# 9.2: Apportionment - Jefferson’s, Adams’s, and Webster’s Methods

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Jefferson’s method was the first method used to apportion the seats in the U.S. House of Representatives in 1792. It was used through 1832. That year, New York had a standard quota of 38.59 but was granted 40 seats by Jefferson’s method. At that time, John Quincy Adams and Daniel Webster each proposed new apportionment methods but the proposals were defeated and Jefferson’s method was still used. Webster’s method was later chosen to be used in 1842 but Adams’s method was never used. Webster’s method and Hamilton’s method often give the same result. For many of the years between 1852 and 1901, Congress used a number of seats for the House that would result in the same apportionment by either Webster’s or Hamilton’s methods. After Hamilton’s method was finally scrapped in 1901, Webster’s method was used in 1901, 1911, and 1931. There were irregularities in the process in 1872 and just after the 1920 census. In 1941, the House size was fixed at 435 seats and the Huntington-Hill method became the permanent method of apportionment.

Jefferson’s, Adams’s, and Webster’s methods are all based on the idea of finding a divisor that will apportion all the seats under the appropriate rounding rule. There should be no seats left over after the number of seats are rounded off. For this to happen we have to adjust the standard divisor either up or down. The difference between the three methods is the rule for rounding off the quotas. Jefferson’s method rounds the quotas down to their lower quotas, Adams’ method rounds the quotas up to their upper quotas, and Webster’s method rounds the quotas either up or down following the usual rounding rule.

## Jefferson’s Method

Jefferson’s method divides all populations by a modified divisor and then rounds the results down to the lower quota. Sometimes the total number of seats will be too large and other times it will be too small. We keep guessing modified divisors until the method assigns the correct total number of seats. Dividing by a larger modified divisor will make each quota smaller so the sum of the lower quotas will be smaller. It is easy to remember which way to go. If the sum is too large, make the divisor larger. If the sum is too small, make the divisor smaller. All the quotas are rounded down so using the standard divisor will give a sum that is too small. Our guess for the first modified divisor should be a number smaller than the standard divisor.

**Summary of Jefferson’s Method:**

- Find the standard divisor, .
- Pick a modified divisor, d, that is slightly less than the standard divisor.
- Divide each state’s population by the modified divisor to get its modified quota.
- Round each modified quota down to its lower quota.
- Find the sum of the lower quotas.
- If the sum is the same as the number of seats to be apportioned, you are done. If the sum is too large, pick a new modified divisor that is larger than d. If the sum is too small, pick a new modified divisor that is smaller than d. Repeat steps three through six until the correct number of seats are apportioned.

**Example \(\PageIndex{1}\): Jefferson’s Method**

Use Jefferson’s method to apportion the 25 seats in Hamiltonia from Example \(\PageIndex{2}\).

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |
---|---|---|---|---|---|---|---|

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

From Example \(\PageIndex{2}\) we know the standard divisor is 9480 and the sum of the lower quotas is 20. In Jefferson’s method the standard divisor will always give us a sum that is too small so we begin by making the standard divisor smaller. There is no formula for this, just guess something. Let’s try the modified divisor, d = 9000.

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

d = 9000 | 2.67 | 6.22 | 3.11 | 1.89 | 7.22 | 5.22 | |

Lower Quota | 2 | 6 | 3 | 1 | 7 | 5 | 24 |

The sum of 24 is too small so we need to try again by making the modified divisor smaller. Let’s try d = 8000.

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

d = 9000 | 2.67 | 6.22 | 3.11 | 1.89 | 7.22 | 5.22 | |

Lower Quota | 2 | 6 | 3 | 1 | 7 | 5 | 24 |

d = 8000 | 3.00 | 7.00 | 3.50 | 2.13 | 8.13 | 5.88 | |

Lower Quota | 3 | 7 | 3 | 2 | 8 | 5 | 28 |

This time the sum of 28 is too big. Try again making the modified divisor larger. We know the divisor must be between 8000 and 9000 so let’s try 8500.

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

d = 9000 | 2.67 | 6.22 | 3.11 | 1.89 | 7.22 | 5.22 | |

Lower Quota | 2 | 6 | 3 | 1 | 7 | 5 | 24 |

d = 8000 | 3.00 | 7.00 | 3.50 | 2.13 | 8.13 | 5.88 | |

Lower Quota | 3 | 7 | 3 | 2 | 8 | 5 | 28 |

d = 8500 | 2.82 | 6.59 | 3.29 | 2.00 | 7.65 | 5.53 | |

Lower Quota | 2 | 6 | 3 | 2 | 7 | 5 | 25 |

This time the sum is 25 so we are done. Alpha gets two senators, Beta gets six senators, Gamma gets three senators, Delta gets two senators, Epsilon gets seven senators, and Zeta gets five senators.

*Note: This is the same result as we got using Hamilton’s method in Example \(\PageIndex{4}\). The two methods do not always give the same result.*

## Adams’s Method

Adams’s method divides all populations by a modified divisor and then rounds the results up to the upper quota. Just like Jefferson’s method we keep guessing modified divisors until the method assigns the correct number of seats. All the quotas are rounded up so the standard divisor will give a sum that is too large. Our guess for the first modified divisor should be a number larger than the standard divisor.

**Summary of Adams’s Method:**

- Find the standard divisor, .
- Pick a modified divisor, d, that is slightly more than the standard divisor.
- Divide each state’s population by the modified divisor to get the modified quota.
- Round each modified quota up to the upper quota.
- Find the sum of the upper quotas.
- If the sum is the same as the number of seats to be apportioned, you are done. If the sum is too big, pick a new modified divisor that is larger than d. If the sum is too small, pick a new modified divisor that is smaller than d. Repeat steps three through six until the correct number of seats are apportioned.

**Example \(\PageIndex{2}\): Adams’s Method**

Use Adams’s method to apportion the 25 seats in Hamiltonia from Example \(\PageIndex{2}\).

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

From Example \(\PageIndex{2}\) we know the standard divisor is 9480 and the sum of the upper quotas is 26. In Adams’s method the standard divisor will always give us a sum that is too big so we begin by making the standard divisor larger. There is no formula for this, just guess something. Let’s try the modified divisor, d = 10,000.

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

d = 10,000 | 2.40 | 5.60 | 2.80 | 1.70 | 6.50 | 4.70 | |

Upper Quota | 3 | 6 | 3 | 2 | 7 | 5 | 26 |

The total number of seats, 26, is too big so we need to try again by making the modified divisor larger. Try d = 11,000.

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

d = 10,000 | 2.40 | 5.60 | 2.80 | 1.70 | 6.50 | 4.70 | |

Upper Quota | 3 | 6 | 3 | 2 | 7 | 5 | 26 |

d = 11,000 | 2.18 | 5.09 | 2.55 | 1.55 | 5.91 | 4.27 | |

Upper Quota | 3 | 6 | 3 | 2 | 6 | 5 | 25 |

This time the total number of seats is 25, the correct number of seats to be apportioned. Give Alpha three seats, Beta six seats, Gamma three seats, Delta two seats, Epsilon six seats, and Zeta five seats.

*Note: This is not the same result as we got using Hamilton’s method in Example \(\PageIndex{4}\).*

## Webster’s Method

Webster’s method divides all populations by a modified divisor and then rounds the results up or down following the usual rounding rules. Just like Jefferson’s method we keep guessing modified divisors until the method assigns the correct number of seats. Because some quotas are rounded up and others down we do not know if the standard divisor will give a sum that is too large or too small. Our guess for the first modified divisor should be the standard divisor.

**Summary of Webster’s Method:**

- Find the standard divisor, . Use the standard divisor as the first modified divisor.
- Divide each state’s population by the modified divisor to get the modified quota.
- Round each modified quota to the nearest integer using conventional rounding rules.
- Find the sum of the rounded quotas.
- If the sum is the same as the number of seats to be apportioned, you are done. If the sum is too big, pick a new modified divisor that is larger than d. If the sum is too small, pick a new modified divisor that is smaller than d. Repeat steps two through five until the correct number of seats are apportioned.

**Example \(\PageIndex{3}\): Webster’s Method**

Use Webster’s method to apportion the 25 seats in Hamiltonia from Example \(\PageIndex{2}\).

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

From Example \(\PageIndex{2}\) we know the standard divisor is 9480. Because some quotas will be rounded up and other quotas will be rounded down we do not know immediately whether the total number of seats is too big or too small. Unlike Jefferson’s and Adam’s method, we do not know which way to adjust the modified divisor. This forces us to use the standard divisor as the first modified divisor.

Note that we must use more decimal places in this example than in the last few examples. Using two decimal places gives more information about which way to round correctly. Think about Alpha’s standard quota. Both 2.48 and 2.53 would round off to 2.5. However, 2.48 should be rounded down to 2 while 2.53 should be rounded up to 3 according to Webster’s method. This situation has not happened in any of the previous examples.

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

d = 9480 | 2.53 | 5.91 | 2.95 | 1.79 | 6.86 | 4.96 | |

Rounded Quota | 3 | 6 | 3 | 2 | 7 | 5 | 26 |

Since the total of 26 seats is too big we need to make the modified divisor larger. Try d = 11,000.

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

d = 9480 | 2.53 | 5.91 | 2.95 | 1.79 | 6.86 | 4.96 | |

Rounded Quota | 3 | 6 | 3 | 2 | 7 | 5 | 26 |

d = 11,000 | 2.18 | 5.09 | 2.55 | 1.55 | 5.91 | 4.27 | |

Rounded Quota | 2 | 5 | 3 | 2 | 6 | 4 | 22 |

The total number of seats is smaller like we hoped but 22 is way too small. That means that d = 11,000 is much too big. We need to pick a new modified divisor between 9480 and 11,000. Try a divisor closer to 9480 such as d = 10,000.

State | Alpha | Beta | Gamma | Delta | Epsilon | Zeta | Total |

Population | 24,000 | 56,000 | 28,000 | 17,000 | 65,000 | 47,000 | 237,000 |

d = 9480 | 2.53 | 5.91 | 2.95 | 1.79 | 6.86 | 4.96 | |

Rounded Quota | 3 | 6 | 3 | 2 | 7 | 5 | 26 |

d = 11,000 | 2.18 | 5.09 | 2.55 | 1.55 | 5.91 | 4.27 | |

Rounded Quota | 2 | 5 | 3 | 2 | 6 | 4 | 22 |

d = 10,000 | 2.40 | 5.60 | 2.80 | 1.70 | 6.50 | 4.70 | |

Rounded Quota | 2 | 6 | 3 | 2 | 7 | 5 | 25 |

*Note: This is the same apportionment we found using Hamilton’s and Jefferson’s methods, but not Adam’s method.*