rational number. + Since 4.556 is a decimal that terminates, this is a rational number. + #math.equation(block: false, alt: "3 the fraction 1 over 5")[$3 frac(1, 5)$] is a mixed number, so it is a rational number. + #math.equation(block: false, alt: "the fraction 41 over 17")[$frac(41, 17)$] is an integer divided by an integer, so it is a rational number. + #math.equation(block: false, alt: "5.646464...")[$5.646464...$] is a decimal that repeats a pattern, so it is a rational number. ] ] === Simplifying Rational Numbers and Expressing in Lowest Terms A rational number is one way to express the division of two integers. As such, there may be multiple ways to express the same value with different rational numbers. For instance, #math.equation(block: false, alt: "the fraction 4 over 5")[$frac(4, 5)$] and #math.equation(block: false, alt: "the fraction 12 over 15")[$frac(12, 15)$] are the same value. If we enter them into a calculator, they both equal 0.8. Another way to understand this is to consider what it looks like in a figure when two fractions are equal. In , we see that #math.equation(block: false, alt: "the fraction 3 over 5")[$frac(3, 5)$] of the rectangle and #math.equation(block: false, alt: "the fraction 9 over 15")[$frac(9, 15)$] of the rectangle are equal areas. #figure( stdfig([#cetz.canvas({ import cetz.draw: * rect((18.18pt, 18.18pt), (36.36pt, 36.36pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((36.36pt, 18.18pt), (54.55pt, 36.36pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((54.55pt, 18.18pt), (72.73pt, 36.36pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((18.18pt, 36.36pt), (36.36pt, 54.55pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((36.36pt, 36.36pt), (54.55pt, 54.55pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((54.55pt, 36.36pt), (72.73pt, 54.55pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((18.18pt, 54.55pt), (36.36pt, 72.73pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((36.36pt, 54.55pt), (54.55pt, 72.73pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((54.55pt, 54.55pt), (72.73pt, 72.73pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) line((36.36pt, 18.18pt), (36.36pt, 109.09pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((54.55pt, 18.18pt), (54.55pt, 109.09pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((18.18pt, 36.36pt), (72.73pt, 36.36pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((18.18pt, 54.55pt), (72.73pt, 54.55pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((18.18pt, 72.73pt), (72.73pt, 72.73pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((18.18pt, 90.91pt), (72.73pt, 90.91pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((18.18pt, 36.36pt), (72.73pt, 36.36pt), stroke: 0.7pt + black) line((18.18pt, 54.55pt), (72.73pt, 54.55pt), stroke: 0.7pt + black) line((18.18pt, 72.73pt), (72.73pt, 72.73pt), stroke: 0.7pt + black) line((18.18pt, 90.91pt), (72.73pt, 90.91pt), stroke: 0.7pt + black) rect((18.18pt, 18.18pt), (72.73pt, 109.09pt), fill: none, stroke: 0.7pt + black) content((45.45pt, 12.18pt), anchor: "north", box(width: 70.00pt)[#align(center)[#text(size: 9pt)[three-fifths of the rectangle is shaded]]]) rect((127.27pt, 18.18pt), (145.45pt, 36.36pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((145.45pt, 18.18pt), (163.64pt, 36.36pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((163.64pt, 18.18pt), (181.82pt, 36.36pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((127.27pt, 36.36pt), (145.45pt, 54.55pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((145.45pt, 36.36pt), (163.64pt, 54.55pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((163.64pt, 36.36pt), (181.82pt, 54.55pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((127.27pt, 54.55pt), (145.45pt, 72.73pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((145.45pt, 54.55pt), (163.64pt, 72.73pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) rect((163.64pt, 54.55pt), (181.82pt, 72.73pt), fill: rgb("#1D56A6").transparentize(68%), stroke: none) line((145.45pt, 18.18pt), (145.45pt, 109.09pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((163.64pt, 18.18pt), (163.64pt, 109.09pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((127.27pt, 36.36pt), (181.82pt, 36.36pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((127.27pt, 54.55pt), (181.82pt, 54.55pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((127.27pt, 72.73pt), (181.82pt, 72.73pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((127.27pt, 90.91pt), (181.82pt, 90.91pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) rect((127.27pt, 18.18pt), (181.82pt, 109.09pt), fill: none, stroke: 0.7pt + black) content((154.55pt, 12.18pt), anchor: "north", box(width: 70.00pt)[#align(center)[#text(size: 9pt)[9 over 15 of the rectangle is shaded]]]) })]), alt: "Two rectangles are plotted on a rectangular grid. The grid is made up of 15 rows of 20 unit squares, each. The first rectangle has 5 rows of 3 unit squares, each. The rectangle is divided into 5 equal pieces. Each piece has 3 unit squares. 3 pieces are shaded and labeled three-fifths of the rectangle is shaded. The second rectangle has 5 rows of 3 unit squares, each. The rectangle is divided into 15 equal pieces. Each piece has a unit square. 9 pieces are shaded and labeled 9 over 15 of the rectangle is shaded.", caption: [Two Rectangles with Equal Areas], ) They are the same proportion of the area of the rectangle. The left rectangle has 5 pieces, three of which are shaded. The right rectangle has 15 pieces, 9 of which are shaded. Each of the pieces of the left rectangle was divided equally into three pieces. This was a multiplication. The numerator describing the left rectangle was 3 but it becomes #math.equation(block: false, alt: "3 times 3")[$3 × 3$], or 9, as each piece was divided into three. Similarly, the denominator describing the left rectangle was 5, but became #math.equation(block: false, alt: "5 times 3")[$5 × 3$], or 15, as each piece was divided into 3. The fractions #math.equation(block: false, alt: "the fraction 3 over 5")[$frac(3, 5)$] and #math.equation(block: false, alt: "the fraction 9 over 15")[$frac(9, 15)$] are #strong[equivalent] because they represent the same portion (often loosely referred to as equal). This understanding of equivalent fractions is very useful for conceptualization, but it isn’t practical, in general, for determining when two fractions are equivalent. Generally, to determine if the two fractions #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$] and #math.equation(block: false, alt: "the fraction c over d")[$frac(c, d)$] are equivalent, we check to see that #math.equation(block: false, alt: "a times d equals b times c")[$a × d = b × c$]. If those two products are equal, then the fractions are equal also. #examplebox("Example 3")[Determining If Two Fractions Are Equivalent][ Determine if #math.equation(block: false, alt: "the fraction 12 over 30")[$frac(12, 30)$] and #math.equation(block: false, alt: "the fraction 14 over 35")[$frac(14, 35)$] are equivalent fractions. #solutionbox[ Applying the definition, #math.equation(block: false, alt: "a equals 12 , b equals 30 , c equals 14")[$a = 12 , b = 30 , c = 14$] and #math.equation(block: false, alt: "d equals 35")[$d = 35$]. So #math.equation(block: false, alt: "a times d equals 12 times 35 equals 420")[$a × d = 12 × 35 = 420$]. Also, #math.equation(block: false, alt: "b times c equals 30 times 14 equals 420")[$b × c = 30 × 14 = 420$]. Since these values are equal, the fractions are equivalent. ] ] That #math.equation(block: false, alt: "a times d equals b times c")[$a × d = b × c$] indicates the fractions #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$] and #math.equation(block: false, alt: "the fraction c over d")[$frac(c, d)$] are equivalent is due to some algebra. One property of natural numbers, integers, and rational numbers (also irrational numbers) is that for any three numbers #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c")[$c$] with #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$], if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "a / c equals b / c")[$a / c = b / c$]. In other words, when two numbers are equal, then dividing both numbers by the same non-zero number, the two newly obtained numbers are also equal. We can apply that to #math.equation(block: false, alt: "a times d")[$a × d$] and #math.equation(block: false, alt: "b times c")[$b × c$], to show that #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$] and #math.equation(block: false, alt: "the fraction c over d")[$frac(c, d)$] are equivalent if #math.equation(block: false, alt: "a times d equals b times c")[$a × d = b × c$]. If #math.equation(block: false, alt: "a times d equals b times c")[$a × d = b × c$], and #math.equation(block: false, alt: "c not equal to 0 , d not equal to 0")[$c ≠ 0 , d ≠ 0$], we can divide both sides by and obtain the following: #math.equation(block: false, alt: "the fraction a times d over c equals the fraction b times c over c")[$frac(a × d, c) = frac(b × c, c)$]. We can divide out the #math.equation(block: false, alt: "c")[$c$] on the right-hand side of the equation, resulting in #math.equation(block: false, alt: "the fraction a times d over c equals b")[$frac(a × d, c) = b$]. Similarly, we can divide both sides of the equation by #math.equation(block: false, alt: "d")[$d$] and obtain the following: #math.equation(block: false, alt: "the fraction a times d over c times d equals the fraction b over d")[$frac(a × d, c × d) = frac(b, d)$]. We can divide out #math.equation(block: false, alt: "d")[$d$] the on the left-hand side of the equation, resulting in #math.equation(block: false, alt: "the fraction a over c equals the fraction b over d")[$frac(a, c) = frac(b, d)$]. So, the rational numbers #math.equation(block: false, alt: "the fraction a over c")[$frac(a, c)$] and #math.equation(block: false, alt: "the fraction b over d")[$frac(b, d)$] are equivalent when #math.equation(block: false, alt: "a times d equals b times c")[$a × d = b × c$]. #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Equivalent_Fractions")[Equivalent Fractions] ] Recall that a #strong[common divisor] or #strong[common factor] of a set of integers is one that divides all the numbers of the set of numbers being considered. In a fraction, when the numerator and denominator have a common divisor, that common divisor can be #strong[divided out]. This is often called #strong[canceling the common factors] or, more colloquially, as #strong[canceling]. To show this, consider the fraction #math.equation(block: false, alt: "the fraction 36 over 63")[$frac(36, 63)$]. The numerator and denominator have the common factor 3. We can rewrite the fraction as #math.equation(block: false, alt: "the fraction 36 over 63 equals the fraction 12 times 3 over 21 times 3")[$frac(36, 63) = frac(12 × 3, 21 × 3)$]. The common divisor 3 is then divided out, or canceled, and we can write the fraction as #math.equation(block: false, alt: "the fraction 12 times 3 over 21 times 3 equals the fraction 12 over 21")[$frac(12 × cancel(3), 21 × cancel(3)) = frac(12, 21)$]. The 3s have been crossed out to indicate they have been divided out. The process of dividing out two factors is also referred to as #strong[reducing the fraction]. If the numerator and denominator have no common positive divisors other than 1, then the rational number is in #strong[lowest terms]. The process of dividing out common divisors of the numerator and denominator of a fraction is called #strong[reducing the fraction]. One way to reduce a fraction to lowest terms is to determine the GCD of the numerator and denominator and divide out the GCD. Another way is to divide out common divisors until the numerator and denominator have no more common factors. #examplebox("Example 4")[Reducing Fractions to Lowest Terms][ Express the following rational numbers in lowest terms: + #math.equation(block: false, alt: "the fraction 36 over 48")[$frac(36, 48)$] + #math.equation(block: false, alt: "the fraction 100 over 250")[$frac(100, 250)$] + #math.equation(block: false, alt: "the fraction 51 over 136")[$frac(51, 136)$] #solutionbox[ + One process to reduce #math.equation(block: false, alt: "the fraction 36 over 48")[$frac(36, 48)$] to lowest terms is to identify the GCD of 36 and 48 and divide out the GCD. The GCD of 36 and 48 is 12. #strong[Step 1:] We can then rewrite the numerator and denominator by factoring 12 from both. #math.equation(block: true, alt: "the fraction 36 over 48 equals the fraction 12 times 3 over 12 times 4")[$frac(36, 48) = frac(12 × 3, 12 × 4)$] #strong[Step 2:] We can now divide out the 12s from the numerator and denominator. #math.equation(block: true, alt: "the fraction 36 over 48 equals the fraction 12 times 3 over 12 times 4 equals the fraction 3 over 4")[$frac(36, 48) = frac(cancel(12) × 3, cancel(12) × 4) = frac(3, 4)$] So, when #math.equation(block: false, alt: "the fraction 36 over 48")[$frac(36, 48)$] is reduced to lowest terms, the result is #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$]. Alternately, you could identify a common factor, divide out that common factor, and repeat the process until the remaining fraction is in lowest terms. #strong[Step 1:] You may notice that 4 is a common factor of 36 and 48. #strong[Step 2:] Divide out the 4, as in #math.equation(block: false, alt: "the fraction 36 over 48 equals the fraction 4 times 9 over 4 times 12 equals the fraction 4 times 9 over 4 times 12 equals the fraction 9 over 12")[$frac(36, 48) = frac(4 × 9, 4 × 12) = frac(cancel(4) × 9, cancel(4) × 12) = frac(9, 12)$]. #strong[Step 3:] Examining the 9 and 12, you identify 3 as a common factor and divide out the 3, as in #math.equation(block: false, alt: "the fraction 9 over 12 equals the fraction 3 times 3 over 3 times 4 equals the fraction 3 over 4")[$frac(9, 12) = frac(cancel(3) × 3, cancel(3) × 4) = frac(3, 4)$]. The 3 and 4 have no common positive factors other than 1, so it is in lowest terms. So, when #math.equation(block: false, alt: "the fraction 36 over 48")[$frac(36, 48)$] is reduced to lowest terms, the result is #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$]. + #strong[Step 1:] To reduce #math.equation(block: false, alt: "the fraction 100 over 250")[$frac(100, 250)$] to lowest terms, identify the GCD of 100 and 250. This GCD is 50. #strong[Step 2:] We can then rewrite the numerator and denominator by factoring 50 from both. #math.equation(block: true, alt: "the fraction 100 over 250 equals the fraction 50 times 2 over 50 times 5")[$frac(100, 250) = frac(50 × 2, 50 × 5)$]. #strong[Step 3:] We can now divide out the 50s from the numerator and denominator. #math.equation(block: true, alt: "the fraction 100 over 250 equals the fraction 50 times 2 over 50 times 5 equals the fraction 2 over 5")[$frac(100, 250) = frac(cancel(50) × 2, cancel(50) × 5) = frac(2, 5)$] So, when #math.equation(block: false, alt: "the fraction 100 over 250")[$frac(100, 250)$] is reduced to lowest terms, the result is #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$]. + #strong[Step 1:] To reduce #math.equation(block: false, alt: "the fraction 51 over 136")[$frac(51, 136)$] to lowest terms, identify the GCD of 51 and 136. This GCD is 17. #strong[Step 2:] We can then rewrite the numerator and denominator by factoring 17 from both. #math.equation(block: true, alt: "the fraction 51 over 136 equals the fraction 17 times 3 over 17 times 8")[$frac(51, 136) = frac(17 × 3, 17 × 8)$] #strong[Step 3:] We can now divide out the 17s from the numerator and denominator. #math.equation(block: true, alt: "the fraction 51 over 136 equals the fraction 17 times 3 over 17 times 8 equals the fraction 3 over 8")[$frac(51, 136) = frac(cancel(17) × 3, cancel(17) × 8) = frac(3, 8)$] So, when #math.equation(block: false, alt: "the fraction 51 over 136")[$frac(51, 136)$] is reduced to lowest terms, the result is #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$]. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Reducing_Fractions_to_Lowest_Terms")[Reducing Fractions to Lowest Terms] ] #notebox("Tech Check", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[Using Desmos to Find Lowest Terms] Desmos is a free #link("https://openstax.org/r/calculator")[online calculator]. Desmos supports reducing fractions to lowest terms. When a fraction is entered, Desmos immediately calculates the decimal representation of the fraction. However, to the left of the fraction, there is a button that, when clicked, shows the fraction in reduced form. ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Using_Desmos_to_Reduce_a_Fraction")[Using Desmos to Reduce a Fraction] ] === Adding and Subtracting Rational Numbers Adding or subtracting rational numbers can be done with a calculator, which often returns a decimal representation, or by finding a common denominator for the rational numbers being added or subtracted. #notebox("Tech Check", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[Using Desmos to Add Rational Numbers in Fractional Form] To create a fraction in Desmos, enter the numerator, then use the division key (/) on your keyboard, and then enter the denominator. The fraction is then entered. Then click the right arrow key to exit the denominator of the fraction. Next, enter the arithmetic operation (+ or –). Then enter the next fraction. The answer is displayed dynamically (calculates as you enter). To change the Desmos result from decimal form to fractional form, use the fraction button on the left of the line that contains the calculation: #figure(figph[A fraction symbol is shown with empty boxes in the numerator and denominator.], alt: "A fraction symbol is shown with empty boxes in the numerator and denominator.", caption: [Fraction button on the Desmos keyboard]) ] #examplebox("Example 5")[Adding Rational Numbers Using Desmos][ Calculate #math.equation(block: false, alt: "the fraction 23 over 42 plus the fraction 9 over 56")[$frac(23, 42) + frac(9, 56)$] using Desmos. #solutionbox[ Enter #math.equation(block: false, alt: "the fraction 23 over 42 plus the fraction 9 over 56")[$frac(23, 42) + frac(9, 56)$] in Desmos. The result is displayed as #math.equation(block: false, alt: "0.70833333333")[$0.70833333333$] (which is #math.equation(block: false, alt: "0.708 3 ¯")[$0.708 overline(3)$]). Clicking the fraction button to the left on the calculation line yields #math.equation(block: false, alt: "the fraction 17 over 24")[$frac(17, 24)$]. ] ] Performing addition and subtraction without a calculator may be more involved. When the two rational numbers have a common denominator, then adding or subtracting the two numbers is straightforward. Add or subtract the numerators, and then place that value in the numerator and the common denominator in the denominator. Symbolically, we write this as #math.equation(block: false, alt: "the fraction a over c plus or minus the fraction b over c equals the fraction a plus or minus b over c")[$frac(a, c) ± frac(b, c) = frac(a ± b, c)$]. This can be seen in the , which shows #math.equation(block: false, alt: "the fraction 3 over 20 plus the fraction 4 over 20 equals the fraction 7 over 20")[$frac(3, 20) + frac(4, 20) = frac(7, 20)$]. #figure( stdfig([#cetz.canvas({ import cetz.draw: * line((30.77pt, 15.38pt), (30.77pt, 92.31pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((46.15pt, 15.38pt), (46.15pt, 92.31pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((61.54pt, 15.38pt), (61.54pt, 92.31pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((15.38pt, 30.77pt), (76.92pt, 30.77pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((15.38pt, 46.15pt), (76.92pt, 46.15pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((15.38pt, 61.54pt), (76.92pt, 61.54pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((15.38pt, 76.92pt), (76.92pt, 76.92pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) rect((15.38pt, 15.38pt), (76.92pt, 92.31pt), fill: none, stroke: 0.7pt + black) content((46.15pt, 9.38pt), anchor: "north", box(width: 70.00pt)[#align(center)[#text(size: 9pt)[20 equal pieces]]]) rect((123.08pt, 15.38pt), (138.46pt, 30.77pt), fill: rgb("#E06CA0").transparentize(68%), stroke: none) rect((138.46pt, 15.38pt), (153.85pt, 30.77pt), fill: rgb("#E06CA0").transparentize(68%), stroke: none) rect((153.85pt, 15.38pt), (169.23pt, 30.77pt), fill: rgb("#E06CA0").transparentize(68%), stroke: none) line((138.46pt, 15.38pt), (138.46pt, 92.31pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((153.85pt, 15.38pt), (153.85pt, 92.31pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((169.23pt, 15.38pt), (169.23pt, 92.31pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((123.08pt, 30.77pt), (184.62pt, 30.77pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((123.08pt, 46.15pt), (184.62pt, 46.15pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((123.08pt, 61.54pt), (184.62pt, 61.54pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) line((123.08pt, 76.92pt), (184.62pt, 76.92pt), stroke: (paint: rgb("#B6D4DA"), thickness: 0.4pt)) rect((123.08pt, 15.38pt), (184.62pt, 92.31pt), fill: none, stroke: 0.7pt + black) content((153.85pt, 9.38pt), anchor: "north", box(width: 70.00pt)[#align(center)[#text(size: 9pt)[3/20 pink + 4/20 green = 7/20]]]) rect((169.23pt, 15.38pt), (184.62pt, 30.77pt), fill: rgb("#6AA84F").transparentize(68%), stroke: none) rect((123.08pt, 30.77pt), (138.46pt, 46.15pt), fill: rgb("#6AA84F").transparentize(68%), stroke: none) rect((138.46pt, 30.77pt), (153.85pt, 46.15pt), fill: rgb("#6AA84F").transparentize(68%), stroke: none) rect((153.85pt, 30.77pt), (169.23pt, 46.15pt), fill: rgb("#6AA84F").transparentize(68%), stroke: none) })]), alt: "Two rectangles are plotted on a rectangular grid. The grid is made up of 15 rows of 20 unit squares, each. The first rectangle has 5 rows of 4 unit squares, each. The rectangle is divided into 20 equal pieces. Each piece has a unit square. The second rectangle has 5 rows of 4 unit squares, each. The rectangle is divided into 20 equal pieces. 3 pieces are shaded in pink and 4 pieces are shaded in green. Text reads, 3 over 20 in pink, 4 over 20 in green. 3 over 20 plus 4 over 20 equals 7 over 20.", caption: [Partially Shaded Rectangle], ) It is customary to then write the result in lowest terms. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ If #math.equation(block: false, alt: "c")[$c$] is a non-zero integer, then #math.equation(block: false, alt: "the fraction a over c plus or minus the fraction b over c equals the fraction a plus or minus b over c")[$frac(a, c) ± frac(b, c) = frac(a ± b, c)$]. ] #examplebox("Example 6")[Adding Rational Numbers with the Same Denominator][ Calculate #math.equation(block: false, alt: "the fraction 13 over 28 plus the fraction 7 over 28")[$frac(13, 28) + frac(7, 28)$]. #solutionbox[ Since the rational numbers have the same denominator, we perform the addition of the numerators, #math.equation(block: false, alt: "13 plus 7")[$13 + 7$], and then place the result in the numerator and the common denominator, 28, in the denominator. #math.equation(block: false, alt: "the fraction 13 over 28 plus the fraction 7 over 28 equals the fraction 13 plus 7 over 28 equals the fraction 20 over 28")[$frac(13, 28) + frac(7, 28) = frac(13 + 7, 28) = frac(20, 28)$] Once we have that result, reduce to lowest terms, which gives #math.equation(block: false, alt: "the fraction 20 over 28 equals the fraction 4 times 5 over 4 times 7 equals the fraction 4 times 5 over 4 times 7 equals the fraction 5 over 7")[$frac(20, 28) = frac(4 × 5, 4 × 7) = frac(cancel(4) × 5, cancel(4) × 7) = frac(5, 7)$]. ] ] #examplebox("Example 7")[Subtracting Rational Numbers with the Same Denominator][ Calculate #math.equation(block: false, alt: "the fraction 45 over 136 minus the fraction 17 over 136")[$frac(45, 136) − frac(17, 136)$]. #solutionbox[ Since the rational numbers have the same denominator, we perform the subtraction of the numerators, #math.equation(block: false, alt: "45 minus 17")[$45 − 17$], and then place the result in the numerator and the common denominator, 136, in the denominator. #math.equation(block: false, alt: "the fraction 45 over 136 minus the fraction 17 over 136 minus the fraction 45 minus 17 over 136 equals the fraction 28 over 136")[$frac(45, 136) − frac(17, 136) − frac(45 − 17, 136) = frac(28, 136)$] Once we have that result, reduce to lowest terms, this gives #math.equation(block: false, alt: "the fraction 28 over 136 equals the fraction 4 times 7 over 4 times 34 equals the fraction 4 times 7 over 4 times 34 equals the fraction 7 over 34")[$frac(28, 136) = frac(4 × 7, 4 × 34) = frac(cancel(4) × 7, cancel(4) × 34) = frac(7, 34)$]. ] ] When the rational numbers do not have common denominators, then we have to transform the rational numbers so that they do have common denominators. The common denominator that reduces work later in the problem is the LCM of the numerator and denominator. When adding or subtracting the rational numbers #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$] and #math.equation(block: false, alt: "the fraction c over d")[$frac(c, d)$], we perform the following steps. #strong[Step 1:] Find #math.equation(block: false, alt: "LCM open parenthesis b , d close parenthesis")[$italic("LCM") ( b , d )$]. #strong[Step 2:] Calculate #math.equation(block: false, alt: "n equals the fraction LCM open parenthesis b , d close parenthesis over b")[$n = frac(italic("LCM") ( b , d ), b)$] and #math.equation(block: false, alt: "m equals the fraction L C M open parenthesis b , d close parenthesis over d")[$m = frac(L C M ( b , d ), d)$]. #strong[Step 3:] Multiply the numerator and denominator of #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$] by #math.equation(block: false, alt: "n")[$n$], yielding #math.equation(block: false, alt: "the fraction a times n over b times n")[$frac(a × n, b × n)$]. #strong[Step 4:] Multiply the numerator and denominator of #math.equation(block: false, alt: "the fraction c over d")[$frac(c, d)$] by #math.equation(block: false, alt: "m")[$m$], yielding #math.equation(block: false, alt: "the fraction c times m over d times m")[$frac(c × m, d × m)$]. #strong[Step 5:] Add or subtract the rational numbers from Steps 3 and 4, since they now have the common denominators. You should be aware that the common denominator is #math.equation(block: false, alt: "LCM open parenthesis b , d close parenthesis")[$italic("LCM") ( b , d )$]. For the first denominator, we have #math.equation(block: false, alt: "b times n equals b times the fraction L C M open parenthesis b , d close parenthesis over b equals L C M open parenthesis b , d close parenthesis")[$b × n = b × frac(L C M ( b , d ), b) = L C M ( b , d )$], since we multiply and divide #math.equation(block: false, alt: "LCM open parenthesis b , d close parenthesis")[$italic("LCM") ( b , d )$] by the same number. For the same reason, #math.equation(block: false, alt: "d times m equals d times the fraction L C M open parenthesis b , d close parenthesis over b equals L C M open parenthesis b , d close parenthesis")[$d × m = d × frac(L C M ( b , d ), b) = L C M ( b , d )$]. #examplebox("Example 8")[Adding Rational Numbers with Unequal Denominators][ Calculate #math.equation(block: false, alt: "the fraction 11 over 18 plus the fraction 2 over 15")[$frac(11, 18) + frac(2, 15)$]. #solutionbox[ The denominators of the fractions are 18 and 15, so we label #math.equation(block: false, alt: "b equals 18")[$b = 18$] and #math.equation(block: false, alt: "d equals 15")[$d = 15$]. #strong[Step 1:] Find LCM(18,15). This is 90. #strong[Step 2]: Calculate #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "m")[$m$]. #math.equation(block: false, alt: "n equals the fraction 90 over 18 equals 5")[$n = frac(90, 18) = 5$] and #math.equation(block: false, alt: "m equals the fraction 90 over 15 equals 6")[$m = frac(90, 15) = 6$]. #strong[Step 3:] Multiplying the numerator and denominator of #math.equation(block: false, alt: "the fraction 11 over 18")[$frac(11, 18)$] by #math.equation(block: false, alt: "n equals 5")[$n = 5$] yields #math.equation(block: false, alt: "the fraction 11 times 5 over 18 times 5 equals the fraction 55 over 90")[$frac(11 × 5, 18 × 5) = frac(55, 90)$]. #strong[Step 4:] Multiply the numerator and denominator of #math.equation(block: false, alt: "the fraction 2 over 15")[$frac(2, 15)$] by #math.equation(block: false, alt: "m equals 6")[$m = 6$] yields #math.equation(block: false, alt: "the fraction 2 times 6 over 15 times 6 equals the fraction 12 over 90")[$frac(2 × 6, 15 × 6) = frac(12, 90)$]. #strong[Step 5:] Now we add the values from Steps 3 and 4: #math.equation(block: false, alt: "the fraction 55 over 90 plus the fraction 12 over 90 equals the fraction 67 over 90")[$frac(55, 90) + frac(12, 90) = frac(67, 90)$]. This is in lowest terms, so we have found that #math.equation(block: false, alt: "the fraction 11 over 18 plus the fraction 2 over 15 equals the fraction 67 over 90")[$frac(11, 18) + frac(2, 15) = frac(67, 90)$]. ] ] #examplebox("Example 9")[Subtracting Rational Numbers with Unequal Denominators][ Calculate #math.equation(block: false, alt: "the fraction 14 over 25 minus the fraction 9 over 70")[$frac(14, 25) − frac(9, 70)$]. #solutionbox[ The denominators of the fractions are 25 and 70, so we label #math.equation(block: false, alt: "b equals 25")[$b = 25$] and #math.equation(block: false, alt: "d equals 70")[$d = 70$]. #strong[Step 1:] Find LCM(25,70). This is 350. #strong[Step 2:] Calculate #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "m")[$m$]: #math.equation(block: false, alt: "n equals the fraction 350 over 25 equals 14")[$n = frac(350, 25) = 14$] and #math.equation(block: false, alt: "m equals the fraction 350 over 70 equals 5")[$m = frac(350, 70) = 5$]. #strong[Step 3:] Multiplying the numerator and denominator of #math.equation(block: false, alt: "the fraction 14 over 25")[$frac(14, 25)$] by #math.equation(block: false, alt: "n equals 14")[$n = 14$] yields #math.equation(block: false, alt: "the fraction 14 times 14 over 25 times 14 equals the fraction 196 over 350")[$frac(14 × 14, 25 × 14) = frac(196, 350)$]. #strong[Step 4:] Multiplying the numerator and denominator of #math.equation(block: false, alt: "the fraction 9 over 70")[$frac(9, 70)$] by #math.equation(block: false, alt: "m equals 5")[$m = 5$] yields #math.equation(block: false, alt: "the fraction 9 times 5 over 70 times 5 equals the fraction 45 over 350")[$frac(9 × 5, 70 × 5) = frac(45, 350)$]. #strong[Step 5:] Now we subtract the value from Step 4 from the value in Step 3: #math.equation(block: false, alt: "the fraction 196 over 350 minus the fraction 45 over 350 equals the fraction 151 over 350")[$frac(196, 350) − frac(45, 350) = frac(151, 350)$]. This is in lowest terms, so we have found that #math.equation(block: false, alt: "the fraction 14 over 25 minus the fraction 9 over 70 equals the fraction 151 over 350")[$frac(14, 25) − frac(9, 70) = frac(151, 350)$]. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Adding_and_Subtracting_Fractions")[Adding and Subtracting Fractions with Different Denominators] ] === Converting Between Improper Fractions and Mixed Numbers One way to visualize a fraction is as parts of a whole, as in #math.equation(block: false, alt: "the fraction 5 over 12")[$frac(5, 12)$] of a pizza. But when the numerator is larger than the denominator, as in #math.equation(block: false, alt: "the fraction 23 over 12")[$frac(23, 12)$], then the idea of parts of a whole seems not to make sense. Such a fraction is an #strong[improper fraction.] That kind of fraction could be written as an integer plus a fraction, which is a #strong[mixed number]. The fraction #math.equation(block: false, alt: "the fraction 23 over 12")[$frac(23, 12)$] rewritten as a mixed number would be #math.equation(block: false, alt: "1 the fraction 11 over 12")[$1 frac(11, 12)$]. Arithmetically, #math.equation(block: false, alt: "1 the fraction 11 over 12")[$1 frac(11, 12)$] is equivalent to #math.equation(block: false, alt: "1 plus the fraction 11 over 12")[$1 + frac(11, 12)$], which is read as “one and 11 twelfths.” Improper fractions can be rewritten as mixed numbers using division and remainders. To find the mixed number representation of an improper fraction, divide the numerator by the denominator. The quotient is the integer part, and the remainder becomes the numerator of the remaining fraction. #examplebox("Example 10")[Rewriting an Improper Fraction as a Mixed Number][ Rewrite #math.equation(block: false, alt: "the fraction 48 over 13")[$frac(48, 13)$] as a mixed number. #solutionbox[ When 48 is divided by 13, the result is 3 with a remainder of 9. So, we can rewrite #math.equation(block: false, alt: "the fraction 48 over 13")[$frac(48, 13)$] as #math.equation(block: false, alt: "3 the fraction 9 over 13")[$3 frac(9, 13)$]. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Improper_Fraction_to_Mixed_Number")[Converting an Improper Fraction to a Mixed Number Using Desmos] ] Similarly, we can convert a mixed number into an improper fraction. To do so, first convert the whole number part to a fraction by writing the whole number as itself divided by 1, and then add the two fractions. Alternately, we can multiply the whole number part and the denominator of the fractional part. Next, add that product to the numerator. Finally, express the number as that product divided by the denominator. #examplebox("Example 11")[Rewriting a Mixed Number as an Improper Fraction][ Rewrite #math.equation(block: false, alt: "5 the fraction 4 over 9")[$5 frac(4, 9)$] as an improper fraction. #solutionbox[ #strong[Step 1:] Multiply the integer part, 5, by the denominator, 9, which gives #math.equation(block: false, alt: "5 times 9 equals 45")[$5 × 9 = 45$]. #strong[Step 2:] Add that product to the numerator, which gives #math.equation(block: false, alt: "45 plus 4 equals 49")[$45 + 4 = 49$]. #strong[Step 3:] Write the number as the sum, 49, divided by the denominator, 9, which gives #math.equation(block: false, alt: "the fraction 49 over 9")[$frac(49, 9)$]. ] ] #notebox("Tech Check", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[Using Desmos to Rewrite a Mixed Number as an Improper Fraction] Desmos can be used to convert from a mixed number to an improper fraction. To do so, we use the idea that a mixed number, such as #math.equation(block: false, alt: "5 the fraction 6 over 11")[$5 frac(6, 11)$], is another way to represent #math.equation(block: false, alt: "5 plus the fraction 6 over 11")[$5 + frac(6, 11)$]. If #math.equation(block: false, alt: "5 plus the fraction 6 over 11")[$5 + frac(6, 11)$] is entered in Desmos, the result is the decimal form of the number. However, clicking the fraction button to the left will convert the decimal to an improper fraction, #math.equation(block: false, alt: "the fraction 61 over 11")[$frac(61, 11)$]. As an added bonus, Desmos will automatically reduce the fraction to lowest terms. ] === Converting Rational Numbers Between Decimal and Fraction Forms Understanding what decimals represent is needed before addressing conversions between the fractional form of a number and its #strong[decimal form], or writing a number in #strong[decimal notation]. The decimal number 4.557 is equal to #math.equation(block: false, alt: "4 the fraction 557 over 1,000")[$4 frac(557, 1","000)$]. The decimal portion, .557, is 557 divided by 1,000. To write any decimal portion of a number expressed as a terminating decimal, divide the decimal number by 10 raised to the power equal to the number of decimal digits. Since there were three decimal digits in 4.557, we divided 557 by #math.equation(block: false, alt: "10 cubed equals 1000")[$10^(3) = 1000$]. Decimal representations may be very long. It is convenient to #strong[round off] the decimal form of the number to a certain number of decimal digits. To round off the decimal form of a number to #math.equation(block: false, alt: "n")[$n$] (decimal) digits, examine the (#math.equation(block: false, alt: "n plus 1")[$n + 1$])st decimal digit. If that digit is 0, 1, 2, 3, or 4, the number is rounded off by writing the number to the #math.equation(block: false, alt: "n")[$n$]th decimal digit and no further. If the (#math.equation(block: false, alt: "n plus 1")[$n + 1$])st decimal digit is 5, 6, 7, 8, or 9, the number is rounded off by writing the number to the #math.equation(block: false, alt: "n")[$n$]th digit, then replacing the #math.equation(block: false, alt: "n")[$n$]th digit by one more than the #math.equation(block: false, alt: "n")[$n$]th digit. #examplebox("Example 12")[Rounding Off a Number in Decimal Form to Three Digits][ Round 5.67849 to three decimal digits. #solutionbox[ The third decimal digit is 8. The digit following the 8 is 4. When the digit is 4, we write the number only to the third digit. So, 5.67849 rounded off to three decimal places is 5.678. ] ] #examplebox("Example 13")[Rounding Off a Number in Decimal Form to Four Digits][ Round 45.11475 to four decimal digits. #solutionbox[ The fourth decimal digit is 7. The digit following the 7 is 5. When the digit is 5, we write the number only to the fourth decimal digit, 45.1147. We then replace the fourth decimal digit by one more than the fourth digit, which yields 45.1148. So, 45.11475 rounded off to four decimal places is 45.1148. ] ] To convert a rational number in fraction form to decimal form, use your calculator to perform the division. #examplebox("Example 14")[Converting a Rational Number in Fraction Form into Decimal Form][ Convert #math.equation(block: false, alt: "the fraction 47 over 25")[$frac(47, 25)$] into decimal form. #solutionbox[ Using a calculator to divide 47 by 25, the result is 1.88. ] ] Converting a terminating decimal to the fractional form may be done in the following way: #strong[Step 1:] Count the number of digits in the decimal part of the number, labeled #math.equation(block: false, alt: "n")[$n$]. #strong[Step 2:] Raise 10 to the #math.equation(block: false, alt: "n")[$n$]th power. #strong[Step 3:] Rewrite the number without the decimal. #strong[Step 4:] The fractional form is the number from Step 3 divided by the result from Step 2. This process works due to what decimals represent and how we work with mixed numbers. For example, we could convert the number 7.4536 to fractional from. The decimal part of the number, the .4536 part of 7.4536, has four digits. By the definition of decimal notation, the decimal portion represents #math.equation(block: false, alt: "the fraction 4,536 over 10 to the power 4 equals the fraction 4,536 over 10,000")[$frac(4","536, 10^(4)) = frac(4","536, 10","000)$]. The decimal number 7.4536 is equal to the improper fraction #math.equation(block: false, alt: "7 the fraction 4,536 over 10,000")[$7 frac(4","536, 10","000)$]. Adding those to fractions yields #math.equation(block: false, alt: "the fraction 74,536 over 10,000")[$frac(74","536, 10","000)$]. #examplebox("Example 15")[Converting from Decimal Form to Fraction Form with Terminating Decimals][ Convert 3.2117 to fraction form. #solutionbox[ #strong[Step 1:] There are four digits after the decimal point, so #math.equation(block: false, alt: "n equals 4")[$n = 4$]. #strong[Step 2:] Raise 10 to the fourth power, #math.equation(block: false, alt: "10 to the power 4 equals 10,000")[$10^(4) = 10,000$]. #strong[Step 3:] When we remove the decimal point, we have 32,117. #strong[Step 4:] The fraction has as its numerator the result from Step 3 and as its denominator the result of Step 2, which is the fraction #math.equation(block: false, alt: "the fraction 32,117 over 10,000")[$frac(32","117, 10","000)$]. ] ] The process is different when converting from the decimal form of a rational number into fraction form when the decimal form is a repeating decimal. This process is not covered in this text. === Multiplying and Dividing Rational Numbers Multiplying rational numbers is less complicated than adding or subtracting rational numbers, as there is no need to find common denominators. To multiply rational numbers, multiply the numerators, then multiply the denominators, and write the numerator product divided by the denominator product. Symbolically, #math.equation(block: false, alt: "the fraction a over b times the fraction c over d equals the fraction a times c over b times d")[$frac(a, b) × frac(c, d) = frac(a × c, b × d)$]. As always, rational numbers should be reduced to lowest terms. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ If #math.equation(block: false, alt: "b")[$b$] and #math.equation(block: false, alt: "d")[$d$] are non-zero integers, then #math.equation(block: false, alt: "the fraction a over b times the fraction c over d equals the fraction a times c over b times d")[$frac(a, b) × frac(c, d) = frac(a × c, b × d)$]. ] #examplebox("Example 16")[Multiplying Rational Numbers][ Calculate #math.equation(block: false, alt: "the fraction 12 over 25 times the fraction 10 over 21")[$frac(12, 25) × frac(10, 21)$]. #solutionbox[ Multiply the numerators and place that in the numerator, and then multiply the denominators and place that in the denominator. #math.equation(block: true, alt: "the fraction 12 over 25 times the fraction 10 over 21 equals the fraction 12 times 10 over 25 times 21 equals the fraction 120 over 525")[$frac(12, 25) × frac(10, 21) = frac(12 × 10, 25 × 21) = frac(120, 525)$] This is not in lowest terms, so this needs to be reduced. The GCD of 120 and 525 is 15. #math.equation(block: true, alt: "the fraction 120 over 525 equals the fraction 15 times 8 over 15 times 35 equals the fraction 8 over 35")[$frac(120, 525) = frac(15 × 8, 15 × 35) = frac(8, 35)$] ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Multiplying_Fractions")[Multiplying Fractions] ] As with multiplication, division of rational numbers can be done using a calculator. #examplebox("Example 17")[Dividing Decimals with a Calculator][ Calculate 3.45 ÷ 2.341 using a calculator. Round to three decimal places if necessary. #solutionbox[ Using a calculator, we obtain 1.473729175565997. Rounding to three decimal places we have 1.474. ] ] Before discussing division of fractions without a calculator, we should look at the reciprocal of a number. The #strong[reciprocal] of a number is 1 divided by the number. For a fraction, the reciprocal is the fraction formed by switching the numerator and denominator. For the fraction #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$], the reciprocal is #math.equation(block: false, alt: "the fraction b over a")[$frac(b, a)$]. An important feature for a number and its reciprocal is that their product is 1. When dividing two fractions by hand, find the reciprocal of the #strong[divisor] (the number that is being divided into the other number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the multiplication. Symbolically,#math.equation(block: false, alt: "the fraction b over a divided by the fraction c over d equals the fraction a over b times the fraction d over c equals the fraction a times d over b times c")[$frac(b, a) ÷ frac(c, d) = frac(a, b) × frac(d, c) = frac(a × d, b × c)$]. As before, reduce to lowest terms. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ If #math.equation(block: false, alt: "b , c")[$b , c$] and #math.equation(block: false, alt: "d")[$d$] are non-zero integers, then #math.equation(block: false, alt: "the fraction b over a divided by the fraction c over d equals the fraction a over b times the fraction d over c equals the fraction a times d over b times c")[$frac(b, a) ÷ frac(c, d) = frac(a, b) × frac(d, c) = frac(a × d, b × c)$]. ] #examplebox("Example 18")[Dividing Rational Numbers][ + Calculate #math.equation(block: false, alt: "the fraction 4 over 21 divided by the fraction 6 over 35")[$frac(4, 21) ÷ frac(6, 35)$]. + Calculate #math.equation(block: false, alt: "the fraction 1 over 8 divided by the fraction 5 over 28")[$frac(1, 8) ÷ frac(5, 28)$]. #solutionbox[ + #strong[Step 1:] Find the reciprocal of the number being divided by #math.equation(block: false, alt: "the fraction 6 over 35")[$frac(6, 35)$]. The reciprocal of that is #math.equation(block: false, alt: "the fraction 35 over 6")[$frac(35, 6)$]. #strong[Step 2:] Multiply the first fraction by that reciprocal. #math.equation(block: true, alt: "the fraction 4 over 21 divided by the fraction 6 over 35 equals the fraction 4 over 21 times the fraction 35 over 6 equals the fraction 140 over 126")[$frac(4, 21) ÷ frac(6, 35) = frac(4, 21) × frac(35, 6) = frac(140, 126)$] The answer, #math.equation(block: false, alt: "the fraction 140 over 126")[$frac(140, 126)$] is not in lowest terms. The GCD of 140 and 126 is 14. Factoring and canceling gives #math.equation(block: false, alt: "the fraction 140 over 126 equals the fraction 14 times 10 over 14 times 9 equals the fraction 10 over 9")[$frac(140, 126) = frac(14 × 10, 14 × 9) = frac(10, 9)$]. + #strong[Step 1:] Find the reciprocal of the number being divided by, which is #math.equation(block: false, alt: "the fraction 5 over 28")[$frac(5, 28)$]. The reciprocal of that is #math.equation(block: false, alt: "the fraction 28 over 5")[$frac(28, 5)$]. #strong[Step 2:] Multiply the first fraction by that reciprocal: #math.equation(block: false, alt: "the fraction 1 over 8 divided by the fraction 5 over 28 equals the fraction 1 over 8 times the fraction 28 over 5 equals the fraction 28 over 40")[$frac(1, 8) ÷ frac(5, 28) = frac(1, 8) × frac(28, 5) = frac(28, 40)$] The answer, #math.equation(block: false, alt: "the fraction 28 over 40")[$frac(28, 40)$], is not in lowest reduced form. The GCD of 28 and 40 is 4. Factoring and canceling gives #math.equation(block: false, alt: "the fraction 28 over 40 equals the fraction 4 times 7 over 4 times 10 equals the fraction 7 over 10")[$frac(28, 40) = frac(4 × 7, 4 × 10) = frac(7, 10)$]. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Dividing_Fractions")[Dividing Fractions] ] === Applying the Order of Operations to Simplify Expressions The order of operations for rational numbers is the same as for integers, as discussed in Order of Operations. The order of operations makes it easier for anyone to correctly calculate and represent. The order follows the well-known acronym PEMDAS: #figure(table( columns: 2, align: left, inset: 6pt, table.header([P], [Parentheses]), [E], [Exponents], [M/D], [Multiplication and division], [A/S], [Addition and subtraction], )) The first step in calculating using the order of operations is to perform operations inside the parentheses. Moving down the list, next perform all exponent operations moving from left to right. Next (left to right once more), perform all multiplications and divisions. Finally, perform the additions and subtractions. #examplebox("Example 19")[Applying the Order of Operations with Rational Numbers][ Correctly apply the rules for the order of operations to accurately compute #math.equation(block: false, alt: "open parenthesis the fraction 5 over 7 minus the fraction 2 over 7 close parenthesis times 2 cubed")[$( frac(5, 7) − frac(2, 7) ) × 2^(3)$]. #solutionbox[ #strong[Step 1:] To calculate this, perform all calculations within the parentheses before other operations. #math.equation(block: true, alt: "open parenthesis the fraction 5 over 7 minus the fraction 2 over 7 close parenthesis times 2 cubed equals open parenthesis the fraction 3 over 7 close parenthesis times 2 cubed")[$( frac(5, 7) − frac(2, 7) ) × 2^(3) = ( frac(3, 7) ) × 2^(3)$] #strong[Step 2:] Since all parentheses have been cleared, we move left to right, and compute all the exponents next. #math.equation(block: true, alt: "open parenthesis the fraction 3 over 7 close parenthesis times 2 cubed equals open parenthesis the fraction 3 over 7 close parenthesis times 8")[$( frac(3, 7) ) × 2^(3) = ( frac(3, 7) ) × 8$] #strong[Step 3:] Now, perform all multiplications and divisions, moving left to right. #math.equation(block: true, alt: "open parenthesis the fraction 3 over 7 close parenthesis times 8 equals the fraction 24 over 7")[$( frac(3, 7) ) × 8 = frac(24, 7)$] ] ] #examplebox("Example 20")[Applying the Order of Operations with Rational Numbers][ Correctly apply the rules for the order of operations to accurately compute #math.equation(block: false, alt: "4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 5 over 9 close parenthesis squared minus open parenthesis the fraction 2 over 3 plus 5 close parenthesis close parenthesis squared")[$4 + frac(2, 3) ÷ attach(( attach(( frac(5, 9) ), t: 2) − ( frac(2, 3) + 5 ) ), t: 2)$]. #solutionbox[ To calculate this, perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time. #strong[Step 1:] The innermost parentheses contain #math.equation(block: false, alt: "the fraction 2 over 3 plus 5")[$frac(2, 3) + 5$]. Calculate that first, dividing after finding the common denominator. #math.equation(block: true, alt: "4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 5 over 9 close parenthesis squared minus open parenthesis the fraction 2 over 3 plus 5 close parenthesis close parenthesis squared; equals 4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 5 over 9 close parenthesis squared minus open parenthesis the fraction 2 over 3 plus the fraction 5 over 1 close parenthesis close parenthesis squared; equals 4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 5 over 9 close parenthesis squared minus open parenthesis the fraction 2 over 3 plus the fraction 15 over 3 close parenthesis close parenthesis squared; equals 4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fra")[$4 + frac(2, 3) ÷ attach(( attach(( frac(5, 9) ), t: 2) − ( frac(2, 3) + 5 ) ), t: 2) \ = 4 + frac(2, 3) ÷ attach(( attach(( frac(5, 9) ), t: 2) − ( frac(2, 3) + frac(5, 1) ) ), t: 2) \ = 4 + frac(2, 3) ÷ attach(( attach(( frac(5, 9) ), t: 2) − ( frac(2, 3) + frac(15, 3) ) ), t: 2) \ = 4 + frac(2, 3) ÷ attach(( attach(( frac(5, 9) ), t: 2) − ( frac(17, 3) ) ), t: 2)$] #strong[Step 2:] Calculate the exponent in the parentheses, #math.equation(block: false, alt: "open parenthesis the fraction 5 over 9 close parenthesis squared")[$attach(( frac(5, 9) ), t: 2)$]. #math.equation(block: true, alt: "4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 5 over 9 close parenthesis squared minus open parenthesis the fraction 17 over 3 close parenthesis close parenthesis squared; equals 4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 25 over 81 close parenthesis minus open parenthesis the fraction 17 over 3 close parenthesis close parenthesis squared")[$4 + frac(2, 3) ÷ attach(( attach(( frac(5, 9) ), t: 2) − ( frac(17, 3) ) ), t: 2) \ = 4 + frac(2, 3) ÷ attach(( ( frac(25, 81) ) − ( frac(17, 3) ) ), t: 2)$] #strong[Step 3:] Subtract inside the parentheses is done, using a common denominator. #math.equation(block: true, alt: "4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 25 over 81 close parenthesis minus open parenthesis the fraction 17 over 3 close parenthesis close parenthesis squared; 4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 25 over 81 close parenthesis minus open parenthesis the fraction 17 times 27 over 3 times 27 close parenthesis close parenthesis squared; 4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction 25 over 81 close parenthesis minus open parenthesis the fraction 459 over 81 close parenthesis close parenthesis squared; 4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction minus 434 over 81 close parenthesis close parenthesis squared")[$4 + frac(2, 3) ÷ attach(( ( frac(25, 81) ) − ( frac(17, 3) ) ), t: 2) \ 4 + frac(2, 3) ÷ attach(( ( frac(25, 81) ) − ( frac(17 × 27, 3 × 27) ) ), t: 2) \ 4 + frac(2, 3) ÷ attach(( ( frac(25, 81) ) − ( frac(459, 81) ) ), t: 2) \ 4 + frac(2, 3) ÷ attach(( ( frac(− 434, 81) ) ), t: 2)$] #strong[Step 4:] At this point, evaluate the exponent and divide. #math.equation(block: true, alt: "4 plus the fraction 2 over 3 divided by open parenthesis open parenthesis the fraction minus 434 over 81 close parenthesis close parenthesis squared; 4 plus the fraction 2 over 3 divided by open parenthesis the fraction 188,356 over 6,561 close parenthesis; equals 4 plus the fraction 2 over 3 times open parenthesis the fraction 6,561 over 188,356 close parenthesis; equals 4 plus the fraction 2,187 over 94,178")[$4 + frac(2, 3) ÷ attach(( ( frac(− 434, 81) ) ), t: 2) \ 4 + frac(2, 3) ÷ ( frac(188","356, 6","561) ) \ = 4 + frac(2, 3) × ( frac(6","561, 188","356) ) \ = 4 + frac(2","187, 94","178)$] #strong[Step 5:] Add. #math.equation(block: true, alt: "4 plus the fraction 2,187 over 94,178; equals the fraction 378,899 over 94,178")[$4 + frac(2","187, 94","178) \ = frac(378","899, 94","178)$] Had this been done on a calculator, the decimal form of the answer would be 4.0232 (rounded to four decimal places). ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Operations_Using_Fractions")[Order of Operations Using Fractions] ] === Applying the Density Property of Rational Numbers Between any two rational numbers, there is another rational number. This is called the #strong[density property] of the rational numbers. Finding a rational number between any two rational numbers is very straightforward. #strong[Step 1:] Add the two rational numbers. #strong[Step 2:] Divide that result by 2. The result is always a rational number. This follows what we know about rational numbers. If two fractions are added, then the result is a fraction. Also, when a fraction is divided by a fraction (and 2 is a fraction), then we get another fraction. This two-step process will give a rational number, provided the first two numbers were rational. #examplebox("Example 21")[Applying the Density Property of Rational Numbers][ Demonstrate the density property of rational numbers by finding a rational number between #math.equation(block: false, alt: "the fraction 4 over 11")[$frac(4, 11)$] and #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$]. #solutionbox[ To find a rational number between #math.equation(block: false, alt: "the fraction 4 over 11")[$frac(4, 11)$] and #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$]: #strong[Step 1:] Add the fractions. #math.equation(block: true, alt: "the fraction 4 over 11 plus the fraction 7 over 12 equals the fraction 4 times 12 over 11 times 12 plus the fraction 7 times 11 over 12 times 11 equals the fraction 48 over 132 plus the fraction 77 over 132 equals the fraction 125 over 132")[$frac(4, 11) + frac(7, 12) = frac(4 × 12, 11 × 12) + frac(7 × 11, 12 × 11) = frac(48, 132) + frac(77, 132) = frac(125, 132)$] #strong[Step 2:] Divide the result by 2. Recall that to divide by 2, you multiply by the reciprocal of 2. The reciprocal of 2 is #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$], as seen below. #math.equation(block: true, alt: "the fraction 125 over 132 divided by 2 equals the fraction 125 over 132 times the fraction 1 over 2 equals the fraction 125 over 264")[$frac(125, 132) ÷ 2 = frac(125, 132) × frac(1, 2) = frac(125, 264)$] So, one rational number between #math.equation(block: false, alt: "the fraction 4 over 11")[$frac(4, 11)$] and #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$] is #math.equation(block: false, alt: "the fraction 125 over 264")[$frac(125, 264)$]. We could check that the number we found is between the other two by finding the decimal representation of the numbers. Using a calculator, the decimal representations of the rational numbers are 0.363636…, 0.473484848…, and 0.5833333…. Here it is clear that #math.equation(block: false, alt: "the fraction 125 over 264")[$frac(125, 264)$] is between #math.equation(block: false, alt: "the fraction 4 over 11")[$frac(4, 11)$] and #math.equation(block: false, alt: "the fraction 7 over 12")[$frac(7, 12)$]. ] ] === Solving Problems Involving Rational Numbers Rational numbers are used in many situations, sometimes to express a portion of a whole, other times as an expression of a ratio between two quantities. For the sciences, converting between units is done using rational numbers, as when converting between gallons and cubic inches. In chemistry, mixing a solution with a given concentration of a chemical per unit volume can be solved with rational numbers. In demographics, rational numbers are used to describe the distribution of the population. In dietetics, rational numbers are used to express the appropriate amount of a given ingredient to include in a recipe. As discussed, the application of rational numbers crosses many disciplines. #examplebox("Example 22")[Mixing Soil for Vegetables][ James is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he determines that #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$] of the soil can be topsoil, but #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$] needs to be peat moss and #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] has to be compost. To fill the raised garden bed with 60 cubic feet of soil, how much of each component does James need to use? #solutionbox[ In this example, we know the proportion of each component to mix, and we know the total amount of the mix we need. In this kind of situation, we need to determine the appropriate amount of each component to include in the mixture. For each component of the mixture, multiply 60 cubic feet, which is the total volume of the mixture we want, by the fraction required of the component. #strong[Step 1:] The required fraction of topsoil is #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$], so James needs #math.equation(block: false, alt: "60 times the fraction 2 over 5")[$60 × frac(2, 5)$] cubic feet of topsoil. Performing the multiplication, James needs #math.equation(block: false, alt: "60 times the fraction 2 over 5 equals the fraction 120 over 5 equals 24")[$60 × frac(2, 5) = frac(120, 5) = 24$] (found by treating the fraction as division, and 120 divided by 5 is 24) cubic feet of topsoil. #strong[Step 2:] The required fraction of peat moss is also #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$], so he also needs #math.equation(block: false, alt: "60 times the fraction 2 over 5")[$60 × frac(2, 5)$] cubic feet, or #math.equation(block: false, alt: "60 times the fraction 2 over 5 equals the fraction 120 over 5 equals 24")[$60 × frac(2, 5) = frac(120, 5) = 24$] cubic feet of peat moss. #strong[Step 3:] The required fraction of compost is #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$]. For the compost, he needs #math.equation(block: false, alt: "60 times the fraction 1 over 5 equals the fraction 60 over 5 equals 12")[$60 × frac(1, 5) = frac(60, 5) = 12$] cubic feet. ] ] #examplebox("Example 23")[Determining the Number of Specialty Pizzas][ At Bella’s Pizza, one-third of the pizzas that are ordered are one of their specialty varieties. If there are 273 pizzas ordered, how many were specialty pizzas? #solutionbox[ One-third of the whole are specialty pizzas, so we need one-third of 273, which gives #math.equation(block: false, alt: "the fraction 1 over 3 times 273 equals the fraction 273 over 3 equals 91")[$frac(1, 3) × 273 = frac(273, 3) = 91$], found by dividing 273 by 3. So, 91 of the pizzas that were ordered were specialty pizzas. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Finding_Fraction_of_Total")[Finding a Fraction of a Total] ] === Using Fractions to Convert Between Units A common application of fractions is called #strong[unit conversion], or #strong[converting units], which is the process of changing from the units used in making a measurement to different units of measurement. For instance, 1 inch is (approximately) equal to 2.54 cm. To convert between units, the two equivalent values are made into a fraction. To convert from the first type of unit to the second type, the fraction has the second unit as the numerator, and the first unit as the denominator. From the inches and centimeters example, to change from inches to centimeters, we use the fraction #math.equation(block: false, alt: "the fraction 2.54 cm over 1 in")[$frac(2.54 "cm", 1 "in")$]. If, on the other hand, we wanted to convert from centimeters to inches, we’d use the fraction #math.equation(block: false, alt: "the fraction 1 in over 2.54 cm")[$frac(1 "in", 2.54 "cm")$]. This fraction is multiplied by the number of units of the type you are converting #emph[from], which means the units of the denominator are the same as the units being multiplied. #examplebox("Example 24")[Converting Liters to Gallons][ It is known that 1 liter (L) is 0.264172 gallons (gal). Use this to convert 14 liters into gallons. #solutionbox[ We know that 1 liter = 0.264172 gal. Since we are converting from liters, when we create the fraction we use, make sure the liter part of the equivalence is in the denominator. So, to convert the 14 liters to gallons, we multiply 14 by #math.equation(block: false, alt: "the fraction 1 gal over 0.264172 gal / 1 liter")[$frac(1 "gal", 0.264172 "gal" / 1 "liter")$]. Notice the gallon part is in the numerator since we’re converting #emph[to] gallons, and the liter part is in the denominator since we are converting #emph[from] liters. Performing this and rounding to three decimal places, we find that 14 liters is #math.equation(block: false, alt: "14 liter times the fraction 0.264172 gal over 1 liter equals 3.69841 gal")[$14 #h(0.28em) "liter" × frac(0.264172 "gal", 1 "liter") = 3.69841 "gal"$]. ] ] #examplebox("Example 25")[Converting Centimeters to Inches][ It is known that 1 inch is 2.54 centimeters. Use this to convert 100 centimeters into inches. #solutionbox[ We know that 1 inch = 2.54 cm. Since we are converting from centimeters, when we create the fraction we use, make sure the centimeter part of the equivalence is in the denominator, #math.equation(block: false, alt: "the fraction 1 in over 2.54 cm")[$frac(1 #h(0.28em) "in", 2.54 #h(0.28em) "cm")$]. To convert the 100 cm to inches, multiply 100 by #math.equation(block: false, alt: "the fraction 1 in over 2.54 cm")[$frac(1 #h(0.28em) "in", 2.54 #h(0.28em) "cm")$]. Notice the inch part is in the numerator since we’re converting #emph[to] inches, and the centimeter part is in the denominator since we are converting #emph[from] centimeters. Performing this and rounding to three decimal places, we obtain #math.equation(block: false, alt: "100 cm times the fraction 1 in over 2.54 cm equals 39.370 in")[$100 #h(0.28em) "cm" × frac(1 #h(0.28em) "in", 2.54 #h(0.28em) "cm") = 39.370 #h(0.28em) "in"$]. This means 100 cm equals 39.370 in. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Converting_Units")[Converting Units] ] === Defining and Applying Percent A #strong[percent] is a specific rational number and is literally per 100. #math.equation(block: false, alt: "n")[$n$] percent, denoted #math.equation(block: false, alt: "n")[$n$]%, is the fraction #math.equation(block: false, alt: "the fraction n over 100")[$frac(n, 100)$]. #examplebox("Example 26")[Rewriting a Percentage as a Fraction][ Rewrite the following as fractions: + 31% + 93% #solutionbox[ + Using the definition and #math.equation(block: false, alt: "n equals 31")[$n = 31$], 31% in fraction form is #math.equation(block: false, alt: "the fraction 31 over 100")[$frac(31, 100)$]. + Using the definition and #math.equation(block: false, alt: "n equals 93")[$n = 93$], 93% in fraction form is #math.equation(block: false, alt: "the fraction 93 over 100")[$frac(93, 100)$]. ] ] #examplebox("Example 27")[Rewriting a Percentage as a Decimal][ Rewrite the following percentages in decimal form: + 54% + 83% #solutionbox[ + Using the definition and #math.equation(block: false, alt: "n equals 54")[$n = 54$], 54% in fraction form is #math.equation(block: false, alt: "the fraction 54 over 100")[$frac(54, 100)$]. Dividing a number by 100 moves the decimal two places to the left; 54% in decimal form is then 0.54. + Using the definition and #math.equation(block: false, alt: "n equals 83")[$n = 83$], 83% in fraction form is #math.equation(block: false, alt: "the fraction 83 over 100")[$frac(83, 100)$]. Dividing a number by 100 moves the decimal two places to the left; 83% in decimal form is then 0.83. ] ] You should notice that you can simply move the decimal two places to the left without using the fractional definition of percent. Percent is used to indicate a fraction of a total. If we want to find 30% of 90, we would perform a multiplication, with 30% written in either decimal form or fractional form. The 90 is the #strong[total], 30 is the #strong[percentage], and 27 (which is #math.equation(block: false, alt: "0.30 times 90")[$0.30 × 90$]) is the #strong[percentage of the total]. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #math.equation(block: false, alt: "n %")[$n %$] of #math.equation(block: false, alt: "x")[$x$] items is #math.equation(block: false, alt: "the fraction n over 100 times x")[$frac(n, 100) × x$]. The #math.equation(block: false, alt: "x")[$x$] is referred to as the #strong[total], the #math.equation(block: false, alt: "n")[$n$] is referred to as the #strong[percent] or #strong[percentage], and the value obtained from #math.equation(block: false, alt: "the fraction n over 100 times x")[$frac(n, 100) × x$] is the #strong[part] of the total and is also referred to as the #strong[percentage of the total]. ] #examplebox("Example 28")[Finding a Percentage of a Total][ + Determine 40% of 300. + Determine 64% of 190. #solutionbox[ + The total is 300, and the percentage is 40. Using the decimal form of 40% and multiplying we obtain #math.equation(block: false, alt: "0.40 times 300 equals 120")[$0.40 × 300 = 120$]. + The total is 190, and the percentage is 64. Using the decimal form of 64% and multiplying we obtain #math.equation(block: false, alt: "0.64 times 190 equals 121.6")[$0.64 × 190 = 121.6$]. ] ] In the previous situation, we knew the total and we found the percentage of the total. It may be that we know the percentage of the total, and we know the percent, but we don't know the total. To find the total if we know the percentage of the total, use the following formula. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ If we know that #math.equation(block: false, alt: "n")[$n$]% of the total is #math.equation(block: false, alt: "x")[$x$], then the total is #math.equation(block: false, alt: "the fraction 100 times x over n")[$frac(100 × x, n)$]. ] #examplebox("Example 29")[Finding the Total When the Percentage and Percentage of the Total Are Known][ + What is the total if 28% of the total is 140? + What is the total if 6% of the total is 91? #solutionbox[ + 28 is the percentage, so #math.equation(block: false, alt: "n equals 28")[$n = 28$]. 28% of the total is 140, so #math.equation(block: false, alt: "x equals 140")[$x = 140$]. Using those we find that the total was #math.equation(block: false, alt: "the fraction 100 times 140 over 28 equals 500")[$frac(100 × "140", "28") = 500$]. + 6 is the percentage, so #math.equation(block: false, alt: "n equals 6")[$n = 6$]. 6% of the total is 91, so #math.equation(block: false, alt: "x equals 91")[$x = 91$]. Using those we find that the total was #math.equation(block: false, alt: "the fraction 100 times 91 over 6 equals 1,516.6")[$frac(100 × "91", "6") = 1,516.6$]. ] ] The percentage can be found if the total and the percentage of the total is known. If you know the total, and the percentage of the total, first divide the part by the total. Move the decimal two places to the right and append the symbol %. The percentage may be found using the following formula. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The percentage, #math.equation(block: false, alt: "n")[$n$], of #math.equation(block: false, alt: "b")[$b$] that is #math.equation(block: false, alt: "a")[$a$] is #math.equation(block: false, alt: "the fraction a over b times 100 %")[$frac(a, b) × 100 %$]. ] #examplebox("Example 30")[Finding the Percentage When the Total and Percentage of the Total Are Known][ Find the percentage in the following: + Total is 300, percentage of the total is 60. + Total is 440, percentage of the total is 176. #solutionbox[ + The total is 300; the percentage of the total is 60. Calculating yields 0.2. Moving the decimal two places to the right gives 20. Appending the percentage to this number results in 20%. So, 60 is 20% of 300. + The total is 440; the percentage of the total is 176. Calculating yields 0.4. Moving the decimal two places to the right gives 40. Appending the percentage to this number results in 40%. So, 176 is 40% of 440. ] ] === Solve Problems Using Percent In the media, in research, and in casual conversation percentages are used frequently to express proportions. Understanding how to use percent is vital to consuming media and understanding numbers. Solving problems using percentages comes down to identifying which of the three components of a percentage you are given, the total, the percentage, or the percentage of the total. If you have two of those components, you can find the third using the methods outlined previously. #examplebox("Example 31")[Percentage of Students Who Are Sleep Deprived][ A study revealed that 70% of students suffer from sleep deprivation, defined to be sleeping less than 8 hours per night. If the survey had 400 participants, how many of those participants had less than 8 hours of sleep per night? #solutionbox[ The percentage of interest is 70%. The total number of students is 400. With that, we can find how many were in the percentage of the total, or, how many were sleep deprived. Applying the formula from above, the number who were sleep deprived was #math.equation(block: false, alt: "0.70 times 400 equals 280")[$0.70 × 400 = 280$]; 280 students on the study were sleep deprived. ] ] #examplebox("Example 32")[Amazon Prime Subscribers][ There are 126 million users who are U.S. Amazon Prime subscribers. If there are 328.2 million residents in the United States, what percentage of U.S. residents are Amazon Prime subscribers? #solutionbox[ We are asked to find the percentage. To do so, we divide the percentage of the total, which is 126 million, by the total, which is 328.2 million. Performing this division and rounding to three decimal places yields #math.equation(block: false, alt: "the fraction 126 over 328.2 equals 0.384")[$frac(126, 328.2) = 0.384$]. The decimal is moved to the right by two places, and a % sign is appended to the end. Doing this shows us that 38.4% of U.S. residents are Amazon Prime subscribers. ] ] #examplebox("Example 33")[Finding the Percentage When the Total and Percentage of the Total Are Known][ Evander plays on the basketball team at their university and 73% of the athletes at their university receive some sort of scholarship for attending. If they know 219 of the student-athletes receive some sort of scholarship, how many student-athletes are at the university? #solutionbox[ We need to find the total number of student-athletes at Evander’s university. #strong[Step 1:] Identify what we know. We know the percentage of students who receive some sort of scholarship, 73%. We also know the number of athletes that form the part of the whole, or 219 student-athletes. #strong[Step 2:] To find the total number of student-athletes, use #math.equation(block: false, alt: "the fraction 100 times x over n")[$frac(100 #h(0.28em) × #h(0.28em) x, n)$], with #math.equation(block: false, alt: "x equals 219")[$x = 219$] and #math.equation(block: false, alt: "n equals 73")[$n = 73$]. Calculating with those values yields #math.equation(block: false, alt: "the fraction 100 times 219 over 73 equals 300")[$frac(100 #h(0.28em) × #h(0.28em) 219, 73) = 300$]. So, there are 300 total student-athletes at Evander’s university ] ] === Key Terms - density property of rational numbers - improper fraction - lowest terms - mixed number - rational number - repeating decimal - terminating decimal === Key Concepts - Rational numbers are fractions of integers, and can always be written as an integer divided by an integer. - The numerator and denominator of a fraction may have common factors. In such cases, the fraction can be reduced by canceling common factors. When the numerator and denominator of a fraction have no common factors, the fraction is said to be reduced. - An improper fraction is one with a numerator larger than the denominator. Such a fraction can be rewritten as an integer plus a proper fraction. This is called a mixed number. - Using division and remainder, an improper fraction may be written as a mixed number. - A mixed number can be converted to an improper fraction by reversing the process for changing an improper fraction to a mixed number. - The arithmetic operations or addition, subtraction, multiplication and division can all be performed on rational numbers. - Addition and subtraction of rational numbers can be performed after a common denominator has been identified, and the fractions have been converted to forms having the common denominator. - Multiplication and division of rational numbers can be performed without regard to common denominators. - Between any two rational numbers, there is always another rational number. This is the density property of the rational numbers. === Formulas - #math.equation(block: false, alt: "the fraction a over c plus or minus the fraction b over c equals the fraction a plus or minus b over c")[$frac(a, c) ± frac(b, c) = frac(a ± b, c)$] - #math.equation(block: false, alt: "the fraction a over b times the fraction c over d equals the fraction a times c over b times d")[$frac(a, b) × frac(c, d) = frac(a × c, b × d)$] - #math.equation(block: false, alt: "the fraction a over b times the fraction c over d equals the fraction a times c over b times d the fraction a over b divided by the fraction c over d equals the fraction a over d times the fraction d over c equals the fraction a times d over b times c")[$frac(a, b) × frac(c, d) = frac(a × c, b × d) frac(a, b) ÷ frac(c, d) = frac(a, d) × frac(d, c) = frac(a × d, b × c)$] === Videos - #link("https://openstax.org/r/Introduction_to_Fractions")[Introduction to Fractions] - #link("https://openstax.org/r/Equivalent_Fractions")[Equivalent Fractions] - #link("https://openstax.org/r/Reducing_Fractions_to_Lowest_Terms")[Reducing Fractions to Lowest Terms] - #link("https://openstax.org/r/Using_Desmos_to_Reduce_a_Fraction")[Using Desmos to Reduce a Fraction] - #link("https://openstax.org/r/Adding_and_Subtracting_Fractions")[Adding and Subtracting Fractions with Different Denominators] - #link("https://openstax.org/r/Improper_Fraction_to_Mixed_Number")[Converting an Improper Fraction to a Mixed Number Using Desmos] - #link("https://openstax.org/r/Multiplying_Fractions")[Multiplying Fractions] - #link("https://openstax.org/r/Dividing_Fractions")[Dividing Fractions] - #link("https://openstax.org/r/Operations_Using_Fractions")[Order of Operations Using Fractions] - #link("https://openstax.org/r/Finding_Fraction_of_Total")[Finding a Fraction of a Total] - #link("https://openstax.org/r/Converting_Units")[Converting Units]