{"id":8304,"date":"2023-09-29T14:32:57","date_gmt":"2023-09-29T14:32:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8304"},"modified":"2024-10-18T20:58:40","modified_gmt":"2024-10-18T20:58:40","slug":"apportionment-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/apportionment-fresh-take\/","title":{"raw":"Apportionment: Fresh Take","rendered":"Apportionment: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Define apportionment and its importance in political representation in the United States and other representative democracies<\/li>\r\n\t<li>Identify various methods of apportionment, including Hamilton's method, Jefferson's method, Webster's method, Huntington-Hill method, and Lowndes' method<\/li>\r\n\t<li>Understand the apportionment process for legislative districts<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Apportionment<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Apportionment:<\/strong> The process of fairly dividing a fixed number of resources (like seats in a legislative body) among groups of varying sizes.<\/p>\r\n<p><strong>Political Importance:<\/strong> Apportionment in politics ensures fair representation by allocating elected officials based on area and population.<\/p>\r\n<p><strong>Constraints:<\/strong> Whole numbers only, must use all resources, each group gets at least one, and division should be proportional to population size.<\/p>\r\n\r\n\r\nBelow are important things to consider when thinking about apportionment.\r\n\r\n\r\n<ul>\r\n\t<li><strong>Zero-Sum Game:<\/strong> If one state gains a representative, another must lose one.<\/li>\r\n\t<li><strong>Minimum Allocation:<\/strong> Every state or group must receive at least one of the resources being divided.<\/li>\r\n\t<li><strong>Whole Numbers:<\/strong> Apportionment deals with whole numbers; you can't have [latex]3.4[\/latex] representatives.<\/li>\r\n\t<li><strong>Proportional Representation:<\/strong> While exact proportionality isn't possible due to the whole number constraint, strive for as close to proportional as possible.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>For more information on the basics of apportionment, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/w_0XwyXgJvk?si=ascUMuo7xe55L-Rt\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Introduction+to+Apportionment.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Apportionment\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Hamilton\u2019s Method<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Hamilton's Method:<\/strong> A step-by-step approach to apportionment, initially proposed by Alexander Hamilton and used in the U.S. from 1852 to 1911.<\/p>\r\n<p>The steps to applying Hamilton's method are:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>divisor<\/strong>.<\/li>\r\n\t<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>. Since we can only allocate whole representatives, Hamilton resolves the whole number problem, as follows:<\/li>\r\n\t<li>Cut off all the decimal parts of all the quotas (but don\u2019t forget what the decimals were). These are called the <strong>lower quotas<\/strong>. Add up the remaining whole numbers. This answer will always be less than or equal to the total number of representatives (and the \u201cor equal to\u201d part happens only in very specific circumstances that are incredibly unlikely to turn up).<\/li>\r\n\t<li>Assuming that the total from Step 3 was less than the total number of representatives, assign the remaining representatives, one each, to the states whose decimal parts of the quota were largest, until the desired total is reached.<\/li>\r\n<\/ol>\r\n<p><strong>Quota Rule:<\/strong> A guideline suggesting that the final number of representatives should be within one of the state's quota.<\/p>\r\n<p><strong>Paradoxes:<\/strong> Hamilton's method is subject to several paradoxes, such as the Alabama Paradox, the New States Paradox, and the Population Paradox.<\/p>\r\n\r\n\r\nTerms to remember:\r\n\r\n\r\n<ul>\r\n\t<li><strong>Divisor:<\/strong> Calculated by dividing the total population by the total number of representatives. It's the baseline for apportionment.<\/li>\r\n\t<li><strong>Quota:<\/strong> Each state's population divided by the divisor, kept to several decimal places.<\/li>\r\n\t<li><strong>Lower Quotas:<\/strong> The whole number parts of the quotas. Summing these will always be less than or equal to the total number of representatives.<\/li>\r\n\t<li><strong>Quota Rule:<\/strong> Aim for the final number of representatives to be within one of the state's quota for fairness.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Use Hamilton\u2019s method to apportion the [latex]75[\/latex] seats of Rhode Island\u2019s House of Representatives among its five counties.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } &amp; \\text { Population }\\\\ \\hline \\text { Bristol } &amp; 49,875\\\\ \\text { Kent } &amp; 166,158\\\\ \\text { Newport } &amp; 82,888\\\\ \\text { Providence } &amp; 626,667\\\\ \\text { Washington } &amp; 126,979\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 }\\end{array} [\/latex]<\/p>\r\n\r\n\r\n[reveal-answer q=\"4331\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4331\"]\r\n\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>\r\n<p>First, we determine the divisor:<\/p>\r\n<p style=\"text-align: center;\">[latex]1,052,567\/75 = 14,034.22667[\/latex]<\/p>\r\n<\/li>\r\n\t<li>\r\n<p>Now we determine each county\u2019s quota by dividing the county\u2019s population by the divisor:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrr} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota }\\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538\\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395\\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061\\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528\\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; \\end{array} [\/latex]<\/p>\r\n<\/li>\r\n\t<li>\r\n<p>Removing the decimal parts of the quotas gives:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial }\\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538 &amp; 3 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395 &amp; 11 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061 &amp; 5 \\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528 &amp; 44\\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 72 }\\end{array} [\/latex]<\/p>\r\n<\/li>\r\n\t<li>\r\n<p>We need [latex]75[\/latex] representatives and we only have [latex]72[\/latex], so we assign the remaining three, one each, to the three counties with the largest decimal parts, which are Newport, Kent, and Providence:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrcc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial } &amp; \\text{ Final } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538 &amp; 3 &amp; 3 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395 &amp; 11 &amp; 12 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061 &amp; 5 &amp; 6 \\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528 &amp; 44 &amp; 45 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478 &amp; 9 &amp; 9 \\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 72 } &amp; \\bf{ 75 }\\end{array} [\/latex]<\/p>\r\n<p>Note that even though Bristol County\u2019s decimal part is greater than [latex].5[\/latex], it isn\u2019t big enough to get an additional representative, because three other counties have greater decimal parts.<\/p>\r\n<\/li>\r\n<\/ol>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>For more information on Hamilton's method, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/YWfEqWLz9pc?si=fsVWIdGG1J8fMJiE\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Hamilton's+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Hamilton's Method\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Jefferson\u2019s Method<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Jefferson\u2019s Method:<\/strong> An alternative to Hamilton's method, used in Congress from 1791 through 1842, which favors larger states.<\/p>\r\n<p>The steps to applying Jefferson\u2019s method are:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>standard divisor<\/strong>.<\/li>\r\n\t<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>.<\/li>\r\n\t<li>Cut off all the decimal parts of all the quotas (but don\u2019t forget what the decimals were). These are the <strong>lower quotas<\/strong>. Add up the remaining whole numbers. This answer will always be less than or equal to the total number of representatives.<\/li>\r\n\t<li>If the total from Step 3 was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. Continue doing this until the total in Step 3 is equal to the total number of representatives. The divisor we end up using is called the <strong>modified divisor<\/strong> or <strong>adjusted divisor<\/strong>.<\/li>\r\n<\/ol>\r\n<p><strong>Divisor-Adjusting Methods:<\/strong> Jefferson's method is one such method, and it's not guaranteed to follow the quota rule.<\/p>\r\n\r\n\r\nTerms to remember:\r\n\r\n\r\n<ul>\r\n\t<li><strong>Standard Divisor:<\/strong> The total population of all states divided by the total number of representatives. Think of it as the \"benchmark\" for how many people each representative should represent.<\/li>\r\n\t<li><strong>Quota:<\/strong> Each state's population divided by the standard divisor, recorded to several decimal places. It's the \"ideal\" number of representatives for each state, but it's not the final answer.<\/li>\r\n\t<li><strong>Modified Divisor:<\/strong> The divisor that is adjusted until the total number of representatives matches the required total. If the initial total number of representatives is less than required, reduce the divisor and recalculate until they match.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>We\u2019ll apply Jefferson\u2019s method for Rhode Island. Recall, the Rhode Island\u2019s House of Representatives splits [latex]75[\/latex] seats among its five counties.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } &amp; \\text { Population }\\\\ \\hline \\text { Bristol } &amp; 49,875\\\\ \\text { Kent } &amp; 166,158\\\\ \\text { Newport } &amp; 82,888\\\\ \\text { Providence } &amp; 626,667\\\\ \\text { Washington } &amp; 126,979\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 }\\end{array}[\/latex]<\/p>\r\n\r\n\r\n[reveal-answer q=\"4332\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4332\"]\r\n\r\n\r\n<p>The original divisor of [latex]14,034.22667[\/latex] gave these results:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538 &amp; 3 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395 &amp; 11 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061 &amp; 5 \\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528 &amp; 44 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 72 }\\end{array} [\/latex]<\/p>\r\n<p>We need [latex]75[\/latex] representatives and we only have [latex]72[\/latex], so we need to use a smaller divisor. Let\u2019s try [latex]13,500[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.6944 &amp; 3 \\\\ \\text { Kent } &amp; 166,158 &amp; 12.3080 &amp; 12 \\\\ \\text { Newport } &amp; 82,888 &amp; 6.1399 &amp; 6 \\\\ \\text { Providence } &amp; 626,667 &amp; 46.4198 &amp; 46 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.4059 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 76 }\\end{array} [\/latex]<\/p>\r\n<p>We\u2019ve gone too far. We need a divisor that\u2019s greater than [latex]13,500[\/latex] but less than [latex]14,034.22667[\/latex]. Let\u2019s try [latex]13,700[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.6405 &amp; 3 \\\\ \\text { Kent } &amp; 166,158 &amp; 12.1283 &amp; 12 \\\\ \\text { Newport } &amp; 82,888 &amp; 6.0502 &amp; 6 \\\\ \\text { Providence } &amp; 626,667 &amp; 45.7421 &amp; 45 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.2685 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 75 }\\end{array} [\/latex]<\/p>\r\n<p>This works.<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>Notice, in comparison to Hamilton\u2019s method, that although the results were the same, they came about in a different way, and the outcome was almost different. Providence County (the largest) almost went up to [latex]46[\/latex] representatives before Kent (which is much smaller) got to [latex]12[\/latex]. Although that didn\u2019t happen here, it can. Divisor-adjusting methods like Jefferson\u2019s are not guaranteed to follow the quota rule! For more information on Jefferson\u2019s method, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/weGGVmy9yLc?si=dakuzc07blcB7dd_\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Jefferson's+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Jefferson's Method\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Webster\u2019s Method<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Webster\u2019s Method:<\/strong> A method similar to Jefferson's but rounds quotas to the nearest whole number. It was adopted by Congress multiple times and is less biased towards larger states.<\/p>\r\n<p>The steps to applying Webster\u2019s method are:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>standard divisor<\/strong>.<\/li>\r\n\t<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>.<\/li>\r\n\t<li>Round all the quotas to the nearest whole number (but don\u2019t forget what the decimals were). Add up the remaining whole numbers.<\/li>\r\n\t<li>If the total from Step 3 was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. If the total from step 3 was larger than the total number of representatives, increase the divisor and recalculate the quota and allocation. Continue doing this until the total in Step 3 is equal to the total number of representatives. The divisor we end up using is called the <strong>modified divisor<\/strong> or <strong>adjusted divisor<\/strong>.<\/li>\r\n<\/ol>\r\n<p><strong>Balinski-Young Impossibility Theorem:<\/strong> A theorem that states no apportionment method can always follow the quota rule without being subject to paradoxes. Understand that no method is perfect; each has its own set of trade-offs and potential paradoxes.<\/p>\r\n\r\n\r\nTerms to remember:\r\n\r\n\r\n<ul>\r\n\t<li><strong>Standard Divisor:<\/strong> The total population divided by the total number of representatives, serving as the benchmark for representation. It's your \"starting point\" for figuring out how many people each representative should ideally represent.<\/li>\r\n\t<li><strong>Quota:<\/strong> The result of dividing each state's population by the standard divisor, rounded to several decimal places. It's like your \"first draft\" of how many representatives each state should have. Don't forget the decimals; they matter!<\/li>\r\n\t<li><strong>Modified Divisor:<\/strong> The adjusted divisor used when the total number of representatives doesn't match the required total. If your total number of representatives is off, tweak this number and recalculate.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>We\u2019ll apply Webster\u2019s method for Rhode Island. Recall, the Rhode Island\u2019s House of Representatives splits [latex]75[\/latex] seats among its five counties.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } &amp; \\text { Population }\\\\ \\hline \\text { Bristol } &amp; 49,875\\\\ \\text { Kent } &amp; 166,158\\\\ \\text { Newport } &amp; 82,888\\\\ \\text { Providence } &amp; 626,667\\\\ \\text { Washington } &amp; 126,979\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 }\\end{array}[\/latex] [reveal-answer q=\"4333\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4333\"]<\/p>\r\n<p>The original divisor of [latex]14,034.22667[\/latex] gave these results:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061 \\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478 \\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; \\end{array} [\/latex]<\/p>\r\n<p>Rounding the quotas up we get:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538 &amp; 4 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395 &amp; 12 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061 &amp; 6 \\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528 &amp; 45 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 76 }\\end{array} [\/latex]<\/p>\r\n<p>This is too many, so we need to increase the divisor. Let\u2019s try [latex]14,100[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5372 &amp; 4 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.7843 &amp; 12 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.8786 &amp; 6 \\\\ \\text { Providence } &amp; 626,667 &amp; 5.8786 &amp; 44 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.0056 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 75 }\\end{array} [\/latex]<\/p>\r\n<p>This works, so we\u2019re done.<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>Like Jefferson\u2019s method, Webster\u2019s method carries a bias in favor of states with large populations, but rounding the quotas to the nearest whole number greatly reduces this bias. (Notice that Providence County, the largest, is the one that gets a representative trimmed because of the increased quota.) Also like Jefferson\u2019s method, Webster\u2019s method does not always follow the quota rule, but it follows the quota rule much more often than Jefferson\u2019s method does.<\/p>\r\n<p>For more information on Webster\u2019s method, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ZNybGTvz_hQ?si=V8VqSUwD-YtddGFZ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Webster's+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Webster's Method\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Huntington-Hill Method<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Huntington-Hill Method<\/strong>: The current method of apportionment used in Congress, aiming to minimize the percent differences in representation.<\/p>\r\n<p>The steps to applying the Huntington-Hill method are:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>standard divisor<\/strong>.<\/li>\r\n\t<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>.<\/li>\r\n\t<li>Cut off the decimal part of the quota to obtain the lower quota, which we\u2019ll call [latex]n[\/latex]. Compute [latex]\\sqrt{n(n+1)}[\/latex], which is the <strong>geometric mean<\/strong> of the lower quota and one value higher.<\/li>\r\n\t<li>If the quota is larger than the geometric mean, round up the quota; if the quota is smaller than the geometric mean, round down the quota. Add up the resulting whole numbers to get the <strong>initial allocation<\/strong>.<\/li>\r\n\t<li>If the total from Step 4 was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. If the total from step 4 was larger than the total number of representatives, increase the divisor and recalculate the quota and allocation. Continue doing this until the total in Step 4 is equal to the total number of representatives. The divisor we end up using is called the <strong>modified divisor<\/strong> or <strong>adjusted divisor<\/strong>.<\/li>\r\n<\/ol>\r\n\r\n\r\nTerms to remember:\r\n\r\n\r\n<ul>\r\n\t<li><strong>Standard Divisor:<\/strong> The total population divided by the total number of representatives, serving as the initial benchmark for representation. Think of it as the \"ideal\" number of people each representative should stand for.<\/li>\r\n\t<li><strong>Quota:<\/strong> The result of dividing each state's population by the standard divisor, rounded to several decimal places. Keep the decimals; they're crucial for the next steps.<\/li>\r\n\t<li><strong>Modified Divisor:<\/strong> The adjusted divisor used when the total number of representatives doesn't match the required total. If your total reps are off, you'll need to adjust this number and redo your calculations.<\/li>\r\n\t<li><strong>Geometric Mean:<\/strong> Calculated as [latex]\\sqrt{n(n+1)}[\/latex], where [latex]n[\/latex] is the lower quota. It helps in rounding the quota. This is your \"rounding guide.\" If the quota is larger, round up; if smaller, round down.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>We\u2019ll apply the Huntington-Hill method for Rhode Island. Recall, the Rhode Island\u2019s House of Representatives splits [latex]75[\/latex] seats among its five counties.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } &amp; \\text { Population }\\\\ \\hline \\text { Bristol } &amp; 49,875\\\\ \\text { Kent } &amp; 166,158\\\\ \\text { Newport } &amp; 82,888\\\\ \\text { Providence } &amp; 626,667\\\\ \\text { Washington } &amp; 126,979\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 }\\end{array}[\/latex]<\/p>\r\n\r\n\r\n[reveal-answer q=\"4334\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4334\"]\r\n\r\n\r\n<p>The original divisor of [latex]14,034.22667[\/latex] gives these results:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrccc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Lower Quota } &amp; \\text{ Geom Mean } &amp; \\text{ Initial } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538 &amp; 3 &amp; 3.46 &amp; 4\\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395 &amp; 11 &amp; 11.49 &amp; 12 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061 &amp; 5 &amp; 5.48 &amp; 6\\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528 &amp; 44 &amp; 44.50 &amp; 45 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478 &amp; 9 &amp; 9.49 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; &amp; &amp; \\bf{ 76 }\\end{array} [\/latex]<\/p>\r\n<p>This is too many, so we need to increase the divisor. Let\u2019s try [latex]14,100[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrccc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Lower Quota } &amp; \\text{ Geom Mean } &amp; \\text{ Initial } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5372 &amp; 3 &amp; 3.46 &amp; 4 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.7843 &amp; 11 &amp; 11.49 &amp; 12 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.8786 &amp; 5 &amp; 5.48 &amp; 6 \\\\ \\text { Providence } &amp; 626,667 &amp; 5.8786 &amp; 5 &amp; 44.50 &amp; 44 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.0056 &amp; 9 &amp; 9.49 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; &amp; &amp; \\bf{ 75 }\\end{array} [\/latex]<\/p>\r\n<p>This works, so we\u2019re done.<\/p>\r\n<p>In this case, the apportionment produced by the Huntington-Hill method was the same as those from Webster\u2019s method.<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>For more information on the Huntington-Hill method, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/l74j-auLjZE?si=dBmYTY4O-oLg1NtR\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Huntington-Hill+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Huntington-Hill Method\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Lowndes\u2019 Method<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Lowndes\u2019 Method<\/strong>: A method proposed by William Lowndes, aiming to favor smaller states in apportionment.<\/p>\r\n<p>The steps to applying Lowndes\u2019 method are:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>divisor<\/strong>.<\/li>\r\n\t<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>.<\/li>\r\n\t<li>Cut off all the decimal parts of all the quotas (but don\u2019t forget what the decimals were). Add up the remaining whole numbers.<\/li>\r\n\t<li>Assuming that the total from Step 3 was less than the total number of representatives, divide the decimal part of each state\u2019s quota by the whole number part. Assign the remaining representatives, one each, to the states whose <strong>ratio <\/strong>of decimal part to whole part were largest, until the desired total is reached.<\/li>\r\n<\/ol>\r\n\r\n\r\nTerms to remember:\r\n\r\n\r\n<ul>\r\n\t<li><strong>Divisor:<\/strong> The total population divided by the total number of representatives, serving as the initial benchmark for representation. This is the \"average\" representation. Divide the total population by the total number of reps to find it.<\/li>\r\n\t<li><strong>Quota:<\/strong> The result of dividing each state's population by the divisor, rounded to several decimal places. Record this number to several decimal places; they play a crucial role in the next steps.<\/li>\r\n\t<li><strong>Decimal to Whole Ratio:<\/strong> The ratio of the decimal part of each state's quota to the whole number part, used to decide where the remaining representatives should go. This ratio helps determine where additional representatives will make the most impact, especially for smaller states.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>We\u2019ll apply Lowndes\u2019 method for Rhode Island. Recall, the Rhode Island\u2019s House of Representatives splits [latex]75[\/latex] seats among its five counties.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } &amp; \\text { Population }\\\\ \\hline \\text { Bristol } &amp; 49,875\\\\ \\text { Kent } &amp; 166,158\\\\ \\text { Newport } &amp; 82,888\\\\ \\text { Providence } &amp; 626,667\\\\ \\text { Washington } &amp; 126,979\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 }\\end{array}[\/latex]<\/p>\r\n\r\n\r\n[reveal-answer q=\"4335\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4335\"]\r\n\r\n\r\n<p>Rhode Island, again beginning in the same way as Hamilton:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial }\\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538 &amp; 3 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395 &amp; 11 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061 &amp; 5 \\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528 &amp; 44\\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478 &amp; 9\\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 72 }\\end{array} [\/latex]<\/p>\r\n<p>We divide each county\u2019s quota\u2019s decimal part by its whole number part to determine which three should get the remaining representatives:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} {\\text {Bristol: }} &amp; {0.5538\/3 \\approx 0.1846} \\\\ {\\text{Kent: }} &amp; {0.8395\/11 \\approx 0.0763} \\\\ {\\text{Newport: }} &amp; {0.9061\/5 \\approx 0.1812} \\\\ {\\text{Providence: }} &amp; {0.6528\/44 \\approx 0.0148} \\\\ {\\text{Washington: }} &amp; {0.0478\/9 \\approx 0.0053} \\\\ \\end{array} [\/latex]<\/p>\r\n<p>The three largest of these are Bristol, Newport, and Kent, so they get the remaining three representatives:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrcc} \\text { County } &amp; \\text { Population } &amp; \\text{ Quota } &amp; \\text{ Initial } &amp; \\text{ Ratio } &amp; \\text{ Final } \\\\ \\hline \\text { Bristol } &amp; 49,875 &amp; 3.5538 &amp; 3 &amp; 0.1846 &amp; 4 \\\\ \\text { Kent } &amp; 166,158 &amp; 11.8395 &amp; 11 &amp; 0.0763 &amp; 12 \\\\ \\text { Newport } &amp; 82,888 &amp; 5.9061 &amp; 5 &amp; 0.1812 &amp; 6 \\\\ \\text { Providence } &amp; 626,667 &amp; 44.6528 &amp; 44 &amp; 0.0148 &amp; 44 \\\\ \\text { Washington } &amp; 126,979 &amp; 9.0478 &amp; 9 &amp; 0.0053 &amp; 9 \\\\ \\textbf{ Total } &amp; \\bf{ 1,052,567 } &amp; &amp; \\bf{ 72 } &amp; &amp; \\bf{ 75 }\\end{array} [\/latex]<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>For more information on Lowndes\u2019 method, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/sNaRK6n7vRw?si=xt-cbPjvphgMtY7f\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Lowndes'+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Lowndes' Method\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Apportionment of Legislative Districts<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Legislative Districts:<\/strong> Areas drawn to represent a fixed number of constituents, aiming for equal representation. These are not always aligned with county or city boundaries; they are drawn to equalize the number of constituents.<\/p>\r\n<p><strong>Redistricting:<\/strong> The process of redefining legislative districts to accommodate population changes. This is usually done by the legislature itself, so be aware of potential biases.<\/p>\r\n<p><strong>Gerrymandering:<\/strong> Manipulating district boundaries to favor a particular political party or group. Keep an eye out for oddly shaped districts; they might be a sign of gerrymandering.<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>The map to the right shows the [latex]38^{th}[\/latex] congressional district in California in 2004[footnote]https:\/\/en.wikipedia.org\/wiki\/File:California_District_38_2004.png[\/footnote]. This district was created through a bi-partisan committee of incumbent legislators. This gerrymandering leads to districts that are not competitive; the prevailing party almost always wins with a large margin.<\/p>\r\n<center><img class=\"aligncenter wp-image-8947 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12162951\/clipboard_ee4311d50ec91008123431231b4687a5e-1.png\" alt=\"A map showing California's 38th congressional district in 2004. It is long and narrow, and bent like an L.\" width=\"509\" height=\"455\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The map to the right shows the [latex]4^{th}[\/latex] congressional district in Illinois in 2004.[footnote]https:\/\/en.wikipedia.org\/wiki\/File:Illinois_District_4_2004.png[\/footnote] This district was drawn to contain the two predominantly Hispanic areas of Chicago. The largely Puerto Rican area to the north and the southern Mexican areas are only connected in this districting by a piece of the highway to the west.<\/p>\r\n<center><img class=\"aligncenter wp-image-8948 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12170621\/clipboard_e95e8bc37113ae16635b56d5c17c10917-1.png\" alt=\"A map showing Illinois 4th congressional district in 2004. It consists of two long and extremely narrow segments, connected on one end by a very narrow segment.\" width=\"479\" height=\"449\" \/><\/center><\/section>\r\n<p>For more information on gerrymandering, watch the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/YcUDBgYodIE?si=-iG9HGa_TutxjMcZ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Gerrymandering_+How+drawing+jagged+lines+can+impact+an+election+-+Christina+Greer.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGerrymandering: How drawing jagged lines can impact an election - Christina Greer\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Define apportionment and its importance in political representation in the United States and other representative democracies<\/li>\n<li>Identify various methods of apportionment, including Hamilton&#8217;s method, Jefferson&#8217;s method, Webster&#8217;s method, Huntington-Hill method, and Lowndes&#8217; method<\/li>\n<li>Understand the apportionment process for legislative districts<\/li>\n<\/ul>\n<\/section>\n<h2>Apportionment<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Apportionment:<\/strong> The process of fairly dividing a fixed number of resources (like seats in a legislative body) among groups of varying sizes.<\/p>\n<p><strong>Political Importance:<\/strong> Apportionment in politics ensures fair representation by allocating elected officials based on area and population.<\/p>\n<p><strong>Constraints:<\/strong> Whole numbers only, must use all resources, each group gets at least one, and division should be proportional to population size.<\/p>\n<p>Below are important things to consider when thinking about apportionment.<\/p>\n<ul>\n<li><strong>Zero-Sum Game:<\/strong> If one state gains a representative, another must lose one.<\/li>\n<li><strong>Minimum Allocation:<\/strong> Every state or group must receive at least one of the resources being divided.<\/li>\n<li><strong>Whole Numbers:<\/strong> Apportionment deals with whole numbers; you can&#8217;t have [latex]3.4[\/latex] representatives.<\/li>\n<li><strong>Proportional Representation:<\/strong> While exact proportionality isn&#8217;t possible due to the whole number constraint, strive for as close to proportional as possible.<\/li>\n<\/ul>\n<\/div>\n<p>For more information on the basics of apportionment, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/w_0XwyXgJvk?si=ascUMuo7xe55L-Rt\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Introduction+to+Apportionment.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Apportionment\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Hamilton\u2019s Method<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Hamilton&#8217;s Method:<\/strong> A step-by-step approach to apportionment, initially proposed by Alexander Hamilton and used in the U.S. from 1852 to 1911.<\/p>\n<p>The steps to applying Hamilton&#8217;s method are:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>divisor<\/strong>.<\/li>\n<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>. Since we can only allocate whole representatives, Hamilton resolves the whole number problem, as follows:<\/li>\n<li>Cut off all the decimal parts of all the quotas (but don\u2019t forget what the decimals were). These are called the <strong>lower quotas<\/strong>. Add up the remaining whole numbers. This answer will always be less than or equal to the total number of representatives (and the \u201cor equal to\u201d part happens only in very specific circumstances that are incredibly unlikely to turn up).<\/li>\n<li>Assuming that the total from Step 3 was less than the total number of representatives, assign the remaining representatives, one each, to the states whose decimal parts of the quota were largest, until the desired total is reached.<\/li>\n<\/ol>\n<p><strong>Quota Rule:<\/strong> A guideline suggesting that the final number of representatives should be within one of the state&#8217;s quota.<\/p>\n<p><strong>Paradoxes:<\/strong> Hamilton&#8217;s method is subject to several paradoxes, such as the Alabama Paradox, the New States Paradox, and the Population Paradox.<\/p>\n<p>Terms to remember:<\/p>\n<ul>\n<li><strong>Divisor:<\/strong> Calculated by dividing the total population by the total number of representatives. It&#8217;s the baseline for apportionment.<\/li>\n<li><strong>Quota:<\/strong> Each state&#8217;s population divided by the divisor, kept to several decimal places.<\/li>\n<li><strong>Lower Quotas:<\/strong> The whole number parts of the quotas. Summing these will always be less than or equal to the total number of representatives.<\/li>\n<li><strong>Quota Rule:<\/strong> Aim for the final number of representatives to be within one of the state&#8217;s quota for fairness.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use Hamilton\u2019s method to apportion the [latex]75[\/latex] seats of Rhode Island\u2019s House of Representatives among its five counties.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } & \\text { Population }\\\\ \\hline \\text { Bristol } & 49,875\\\\ \\text { Kent } & 166,158\\\\ \\text { Newport } & 82,888\\\\ \\text { Providence } & 626,667\\\\ \\text { Washington } & 126,979\\\\ \\textbf{ Total } & \\bf{ 1,052,567 }\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4331\">Show Solution<\/button> <\/p>\n<div id=\"q4331\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>\n<p>First, we determine the divisor:<\/p>\n<p style=\"text-align: center;\">[latex]1,052,567\/75 = 14,034.22667[\/latex]<\/p>\n<\/li>\n<li>\n<p>Now we determine each county\u2019s quota by dividing the county\u2019s population by the divisor:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrr} \\text { County } & \\text { Population } & \\text{ Quota }\\\\ \\hline \\text { Bristol } & 49,875 & 3.5538\\\\ \\text { Kent } & 166,158 & 11.8395\\\\ \\text { Newport } & 82,888 & 5.9061\\\\ \\text { Providence } & 626,667 & 44.6528\\\\ \\text { Washington } & 126,979 & 9.0478\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & \\end{array}[\/latex]<\/p>\n<\/li>\n<li>\n<p>Removing the decimal parts of the quotas gives:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial }\\\\ \\hline \\text { Bristol } & 49,875 & 3.5538 & 3 \\\\ \\text { Kent } & 166,158 & 11.8395 & 11 \\\\ \\text { Newport } & 82,888 & 5.9061 & 5 \\\\ \\text { Providence } & 626,667 & 44.6528 & 44\\\\ \\text { Washington } & 126,979 & 9.0478 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 72 }\\end{array}[\/latex]<\/p>\n<\/li>\n<li>\n<p>We need [latex]75[\/latex] representatives and we only have [latex]72[\/latex], so we assign the remaining three, one each, to the three counties with the largest decimal parts, which are Newport, Kent, and Providence:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrcc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } & \\text{ Final } \\\\ \\hline \\text { Bristol } & 49,875 & 3.5538 & 3 & 3 \\\\ \\text { Kent } & 166,158 & 11.8395 & 11 & 12 \\\\ \\text { Newport } & 82,888 & 5.9061 & 5 & 6 \\\\ \\text { Providence } & 626,667 & 44.6528 & 44 & 45 \\\\ \\text { Washington } & 126,979 & 9.0478 & 9 & 9 \\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 72 } & \\bf{ 75 }\\end{array}[\/latex]<\/p>\n<p>Note that even though Bristol County\u2019s decimal part is greater than [latex].5[\/latex], it isn\u2019t big enough to get an additional representative, because three other counties have greater decimal parts.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>For more information on Hamilton&#8217;s method, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/YWfEqWLz9pc?si=fsVWIdGG1J8fMJiE\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Hamilton's+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Hamilton&#8217;s Method\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Jefferson\u2019s Method<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Jefferson\u2019s Method:<\/strong> An alternative to Hamilton&#8217;s method, used in Congress from 1791 through 1842, which favors larger states.<\/p>\n<p>The steps to applying Jefferson\u2019s method are:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>standard divisor<\/strong>.<\/li>\n<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>.<\/li>\n<li>Cut off all the decimal parts of all the quotas (but don\u2019t forget what the decimals were). These are the <strong>lower quotas<\/strong>. Add up the remaining whole numbers. This answer will always be less than or equal to the total number of representatives.<\/li>\n<li>If the total from Step 3 was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. Continue doing this until the total in Step 3 is equal to the total number of representatives. The divisor we end up using is called the <strong>modified divisor<\/strong> or <strong>adjusted divisor<\/strong>.<\/li>\n<\/ol>\n<p><strong>Divisor-Adjusting Methods:<\/strong> Jefferson&#8217;s method is one such method, and it&#8217;s not guaranteed to follow the quota rule.<\/p>\n<p>Terms to remember:<\/p>\n<ul>\n<li><strong>Standard Divisor:<\/strong> The total population of all states divided by the total number of representatives. Think of it as the &#8220;benchmark&#8221; for how many people each representative should represent.<\/li>\n<li><strong>Quota:<\/strong> Each state&#8217;s population divided by the standard divisor, recorded to several decimal places. It&#8217;s the &#8220;ideal&#8221; number of representatives for each state, but it&#8217;s not the final answer.<\/li>\n<li><strong>Modified Divisor:<\/strong> The divisor that is adjusted until the total number of representatives matches the required total. If the initial total number of representatives is less than required, reduce the divisor and recalculate until they match.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>We\u2019ll apply Jefferson\u2019s method for Rhode Island. Recall, the Rhode Island\u2019s House of Representatives splits [latex]75[\/latex] seats among its five counties.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } & \\text { Population }\\\\ \\hline \\text { Bristol } & 49,875\\\\ \\text { Kent } & 166,158\\\\ \\text { Newport } & 82,888\\\\ \\text { Providence } & 626,667\\\\ \\text { Washington } & 126,979\\\\ \\textbf{ Total } & \\bf{ 1,052,567 }\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4332\">Show Solution<\/button> <\/p>\n<div id=\"q4332\" class=\"hidden-answer\" style=\"display: none\">\n<p>The original divisor of [latex]14,034.22667[\/latex] gave these results:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } \\\\ \\hline \\text { Bristol } & 49,875 & 3.5538 & 3 \\\\ \\text { Kent } & 166,158 & 11.8395 & 11 \\\\ \\text { Newport } & 82,888 & 5.9061 & 5 \\\\ \\text { Providence } & 626,667 & 44.6528 & 44 \\\\ \\text { Washington } & 126,979 & 9.0478 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 72 }\\end{array}[\/latex]<\/p>\n<p>We need [latex]75[\/latex] representatives and we only have [latex]72[\/latex], so we need to use a smaller divisor. Let\u2019s try [latex]13,500[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } \\\\ \\hline \\text { Bristol } & 49,875 & 3.6944 & 3 \\\\ \\text { Kent } & 166,158 & 12.3080 & 12 \\\\ \\text { Newport } & 82,888 & 6.1399 & 6 \\\\ \\text { Providence } & 626,667 & 46.4198 & 46 \\\\ \\text { Washington } & 126,979 & 9.4059 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 76 }\\end{array}[\/latex]<\/p>\n<p>We\u2019ve gone too far. We need a divisor that\u2019s greater than [latex]13,500[\/latex] but less than [latex]14,034.22667[\/latex]. Let\u2019s try [latex]13,700[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } \\\\ \\hline \\text { Bristol } & 49,875 & 3.6405 & 3 \\\\ \\text { Kent } & 166,158 & 12.1283 & 12 \\\\ \\text { Newport } & 82,888 & 6.0502 & 6 \\\\ \\text { Providence } & 626,667 & 45.7421 & 45 \\\\ \\text { Washington } & 126,979 & 9.2685 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 75 }\\end{array}[\/latex]<\/p>\n<p>This works.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Notice, in comparison to Hamilton\u2019s method, that although the results were the same, they came about in a different way, and the outcome was almost different. Providence County (the largest) almost went up to [latex]46[\/latex] representatives before Kent (which is much smaller) got to [latex]12[\/latex]. Although that didn\u2019t happen here, it can. Divisor-adjusting methods like Jefferson\u2019s are not guaranteed to follow the quota rule! For more information on Jefferson\u2019s method, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/weGGVmy9yLc?si=dakuzc07blcB7dd_\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Jefferson's+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Jefferson&#8217;s Method\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Webster\u2019s Method<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Webster\u2019s Method:<\/strong> A method similar to Jefferson&#8217;s but rounds quotas to the nearest whole number. It was adopted by Congress multiple times and is less biased towards larger states.<\/p>\n<p>The steps to applying Webster\u2019s method are:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>standard divisor<\/strong>.<\/li>\n<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>.<\/li>\n<li>Round all the quotas to the nearest whole number (but don\u2019t forget what the decimals were). Add up the remaining whole numbers.<\/li>\n<li>If the total from Step 3 was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. If the total from step 3 was larger than the total number of representatives, increase the divisor and recalculate the quota and allocation. Continue doing this until the total in Step 3 is equal to the total number of representatives. The divisor we end up using is called the <strong>modified divisor<\/strong> or <strong>adjusted divisor<\/strong>.<\/li>\n<\/ol>\n<p><strong>Balinski-Young Impossibility Theorem:<\/strong> A theorem that states no apportionment method can always follow the quota rule without being subject to paradoxes. Understand that no method is perfect; each has its own set of trade-offs and potential paradoxes.<\/p>\n<p>Terms to remember:<\/p>\n<ul>\n<li><strong>Standard Divisor:<\/strong> The total population divided by the total number of representatives, serving as the benchmark for representation. It&#8217;s your &#8220;starting point&#8221; for figuring out how many people each representative should ideally represent.<\/li>\n<li><strong>Quota:<\/strong> The result of dividing each state&#8217;s population by the standard divisor, rounded to several decimal places. It&#8217;s like your &#8220;first draft&#8221; of how many representatives each state should have. Don&#8217;t forget the decimals; they matter!<\/li>\n<li><strong>Modified Divisor:<\/strong> The adjusted divisor used when the total number of representatives doesn&#8217;t match the required total. If your total number of representatives is off, tweak this number and recalculate.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>We\u2019ll apply Webster\u2019s method for Rhode Island. Recall, the Rhode Island\u2019s House of Representatives splits [latex]75[\/latex] seats among its five counties.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } & \\text { Population }\\\\ \\hline \\text { Bristol } & 49,875\\\\ \\text { Kent } & 166,158\\\\ \\text { Newport } & 82,888\\\\ \\text { Providence } & 626,667\\\\ \\text { Washington } & 126,979\\\\ \\textbf{ Total } & \\bf{ 1,052,567 }\\end{array}[\/latex] <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4333\">Show Solution<\/button> <\/p>\n<div id=\"q4333\" class=\"hidden-answer\" style=\"display: none\">\n<p>The original divisor of [latex]14,034.22667[\/latex] gave these results:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } \\\\ \\hline \\text { Bristol } & 49,875 & 3.5538 \\\\ \\text { Kent } & 166,158 & 11.8395 \\\\ \\text { Newport } & 82,888 & 5.9061 \\\\ \\text { Providence } & 626,667 & 44.6528 \\\\ \\text { Washington } & 126,979 & 9.0478 \\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & \\end{array}[\/latex]<\/p>\n<p>Rounding the quotas up we get:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } \\\\ \\hline \\text { Bristol } & 49,875 & 3.5538 & 4 \\\\ \\text { Kent } & 166,158 & 11.8395 & 12 \\\\ \\text { Newport } & 82,888 & 5.9061 & 6 \\\\ \\text { Providence } & 626,667 & 44.6528 & 45 \\\\ \\text { Washington } & 126,979 & 9.0478 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 76 }\\end{array}[\/latex]<\/p>\n<p>This is too many, so we need to increase the divisor. Let\u2019s try [latex]14,100[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } \\\\ \\hline \\text { Bristol } & 49,875 & 3.5372 & 4 \\\\ \\text { Kent } & 166,158 & 11.7843 & 12 \\\\ \\text { Newport } & 82,888 & 5.8786 & 6 \\\\ \\text { Providence } & 626,667 & 5.8786 & 44 \\\\ \\text { Washington } & 126,979 & 9.0056 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 75 }\\end{array}[\/latex]<\/p>\n<p>This works, so we\u2019re done.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Like Jefferson\u2019s method, Webster\u2019s method carries a bias in favor of states with large populations, but rounding the quotas to the nearest whole number greatly reduces this bias. (Notice that Providence County, the largest, is the one that gets a representative trimmed because of the increased quota.) Also like Jefferson\u2019s method, Webster\u2019s method does not always follow the quota rule, but it follows the quota rule much more often than Jefferson\u2019s method does.<\/p>\n<p>For more information on Webster\u2019s method, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ZNybGTvz_hQ?si=V8VqSUwD-YtddGFZ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Webster's+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Webster&#8217;s Method\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Huntington-Hill Method<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Huntington-Hill Method<\/strong>: The current method of apportionment used in Congress, aiming to minimize the percent differences in representation.<\/p>\n<p>The steps to applying the Huntington-Hill method are:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>standard divisor<\/strong>.<\/li>\n<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>.<\/li>\n<li>Cut off the decimal part of the quota to obtain the lower quota, which we\u2019ll call [latex]n[\/latex]. Compute [latex]\\sqrt{n(n+1)}[\/latex], which is the <strong>geometric mean<\/strong> of the lower quota and one value higher.<\/li>\n<li>If the quota is larger than the geometric mean, round up the quota; if the quota is smaller than the geometric mean, round down the quota. Add up the resulting whole numbers to get the <strong>initial allocation<\/strong>.<\/li>\n<li>If the total from Step 4 was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. If the total from step 4 was larger than the total number of representatives, increase the divisor and recalculate the quota and allocation. Continue doing this until the total in Step 4 is equal to the total number of representatives. The divisor we end up using is called the <strong>modified divisor<\/strong> or <strong>adjusted divisor<\/strong>.<\/li>\n<\/ol>\n<p>Terms to remember:<\/p>\n<ul>\n<li><strong>Standard Divisor:<\/strong> The total population divided by the total number of representatives, serving as the initial benchmark for representation. Think of it as the &#8220;ideal&#8221; number of people each representative should stand for.<\/li>\n<li><strong>Quota:<\/strong> The result of dividing each state&#8217;s population by the standard divisor, rounded to several decimal places. Keep the decimals; they&#8217;re crucial for the next steps.<\/li>\n<li><strong>Modified Divisor:<\/strong> The adjusted divisor used when the total number of representatives doesn&#8217;t match the required total. If your total reps are off, you&#8217;ll need to adjust this number and redo your calculations.<\/li>\n<li><strong>Geometric Mean:<\/strong> Calculated as [latex]\\sqrt{n(n+1)}[\/latex], where [latex]n[\/latex] is the lower quota. It helps in rounding the quota. This is your &#8220;rounding guide.&#8221; If the quota is larger, round up; if smaller, round down.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>We\u2019ll apply the Huntington-Hill method for Rhode Island. Recall, the Rhode Island\u2019s House of Representatives splits [latex]75[\/latex] seats among its five counties.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } & \\text { Population }\\\\ \\hline \\text { Bristol } & 49,875\\\\ \\text { Kent } & 166,158\\\\ \\text { Newport } & 82,888\\\\ \\text { Providence } & 626,667\\\\ \\text { Washington } & 126,979\\\\ \\textbf{ Total } & \\bf{ 1,052,567 }\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4334\">Show Solution<\/button> <\/p>\n<div id=\"q4334\" class=\"hidden-answer\" style=\"display: none\">\n<p>The original divisor of [latex]14,034.22667[\/latex] gives these results:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrccc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Lower Quota } & \\text{ Geom Mean } & \\text{ Initial } \\\\ \\hline \\text { Bristol } & 49,875 & 3.5538 & 3 & 3.46 & 4\\\\ \\text { Kent } & 166,158 & 11.8395 & 11 & 11.49 & 12 \\\\ \\text { Newport } & 82,888 & 5.9061 & 5 & 5.48 & 6\\\\ \\text { Providence } & 626,667 & 44.6528 & 44 & 44.50 & 45 \\\\ \\text { Washington } & 126,979 & 9.0478 & 9 & 9.49 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & & & \\bf{ 76 }\\end{array}[\/latex]<\/p>\n<p>This is too many, so we need to increase the divisor. Let\u2019s try [latex]14,100[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrccc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Lower Quota } & \\text{ Geom Mean } & \\text{ Initial } \\\\ \\hline \\text { Bristol } & 49,875 & 3.5372 & 3 & 3.46 & 4 \\\\ \\text { Kent } & 166,158 & 11.7843 & 11 & 11.49 & 12 \\\\ \\text { Newport } & 82,888 & 5.8786 & 5 & 5.48 & 6 \\\\ \\text { Providence } & 626,667 & 5.8786 & 5 & 44.50 & 44 \\\\ \\text { Washington } & 126,979 & 9.0056 & 9 & 9.49 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & & & \\bf{ 75 }\\end{array}[\/latex]<\/p>\n<p>This works, so we\u2019re done.<\/p>\n<p>In this case, the apportionment produced by the Huntington-Hill method was the same as those from Webster\u2019s method.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>For more information on the Huntington-Hill method, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/l74j-auLjZE?si=dBmYTY4O-oLg1NtR\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Huntington-Hill+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Huntington-Hill Method\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Lowndes\u2019 Method<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Lowndes\u2019 Method<\/strong>: A method proposed by William Lowndes, aiming to favor smaller states in apportionment.<\/p>\n<p>The steps to applying Lowndes\u2019 method are:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the <strong>divisor<\/strong>.<\/li>\n<li>Divide each state\u2019s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the <strong>quota<\/strong>.<\/li>\n<li>Cut off all the decimal parts of all the quotas (but don\u2019t forget what the decimals were). Add up the remaining whole numbers.<\/li>\n<li>Assuming that the total from Step 3 was less than the total number of representatives, divide the decimal part of each state\u2019s quota by the whole number part. Assign the remaining representatives, one each, to the states whose <strong>ratio <\/strong>of decimal part to whole part were largest, until the desired total is reached.<\/li>\n<\/ol>\n<p>Terms to remember:<\/p>\n<ul>\n<li><strong>Divisor:<\/strong> The total population divided by the total number of representatives, serving as the initial benchmark for representation. This is the &#8220;average&#8221; representation. Divide the total population by the total number of reps to find it.<\/li>\n<li><strong>Quota:<\/strong> The result of dividing each state&#8217;s population by the divisor, rounded to several decimal places. Record this number to several decimal places; they play a crucial role in the next steps.<\/li>\n<li><strong>Decimal to Whole Ratio:<\/strong> The ratio of the decimal part of each state&#8217;s quota to the whole number part, used to decide where the remaining representatives should go. This ratio helps determine where additional representatives will make the most impact, especially for smaller states.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>We\u2019ll apply Lowndes\u2019 method for Rhode Island. Recall, the Rhode Island\u2019s House of Representatives splits [latex]75[\/latex] seats among its five counties.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} \\text { County } & \\text { Population }\\\\ \\hline \\text { Bristol } & 49,875\\\\ \\text { Kent } & 166,158\\\\ \\text { Newport } & 82,888\\\\ \\text { Providence } & 626,667\\\\ \\text { Washington } & 126,979\\\\ \\textbf{ Total } & \\bf{ 1,052,567 }\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4335\">Show Solution<\/button> <\/p>\n<div id=\"q4335\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rhode Island, again beginning in the same way as Hamilton:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial }\\\\ \\hline \\text { Bristol } & 49,875 & 3.5538 & 3 \\\\ \\text { Kent } & 166,158 & 11.8395 & 11 \\\\ \\text { Newport } & 82,888 & 5.9061 & 5 \\\\ \\text { Providence } & 626,667 & 44.6528 & 44\\\\ \\text { Washington } & 126,979 & 9.0478 & 9\\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 72 }\\end{array}[\/latex]<\/p>\n<p>We divide each county\u2019s quota\u2019s decimal part by its whole number part to determine which three should get the remaining representatives:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lr} {\\text {Bristol: }} & {0.5538\/3 \\approx 0.1846} \\\\ {\\text{Kent: }} & {0.8395\/11 \\approx 0.0763} \\\\ {\\text{Newport: }} & {0.9061\/5 \\approx 0.1812} \\\\ {\\text{Providence: }} & {0.6528\/44 \\approx 0.0148} \\\\ {\\text{Washington: }} & {0.0478\/9 \\approx 0.0053} \\\\ \\end{array}[\/latex]<\/p>\n<p>The three largest of these are Bristol, Newport, and Kent, so they get the remaining three representatives:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lrrcc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } & \\text{ Ratio } & \\text{ Final } \\\\ \\hline \\text { Bristol } & 49,875 & 3.5538 & 3 & 0.1846 & 4 \\\\ \\text { Kent } & 166,158 & 11.8395 & 11 & 0.0763 & 12 \\\\ \\text { Newport } & 82,888 & 5.9061 & 5 & 0.1812 & 6 \\\\ \\text { Providence } & 626,667 & 44.6528 & 44 & 0.0148 & 44 \\\\ \\text { Washington } & 126,979 & 9.0478 & 9 & 0.0053 & 9 \\\\ \\textbf{ Total } & \\bf{ 1,052,567 } & & \\bf{ 72 } & & \\bf{ 75 }\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>For more information on Lowndes\u2019 method, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/sNaRK6n7vRw?si=xt-cbPjvphgMtY7f\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Apportionment_+Lowndes'+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cApportionment: Lowndes&#8217; Method\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Apportionment of Legislative Districts<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Legislative Districts:<\/strong> Areas drawn to represent a fixed number of constituents, aiming for equal representation. These are not always aligned with county or city boundaries; they are drawn to equalize the number of constituents.<\/p>\n<p><strong>Redistricting:<\/strong> The process of redefining legislative districts to accommodate population changes. This is usually done by the legislature itself, so be aware of potential biases.<\/p>\n<p><strong>Gerrymandering:<\/strong> Manipulating district boundaries to favor a particular political party or group. Keep an eye out for oddly shaped districts; they might be a sign of gerrymandering.<\/p>\n<\/div>\n<section class=\"textbox example\">\n<p>The map to the right shows the [latex]38^{th}[\/latex] congressional district in California in 2004<a class=\"footnote\" title=\"https:\/\/en.wikipedia.org\/wiki\/File:California_District_38_2004.png\" id=\"return-footnote-8304-1\" href=\"#footnote-8304-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>. This district was created through a bi-partisan committee of incumbent legislators. This gerrymandering leads to districts that are not competitive; the prevailing party almost always wins with a large margin.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-8947 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12162951\/clipboard_ee4311d50ec91008123431231b4687a5e-1.png\" alt=\"A map showing California's 38th congressional district in 2004. It is long and narrow, and bent like an L.\" width=\"509\" height=\"455\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12162951\/clipboard_ee4311d50ec91008123431231b4687a5e-1.png 509w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12162951\/clipboard_ee4311d50ec91008123431231b4687a5e-1-300x268.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12162951\/clipboard_ee4311d50ec91008123431231b4687a5e-1-65x58.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12162951\/clipboard_ee4311d50ec91008123431231b4687a5e-1-225x201.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12162951\/clipboard_ee4311d50ec91008123431231b4687a5e-1-350x313.png 350w\" sizes=\"(max-width: 509px) 100vw, 509px\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>The map to the right shows the [latex]4^{th}[\/latex] congressional district in Illinois in 2004.<a class=\"footnote\" title=\"https:\/\/en.wikipedia.org\/wiki\/File:Illinois_District_4_2004.png\" id=\"return-footnote-8304-2\" href=\"#footnote-8304-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a> This district was drawn to contain the two predominantly Hispanic areas of Chicago. The largely Puerto Rican area to the north and the southern Mexican areas are only connected in this districting by a piece of the highway to the west.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-8948 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12170621\/clipboard_e95e8bc37113ae16635b56d5c17c10917-1.png\" alt=\"A map showing Illinois 4th congressional district in 2004. It consists of two long and extremely narrow segments, connected on one end by a very narrow segment.\" width=\"479\" height=\"449\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12170621\/clipboard_e95e8bc37113ae16635b56d5c17c10917-1.png 479w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12170621\/clipboard_e95e8bc37113ae16635b56d5c17c10917-1-300x281.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12170621\/clipboard_e95e8bc37113ae16635b56d5c17c10917-1-65x61.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12170621\/clipboard_e95e8bc37113ae16635b56d5c17c10917-1-225x211.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/09\/12170621\/clipboard_e95e8bc37113ae16635b56d5c17c10917-1-350x328.png 350w\" sizes=\"(max-width: 479px) 100vw, 479px\" \/><\/div>\n<\/section>\n<p>For more information on gerrymandering, watch the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/YcUDBgYodIE?si=-iG9HGa_TutxjMcZ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Gerrymandering_+How+drawing+jagged+lines+can+impact+an+election+-+Christina+Greer.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGerrymandering: How drawing jagged lines can impact an election &#8211; Christina Greer\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-8304-1\">https:\/\/en.wikipedia.org\/wiki\/File:California_District_38_2004.png <a href=\"#return-footnote-8304-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-8304-2\">https:\/\/en.wikipedia.org\/wiki\/File:Illinois_District_4_2004.png <a href=\"#return-footnote-8304-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":15,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Introduction to Apportionment\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/w_0XwyXgJvk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Apportionment: Hamilton\\'s Method\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/YWfEqWLz9pc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Apportionment: Jefferson\\'s Method\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/weGGVmy9yLc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Apportionment: Webster\\'s Method\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ZNybGTvz_hQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Apportionment: Huntington-Hill Method\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/l74j-auLjZE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Apportionment: Lowndes\\' Method\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sNaRK6n7vRw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Gerrymandering: How drawing jagged lines can impact an election - Christina Greer\",\"author\":\"TED-Ed\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/YcUDBgYodIE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":90,"module-header":"fresh_take","content_attributions":null,"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8304"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":29,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8304\/revisions"}],"predecessor-version":[{"id":14895,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8304\/revisions\/14895"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/90"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8304\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8304"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8304"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8304"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8304"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}