#set document(title: "6.1 Understanding Percent", 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")) == 6.1#h(0.6em)Understanding Percent #figure(figph[The federal budget in the fiscal year 2021. The two circled flow chart shows total outlays and total revenues. The total outlay is \$6.8 trillion and 30.5 percent of the G D P. Three categories are labeled mandatory, discretionary, and net interest, and the G D P percentages are 21.6, 7.3, and 1.6 respectively. The total revenue is \$4.0 trillion and 18.1 percent of the G D P. Four categories are labeled individual income taxes, payroll taxes, corporate income taxes, and other, and the G D P percentages are 9.1, 5.9, 1.7, and 1.4 respectively.], alt: "The federal budget in the fiscal year 2021. The two circled flow chart shows total outlays and total revenues. The total outlay is $6.8 trillion and 30.5 percent of the G D P. Three categories are labeled mandatory, discretionary, and net interest, and the G D P percentages are 21.6, 7.3, and 1.6 respectively. The total revenue is $4.0 trillion and 18.1 percent of the G D P. Four categories are labeled individual income taxes, payroll taxes, corporate income taxes, and other, and the G D P percentages are 9.1, 5.9, 1.7, and 1.4 respectively.", caption: [The federal budget describes how money is spent and how money is earned.]) === Learning Objectives After completing this section, you should be able to: + Define and calculate percent. + Convert between percent, decimal, and fractional values. + Calculate the total, percent, or part. + Solve application problems involving percents. In 2020, the U.S. federal government #link("https://openstax.org/r/National_Park_Service")[budgeted \$3.5 billion for the National Park Service], which appears to be a very large number (and is!) and a large portion of the total federal budget. However, the #link("https://openstax.org/r/U.S._federal_government")[total outlays from the U.S. federal government in 2020 was \$6.6 trillion]. So, the amount budgeted for the National Park Service was less than one-tenth of 1 percent, or 1/10%, of the total outlays. This #strong[percent] describes a specific number. Understanding that ratio puts the \$3.5 billion budgeted to the National Park Service in perspective. This chapter focuses on percent as a primary tool for understanding money management. The interest paid on debt, the interest earned through investments, and even taxes are entirely determined using percent. This section introduces the basics of working with this invaluable tool. === Define and Calculate Percent The word #strong[percent] comes from the Latin phrase per centum, which means “by the hundred.” So any percent is a number divided by 100. Changing a percent to a fraction is to write the percent in its #strong[fractional form]. To write #math.equation(block: false, alt: "n")[$n$]% in its fractional form is to write the percent as the fraction #math.equation(block: false, alt: "the fraction n over 100")[$frac(n, 100)$]. #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #emph[A percent need not be an integer and does not have to be less than 100.] ] #examplebox("Example 1")[Rewriting a Percent as a Fraction][ Rewrite the following as fractions: + 18% + 84% + 38.7% + 213% #solutionbox[ + Using the definition and #math.equation(block: false, alt: "n")[$n$] = 18, 18% in fractional form is #math.equation(block: false, alt: "the fraction 18 over 100")[$frac(18, 100)$]. + Using the definition and #math.equation(block: false, alt: "n")[$n$] = 84, 84% in fractional form is #math.equation(block: false, alt: "the fraction 84 over 100")[$frac(84, 100)$]. + Using the definition and #math.equation(block: false, alt: "n")[$n$] = 38.7, 38.7% in fractional form is #math.equation(block: false, alt: "the fraction 38.7 over 100")[$frac(38.7, 100)$]. + Using the definition and #math.equation(block: false, alt: "n")[$n$] = 213, 213% in fractional form is #math.equation(block: false, alt: "the fraction 213 over 100")[$frac(213, 100)$]. ] ] === Convert Between Percent, Decimal, and Fractional Values When any calculation with a percent is to be performed, the form of the percent must be changed, either to its fractional form or its #strong[decimal form]. We can change a percent into decimal form by dividing the percent by 100 and representing the result as a decimal. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The decimal form of #math.equation(block: false, alt: "n")[$n$]% is found by calculating the decimal value of #math.equation(block: false, alt: "n divided by 100")[$n ÷ 100$]. ] #examplebox("Example 2")[Converting a Percent to Decimal Form][ Convert the following percents to decimal form: + 17% + 7% + 18.45% #solutionbox[ + To convert 17% to its decimal form, divide 17 by 100. This moves the decimal two places to the left, resulting in 0.17. The decimal form of 17% is 0.17. + To convert 7% to its decimal form, divide 7 by 100. This moves the decimal two places to the left, resulting in 0.07. The decimal form of 7% is 0.07. + To convert 18.45% to its decimal form, divide 18.45 by 100. This moves the decimal two places to the left, resulting in 0.1845. The decimal form of 18.45% is 0.1845. ] ] You should notice that, to convert from percent to decimal form, you can simply move the decimal two places to the left without performing the division. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ To convert the number #math.equation(block: false, alt: "x")[$x$] from decimal form to percent, multiply #math.equation(block: false, alt: "x")[$x$] by 100 and place a percent sign, %, after the number, #math.equation(block: false, alt: "open parenthesis x times 100 close parenthesis %")[$( x × 100 ) %$]. ] #examplebox("Example 3")[Converting the Decimal Form of a Percent to Percent][ Convert each of the following to percent: + 0.34 + 4.15 + 0.0391 #solutionbox[ + Using the formula and #math.equation(block: false, alt: "x")[$x$] = 0.34, we calculate #math.equation(block: false, alt: "open parenthesis 0.34 times 100 close parenthesis %")[$( 0.34 × 100 ) %$], which gives us 34%. + Using the formula and #math.equation(block: false, alt: "x")[$x$] = 4.15, we calculate #math.equation(block: false, alt: "open parenthesis 4.15 times 100 close parenthesis %")[$( 4.15 × 100 ) %$], which gives us 415%. + Using the formula and #math.equation(block: false, alt: "x")[$x$] = 0.0391, we calculate #math.equation(block: false, alt: "open parenthesis 0.0391 times 100 close parenthesis %")[$( 0.0391 × 100 ) %$], which gives us 3.91%. ] ] You should notice that, to convert from decimal form to percent form, you can simply move the decimal two places to the right without performing the multiplication. === Calculate the Total, Percent, or Part The word “of” is used to indicate multiplication using fractions, as in “one-fourth of 56.” To find “one-fourth of 56” we would multiply 56 by one-fourth. We can think of percents as fractions with a specific denominator—100. So, to calculate “25% of 52,” we multiply 52 by 25%. But, first we need to convert the percent to either fractional form (25/100) or decimal form. Using the decimal form of 25% we have 0.25 × 52, which equals 13. In this problem, 52 is the #strong[total] or #strong[base], 25 is the #strong[percentage], and 13 is #strong[the percentage of] 52, or the #strong[part] of 52. This is sometimes referred to as the #strong[amount]. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The mathematical formula relating the total (base), the percent in decimal form, and the part (amount) is #math.equation(block: false, alt: "part equals percent times total")[$italic("part") = italic("percent") × italic("total")$], or, #math.equation(block: false, alt: "amount equals percent times base")[$italic("amount") = italic("percent") × italic("base")$]. ] #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #emph[In all calculations, the percent is expressed in decimal form.] ] Knowing any two of the values in our formula allows us to calculate the third value. In the following example, we know the total and the percent, and are asked to find the percentage of the total. #examplebox("Example 4")[Finding the Percent of a Total][ + Determine 70% of 3,500 + Determine 156% of 720 #solutionbox[ + The total is #math.equation(block: false, alt: "x")[$x$] = 3,500, and the percent is #math.equation(block: false, alt: "n")[$n$] = 70. The decimal form of 70% is 0.70. To find the part, or percent of the total, substitute those values into the formula and calculate. #math.equation(block: true, alt: "part, equals, percent times total; equals, 0.70 times 3500; equals, 2450")[$"part" & = & "percent" × "total" \ & = & 0.70 × 3500 \ & = & 2450$] From this, we say that 70% of 3,500 is 2,450. + The total is #math.equation(block: false, alt: "x")[$x$] = 720, and the percent is #math.equation(block: false, alt: "n")[$n$] = 156. The decimal form of 156% is 1.56. To find the part, or percent of the total, substitute those values into the formula and calculate. #math.equation(block: true, alt: "part, equals, percent times total; equals, 1.56 times 720; equals, 1,123.2")[$"part" & = & "percent" × "total" \ & = & 1.56 × 720 \ & = & 1,123.2$] From this, we say that 156% of 720 is 1,123.2. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/solve_percent_problem1")[Finding Percent of a Total] ] In the previous example, we knew the total and the percent and found the part using our formula. We may instead know the percent and the part, but not the total. We can use our formula again to solve for the total. #examplebox("Example 5")[Finding the Total from the Percent and the Part][ + What is the total if 35% of the total is 70? + What is the total if 10% of the total is 4,000? #solutionbox[ + #strong[Step 1:] The percent is 35, which in decimal form is 0.35. We were given that 35% of the total is 70, so the part is 70. We are to find the total. Substituting into the formula, we have #math.equation(block: true, alt: "part, equals, percent times total; 70, equals, 0.35 times total")[$"part" & = & "percent" × "total" \ 70 & = & 0.35 × "total"$] #linebreak() #strong[Step 2:] To find the total, we solve the equation for the total. #math.equation(block: true, alt: "70, equals, 0.35 times total; the fraction 70 over 0.35, equals, the fraction 0.35 times total over 0.35; 200, equals, the fraction 0.35 times total over 0.35; 200, equals, total")[$70 & = & 0.35 × "total" \ frac(70, 0.35) & = & frac(0.35 × "total", 0.35) \ 200 & = & frac(cancel(0.35) × "total", cancel(0.35)) \ 200 & = & "total"$] #linebreak() From this we see that 200 is the total, or, that 35% of 200 is 70. #linebreak() + #strong[Step 1:] The percent is 10, which in decimal form is 0.1. We were given that 10% of the total is 4,000, so the part is 4,000. Substituting into the formula, we have #math.equation(block: true, alt: "part, equals, percent times total; 4,000, equals, 0.1 times total")[$"part" & = & "percent" × "total" \ 4,000 & = & 0.1 × "total"$] #linebreak() #strong[Step 2:] To find the total, we solve the equation for the total. #math.equation(block: true, alt: "4,000, equals, 0.1 times total; the fraction 4,000 over 0.1, equals, the fraction 0.1 times total over 0.1; 40,000, equals, the fraction 0.1 times total over 0.1; 40,000, equals, total")[$4,000 & = & 0.1 × "total" \ frac(4","000, 0.1) & = & frac(0.1 × "total", 0.1) \ 40,000 & = & frac(cancel(0.1) × "total", cancel(0.1)) \ 40,000 & = & "total"$] #linebreak() From this we see that 40,000 is the total, or that 10% of 40,000 is 4,000. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/solve_percent_problem2")[Finding the Total from the Percent and the Part] ] Similarly, the percent can be found if the total and the percent of the total (the part) are known. This will result in the decimal form of the percent, so it must be converted to percent form. #examplebox("Example 6")[Finding the Percent from the Total and the Part][ + What percent of 500 is 175? + What percent of 228 is 155? #solutionbox[ + #strong[Step 1:] The total is 500, the percent of the total is 175. Substituting into the formula, we have #math.equation(block: true, alt: "part, equals, percent times total; 175, equals, percent times 500")[$"part" & = & "percent" × "total" \ 175 & = & "percent" × 500$] #linebreak() #strong[Step 2:] To find the percent, we solve the equation for the percent. #math.equation(block: true, alt: "175, equals, percent times 500; the fraction 175 over 500, equals, the fraction percent times 500 over 500; 0.35, equals, the fraction percent times 500 over 500; 0.35, equals, percent")[$175 & = & "percent" × 500 \ frac(175, 500) & = & frac("percent" × 500, 500) \ 0.35 & = & frac("percent" × cancel(500), cancel(500)) \ 0.35 & = & "percent"$] #linebreak() We see the percent in decimal form is 0.35. Converting from the decimal form yields 35%. We say that 175 is 35% of 500. #linebreak() + #strong[Step 1:] The total is 228, the percent of the total is 155. Substituting into the formula, we have #math.equation(block: true, alt: "part, equals, percent times total; 155, equals, percent times 228")[$"part" & = & "percent" × "total" \ 155 & = & "percent" × 228$] #linebreak() #strong[Step 2:] To find the percent, we solve the equation for the percent. #math.equation(block: true, alt: "155, equals, percent times 228; the fraction 155 over 228, equals, the fraction percent times 228 over 228; 0.6798, equals, the fraction percent times 228 over 228; 0.6798, equals, percent")[$155 & = & "percent" × 228 \ frac(155, 228) & = & frac("percent" × 228, 228) \ 0.6798 & = & frac("percent" × cancel(228), cancel(228)) \ 0.6798 & = & "percent"$] #linebreak() We see the percent is 0.6798 (rounded to four decimal places). Converting from the decimal form yields 67.98%. We say that 155 is 67.98% of 228. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/solve_percent_problem3")[Finding the Percent When the Total and the Part Are Known] ] === Solve Application Problems Involving Percents Percents are frequently used in finance, research, science experiments, and even casual conversation. Understanding these types of values helps when consuming media or discussing finances, for instance. Effectively working with and interpreting numbers and percents will help you become an informed consumer of this information. In most cases, working through what is presented requires you to identify that you are indeed working with a question of percents, which two of the three values that are related through percents are known, and which of the three values you need to find. #examplebox("Example 7")[Retention Rate at College][ Justine applies to a medium size university outside her hometown and finds out that the retention rate (percent of students who return for their sophomore year) for the 2021 academic year at the university was 84%. During a visit to the registrar’s office, she finds out that 1,350 people had enrolled in academic year 2021. How many students from the academic year 2021 are returning for the 2022 academic year? #solutionbox[ The percent of students who will return for the 2022 academic year (the retention rate) is 84%. The total number of students who enrolled in the 2021 academic year was 1,350. This means the percent is known and the total is known. From this, we can determine the number of students who will return (percent of the total) for the 2022 academic year using the formula #math.equation(block: false, alt: "part equals percent times total")[$"part" = "percent" × "total"$]. Substituting into the formula and calculating, we find that the number of students that are returning is #math.equation(block: true, alt: "part, equals, percent times total; equals, 0.84 times 1,350; equals, 1,134")[$"part" & = & "percent" × "total" \ & = & 0.84 × 1,350 \ & = & 1,134$] So 1,134 students will return for the 2022 academic year. ] ] #examplebox("Example 8")[Percent of Chemistry Majors][ Cameron enrolls in a calculus class. In this class of 45 students, there are 18 chemistry majors. What percent of the class are chemistry majors? #solutionbox[ In this situation, the percent is to be determined. We know the total number of students, 45, and the part of the students that are chemistry majors, 18. Using that information and the formula #math.equation(block: false, alt: "part equals percent times total")[$"part" = "percent" × "total"$], the percent can be found. Substituting and solving, we have #math.equation(block: true, alt: "18, equals, percent times 45; the fraction 18 over 45, equals, the fraction percent times 45 over 45; 0.4, equals, the fraction percent times 45 over 45; 0.4, equals, percent")[$18 & = & "percent" × 45 \ frac(18, 45) & = & frac("percent" × 45, 45) \ 0.4 & = & frac("percent" × cancel(45), cancel(45)) \ 0.4 & = & "percent"$] Converting the 0.4 from decimal form, we find that 40% of the students in the calculus class are chemistry majors. ] ] #examplebox("Example 9")[Total Sales and Commission][ Mariel makes a 20% commission on every sale she makes. One week, her commission check is for \$153.00. What were her total sales that week? #solutionbox[ In this problem, Mariel’s total sales is to be determined. We know the percent she earns is 20%. We also know that her sales commission was \$153.00, which is the percent of the total. Using this information and the formula #math.equation(block: false, alt: "part equals percent times total")[$"part" = "percent" × "total"$] we can find Mariel’s total sales. The decimal form of 20% is 0.2. The part, or percent of the total, is 153. Substituting and solving, we obtain #math.equation(block: true, alt: "part, equals, percent times total; 153, equals, 0.2 times total; the fraction 153 over 0.2, equals, the fraction 0.2 times total over 0.2; 765, equals, the fraction 0.2 times total over 0.2; 765, equals, total")[$"part" & = & "percent" × "total" \ 153 & = & 0.2 × "total" \ frac(153, 0.2) & = & frac(0.2 × "total", 0.2) \ 765 & = & frac(cancel(0.2) × "total", cancel(0.2)) \ 765 & = & "total"$] Mariel’s total sales were \$765.00. ] ] #notebox("Who Knew?", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[LED Lightbulbs] According to the #link("https://openstax.org/r/U.S._government")[energy website from the U.S. government], LED lightbulbs use at least 75% less energy than incandescent bulbs. They also last up to 25 times as long as an incandescent bulb. If lighting is a significant percent of your electrical use, replacing all incandescent bulbs with LED bulbs will significantly reduce your electric bill. ] For any answer, round to two decimal places, if necessary. === Key Terms - Percent - Fractional form - Decimal form - Total - Base - Percent of the total - Part - Amount === Key Concepts - Know what a percent is as a fraction, a decimal, and as a part of the whole. - Use the percent equation to find any of the three values that are related by the equation. - Apply the percent equation in applications. === Video - #link("https://openstax.org/r/solve_percent_problem1")[Finding Percent of a Total] - #link("https://openstax.org/r/solve_percent_problem2")[Finding the Total from the Percent and the Part] - #link("https://openstax.org/r/solve_percent_problem3")[Finding the Percent When the Total and the Part Are Known] === Formulas part = percent x total