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Math 106: Contemporary Math9 chapters | 106 lessons

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Lesson Transcript

Instructor:
*Maria Airth*

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

The Jefferson Method of Apportionment is just one of many different methods of apportionment. In this lesson, we will review the Jefferson Method using examples to solidify the concepts.

Congratulations! You have been elected president of the United States of America! As your first act in office, you have decided to help middle school students all over the U.S. by consolidating the states into just three, easy to remember states. No more memorizing 50 states and capitals. You have, aptly, named these new states Stars, White Stripe, and Red Stripe (the stars and stripes, for short).

Well, these new states need to have representation in your Congress. You've decided to use the **Jefferson Method of Apportionment** to assign seats to each state. This method of apportionment was used by the U.S. Congress from 1790 to 1840.

But, how do you do that? To understand the Jefferson Method of Apportionment, you must first understand that **apportionment** is dividing, or portioning, the total number of voting seats in a body of government between entities. So, what we are doing is deciding how many votes each state will have.

In Jefferson's Method of Apportionment, each state is assigned a lower quota of seats. If the lower quota equals the total to be assigned, you are finished. But, if the lower quota does not match the seats to be assigned, you must modify the standard divisor until the modified quotas for the states does equal the total to be assigned. I may have just mentioned some terms that you may not be familiar with. Let's define those now.

In Jefferson's Method, we first have to find the lower quota. To do this, we start at the very beginning with determining a **standard divisor** by dividing the total population by the number of seats to assign. For example, if the total population is 1,200, the number of seats is 12, then the standard Divisor = 1,200 / 12 = 100.

We use the standard divisor to get the **standard quota**, the number of seats each state should get. This is calculated by dividing the population of the state by the standard divisor. That makes a bit of sense; the **divisor** divides everything, and the **quota** is the number of seats calculated using the divisor.

Earlier, we got a standard divisor of 100, so let's assume we have a state with 120 people. That means that the standard quota = 120 / 100 = 1.2. Notice that the quota is not a whole number. It would not be possible to occupy two-tenths of a seat for voting purposes, so we need to find a method to change that decimal into a whole number.

The nearest whole numbers to 1.2 are 1 and 2. It's pretty obvious that the lower whole number is referred to as the **lower quota** and the higher whole number is called the **upper quota**.

So, so far we have:

**Standard divisor (SD)**: the population / seats, and that's the number to divide with**Standard quota (SQ)**: state population / standard divisor, which is the raw fractional result after using the divisor**Lower quota (LQ)**: the closest, smaller whole number to the standard quota**Upper quota (UQ)**: the closest, larger whole number to the standard quota

It would be great if all the lower quotas of any Jefferson Method procedure added up to the correct number of seats, but that really doesn't happen often. So, we have to go back to the beginning and change the original standard divisor. This creates a **modified divisor (MD)**. You can use trial and error to change the divisor however you like; just make sure you always make it less than the original standard divisor.

Now that you have chosen a new modified divisor, you use the same calculations as before to come up with a **modified quota (MQ)** and the same criteria to determine the **modified lower quota**. Continue modifying the divisor until your calculations result in modified lower quotas that sum to equal your desired number of seats exactly.

This second run through on the procedure has added three new terms to our list:

**Modified divisor (MD)**: a number smaller than the original standard divisor, picked through trial and error**Modified quota (MQ)**: the quota as calculated using the modified divisor**Modified lower quota (MLQ)**: the closest smaller whole number to the modified quota

Can you see how each of these is directly related to its standard counterpart? To return to our example from before, the lower quota was 1. What if we had decided to have two seats in the House? This is where the modification of the divisor comes in. Our original standard divisor of 100 led to a final result of having too few seats apportioned. No problem; we can just make up a new modified divisor smaller than 100. If the modified divisor = 50, then the modified quota = 120 / 50 = 2.4. Here, the modified lower quota = 2, and that is the correct total number of seats, so we're done now.

It might sound a bit suspicious to be modifying numbers just because we want to, but remember, as long as every state's population is being divided by the same divisor, they are all being treated the same. It comes down to needing to apportion all the seats within the calculation without just giving seats away after the fact.

Let's return to your dilemma of apportioning out voting seats in the House of Representatives to your three newly formed states. There are about 320 million people in the U.S. We'll assume that Stars has 120 million of these, White Stripe has 75 million, and Red Stripe has 125 million. You decide that you would be happy with 500 voting seats. So, using Jefferson's Method of Apportionment, how many seats does each state receive?

Well, we start with finding our standard divisor. 320,000,000 / 500 = 640,000. Now we find our standard quotas for each state by dividing each state population by the standard divisor.

Stars = 120,000,000 / 640,000 = 187.5

White Stripe = 75,000,000 / 640,000 = 117.18

Red Stripe = 125,000,000/ 640,000 = 195.3

Remembering that the lower quota is the rounded down whole number of the standard quota, we sum the lower quotas to get: 187 + 117 + 195 = 499. So close! But the rules state we must get exactly the number of seats to be apportioned. So, we have to continue on to the modified versions of these calculations. Don't forget you get to choose whatever modified divisor you want; it just has to be smaller than the original 640,000.

Using trial and error, I chose modified divisor = 638,000.

That gives stars = 120,000,000 / 638,000 = 188.08, modified lower quota = 188

White Stripe = 75,000,000 / 638,000 = 117.55, modified lower quota = 117

Red Stripe = 125,000,000 / 638,000 = 195.92, modified lower quota = 195

Thus, the sum of the modified lower quotas is 500. Perfect! Stars will receive 188 seats, White Stripe will have 117 votes, and Red Stripe will send 195 people to sit in the House of Representatives. Great job, President!

Please note that it can be time consuming to find the correct modified divisor. You might be lucky and hit on the correct one on your first try, but if you don't, just keep doing the calculations until they come out with the lower quotas summing to the total number of seats.

Those were some big numbers we were working with before. Let's try an example with smaller numbers. Now that you have your House of Representatives seats apportioned to the states, you want to create a smaller task force group to work more closely with you. You want only ten people on this task force but want to use Jefferson's Method of Apportionment to apportion the seats on this board to the state representatives already in the House.

Do you remember the first step? Good; we need to get the standard divisor by dividing the total population of the House by the number of seats on the task force. Standard divisor = 500 / 10 = 50. Great work.

What's next? Yes, we find each state's standard quota and then adjust it to its lower quota. You may need to look back to remember how many members each state received in the previous example.

Stars: 188 / 50 = 3.76, lower quota = 3

White Stripe: 117 / 50 = 2.34, lower quota = 2

Red Stripe: 195 / 50 = 3.9, lower quota = 3

The sum of lower quotas is 8, which is not equal to 10, so we have to go down the modified fork in the road.

Choosing modified divisor = 47, the new modified quotas and modified lower quotas are:

Stars: 188 / 47 = 4, modified lower quota = 4

White Stripe: 117 / 47 = 2.48, modified lower quota = 2

Red Stripe: 195 / 47 = 4.18, modified lower quota = 4

Summing the modified lower quotas, we see that now we have all ten seats on the task force assigned. Great job.

This lesson included quite a lot of information. The road to apportioning voting seats using the Jefferson Method is forked, which can lead to some time-consuming calculations. Thankfully, the calculations are fairly simple.

The **Jefferson Method of Apportionment** called for assigning the lower quota to each state. If lower quota was equal to the seats to be assigned, then the process is finished. If not, then by trial and error, choose a modified divisor that will result in modified lower quotas that sum to the exact number of seats to be apportioned.

We learned many terms in this lesson.

**Standard divisor**is the total population divided by the number of seats to be assigned. This number is used to divide all the state populations to determine quotas.**Modified divisor**is a number chosen by trial and error that will be used to divide each state population if the standard version does not result in the correct number of seats to be assigned.**Standard quota**is the state population divided by the standard divisor.**Modified quota**is the state population divided by the modified divisor.**Lower quota**is the closest smaller whole number to the standard quota, and you round down.**Modified lower quota**is the closest smaller whole number to the modified quota.

When the lower quotas or modified lower quotas add up to the exact number of seats to be apportioned, you have finished the Jefferson Method of Apportionment, and you can breathe easy in your new presidency! Thanks for joining me.

Upon completing this lesson, you will be able to:

- Define standard divisor, standard quota, lower quota, and their modified counterparts
- Explain how to use the Jefferson Method of Apportionment

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Math 106: Contemporary Math9 chapters | 106 lessons

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