{"id":8934,"date":"2023-10-12T15:50:14","date_gmt":"2023-10-12T15:50:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=8934"},"modified":"2024-10-18T20:58:38","modified_gmt":"2024-10-18T20:58:38","slug":"apportionment-learn-it-6","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/apportionment-learn-it-6\/","title":{"raw":"Apportionment: Learn It 6","rendered":"Apportionment: Learn It 6"},"content":{"raw":"

Lowndes\u2019 Method<\/h2>\r\n

William Lowndes (1782-1822) was a Congressman from South Carolina (a small state) who proposed a method of apportionment that was more favorable to smaller states. Unlike the methods of Hamilton, Jefferson, and Webster, Lowndes\u2019s method has never been used to apportion Congress.<\/p>\r\n

Lowndes believed that an additional representative was much more valuable to a small state than to a large one. If a state already has [latex]20[\/latex] or [latex]30[\/latex] representatives, getting one more doesn\u2019t matter very much. But if it only has [latex]2[\/latex] or [latex]3[\/latex], one more is a big deal, and he felt that the additional representatives should go where they could make the most difference.<\/p>\r\n

Like Hamilton\u2019s method, Lowndes\u2019s method follows the quota rule. In fact, it arrives at the same quotas as Hamilton and the rest, and like Hamilton and Jefferson, it drops the decimal parts. But in deciding where the remaining representatives should go, we divide the decimal part of each state\u2019s quota by the whole number part (so that the same decimal part with a smaller whole number is worth more, because it matters more to that state).<\/p>\r\n

\r\n
\r\n

Lowndes\u2019 method<\/h3>\r\n
    \r\n\t
  1. 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 divisor<\/strong>.<\/li>\r\n\t
  2. 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 quota<\/strong>.<\/li>\r\n\t
  3. 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
  4. 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 ratio <\/strong>of decimal part to whole part were largest, until the desired total is reached.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section>\r\n
    \r\n

    We\u2019ll return to Delaware and apply Lowndes\u2019 method. As a reminder, the state of Delaware has three counties: Kent, New Castle, and Sussex. The Delaware state House of Representatives has [latex]41[\/latex] members.<\/p>\r\n

    The populations of the counties are as follows (from the 2010 Census):<\/p>\r\n\r\n

    [latex]\\begin{array}{lr} \\text { County } & \\text { Population } \\\\ \\hline \\text { Kent } & 162,310 \\\\ \\text { New Castle } & 538,479 \\\\ \\text { Sussex } & 197,145 \\\\ \\textbf{ Total } & \\bf{ 897,934 }\\end{array}[\/latex]<\/p> [reveal-answer q=\"4331\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"4331\"]\r\n\r\n

    We begin, as we did with Hamilton\u2019s method, by finding the quotas with the original divisor, [latex]21,900.82927[\/latex].<\/p>\r\n\r\n

    [latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial }\\\\ \\hline \\text { Kent } & 162,310 & 7.4111 & 7\\\\ \\text { New Castle } & 538,479 & 24.5872 & 24\\\\ \\text { Sussex } & 197,145 & 9.0017 & 9\\\\ \\textbf{ Total } & \\bf{ 897,934 } & & \\bf{ 40 }\\end{array} [\/latex]<\/p>\r\n\r\n

    We need one more representative. To find out which county should get it, Lowndes says to divide each county\u2019s decimal part by its whole number part, with the largest result getting the extra representative:<\/p>\r\n\r\n

    [latex]\\begin{array}{lr} \\text {Kent: } & 0.4111\/7 \\approx 0.0587 \\\\ \\text{New Castle: } & 0.5872\/24 \\approx 0.0245 \\\\ \\text{ Sussex: } & 0.0017\/9 \\approx 0.0002 \\\\ \\end{array}[\/latex]<\/p>\r\n\r\n

    The largest of these is Kent\u2019s, so Kent gets the [latex]41^{th}[\/latex] representative:<\/p>\r\n\r\n

    [latex]\\begin{array}{lrrcc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } & \\text{ Ratio } & \\text{ Final } \\\\ \\hline \\text { Kent } & 162,310 & 7.4111 & 7 & 0.0587 & 8 \\\\ \\text { New Castle } & 538,479 & 24.5872 & 24 & 0.0245 & 24 \\\\ \\text { Sussex } & 197,145 & 9.0017 & 9 & 0.0002 & 9 \\\\ \\textbf{ Total } & \\bf{ 897,934 } & & \\bf{ 40 } & & \\bf{ 41 }\\end{array} [\/latex]<\/p> [\/hidden-answer]<\/section>\r\n

    \r\n

    [ohm2_question hide_question_numbers=1]13235[\/ohm2_question]<\/p>\r\n<\/section>\r\n

    As you can see, there is no \u201cright answer\u201d when it comes to choosing a method for apportionment. Each method has its virtues, and favors different sized states.<\/p>","rendered":"

    Lowndes\u2019 Method<\/h2>\n

    William Lowndes (1782-1822) was a Congressman from South Carolina (a small state) who proposed a method of apportionment that was more favorable to smaller states. Unlike the methods of Hamilton, Jefferson, and Webster, Lowndes\u2019s method has never been used to apportion Congress.<\/p>\n

    Lowndes believed that an additional representative was much more valuable to a small state than to a large one. If a state already has [latex]20[\/latex] or [latex]30[\/latex] representatives, getting one more doesn\u2019t matter very much. But if it only has [latex]2[\/latex] or [latex]3[\/latex], one more is a big deal, and he felt that the additional representatives should go where they could make the most difference.<\/p>\n

    Like Hamilton\u2019s method, Lowndes\u2019s method follows the quota rule. In fact, it arrives at the same quotas as Hamilton and the rest, and like Hamilton and Jefferson, it drops the decimal parts. But in deciding where the remaining representatives should go, we divide the decimal part of each state\u2019s quota by the whole number part (so that the same decimal part with a smaller whole number is worth more, because it matters more to that state).<\/p>\n

    \n
    \n

    Lowndes\u2019 method<\/h3>\n
      \n
    1. 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 divisor<\/strong>.<\/li>\n
    2. 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 quota<\/strong>.<\/li>\n
    3. 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
    4. 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 ratio <\/strong>of decimal part to whole part were largest, until the desired total is reached.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n
      \n

      We\u2019ll return to Delaware and apply Lowndes\u2019 method. As a reminder, the state of Delaware has three counties: Kent, New Castle, and Sussex. The Delaware state House of Representatives has [latex]41[\/latex] members.<\/p>\n

      The populations of the counties are as follows (from the 2010 Census):<\/p>\n

      [latex]\\begin{array}{lr} \\text { County } & \\text { Population } \\\\ \\hline \\text { Kent } & 162,310 \\\\ \\text { New Castle } & 538,479 \\\\ \\text { Sussex } & 197,145 \\\\ \\textbf{ Total } & \\bf{ 897,934 }\\end{array}[\/latex]<\/p>\n