1. Getting Your Fair Share Jelly beans, student groups, and Alexander Hamilton Everyone wants to make sure that they get their fair share, but dividing assets among a group isn’t as easy as division. Prepared for SSAC by Aaron Montgomery - Central Washington University © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007 SSAC2007.JF1075.AM1.1 1 Core Quantitative Skill: Basic Arithmetic Ratio and proportion; percentage Supporting Skills Estimation Proportion image from www.neopets.com

2. 2 Sometimes, determining how much a group deserves is a simple division problem. Unfortunately, if the items to be distributed are indivisible, there may be leftover items after the division. Such division problems occur when distributing money or assigning legislative seats. This module discusses what constitutes a fair division and illustrates some methods that can be used to allocate the “left over” items in a fair manner. Slide 3 presents the problem. Slides 4 & 5 discuss fair divisions and apportionment. Slides 6–8 discuss proportion and percentages. Slides 9 & 10 discuss different ways of computing quotas. Slides 11–13 present Hamilton’s method of apportionment. Slide 14 contains a homework assignment pertaining to this module. Overview of Module

3. 3 You need to decide how to allocate $5,450 in funds to four student clubs. Funds must be distributed in ten dollar amounts and each organization should receive funding based on the membership size of that organization. Booster Club: 145 students Outdoor Club: 94 students Debate Club: 87 students Math Club: 7 students How much should each group get? Problem Note: the goal is to decide on a fair method of division, one that would work in places where we cannot continue to subdivide indefinitely. The choice of $10 increments for this problem has to do with the man on that bill.

4. 4 Fair Division Although the title of this slide indicates that I will define what it means for a distribution to be fair, I will not. You need to think about what fairness means in this context. One generally accepted aspect of fair distribution, is that you should make a decision about what rule to use before you know how the rule will affect your particular situation. When allocating funds to various clubs, you should decide on which rule is fair before knowing the sizes of the clubs that you will be joining. So before you decide how much each group should get, you need to ask a different question: What rule do I use?

5. 5 Claimants and Claims A Claimant is something that will be given some of the items. Claimants might be people, organizations, or registered voters. Each claimant has a certain number of claims on the items. Two claimants may have different numbers of claims. For example, in our case, the Booster Club is one of the four claimants and its 145 members would entitle it 145 claims on the money.

6. 6 Ratio and Proportion A proportion is the quotient of the number of cases possessing a property (numerator) to the total number of cases observed (denominator). In our example, we can compute the proportion of memberships that are booster club memberships by computing 145/(145 + 94 + 87 + 7) = 145/333 ≈ 0.44. We would say that the proportion of booster club memberships is 0.44. A percentage is obtained when a proportion is multiplied by 100. It can be interpreted as the number of cases possessing a property out of 100 total cases. The name “percent” literally translates from Latin as “per hundred.” In our example, we would say that 44% of the memberships belong to the booster club. Proportions and percentages are useful when converting between quantities with different units. In our example, we can use the percentage of memberships to determine the percentage of money each organization should get.

7. 7 Common Percentage Errors Because 0.44 is to 1 as 44 is to 100, 0.44 and 44% are the same proportion. However, 0.44% is not the same as these two values, 0.44% = 0.0044 is a very small number. And 44 is also a different value, 44 = 4400% is a very large percentage. You will need to be careful how you enter 0.44 = 44% into Excel. If the cell is formatted as a “Number,” you will want to enter 0.44. If the cell is formatted as a “Percentage,” you will want to enter 44. Proportions need to be multiplied by the total number of objects to obtain an actual count on the number of objects. There are not 0.44 of anything in the previous examples. The number of booster club memberships is (0.44)(333) = 146.52 ≈ 147 booster club memberships. Percentages are always “of” some larger group. For example, you say “19% of the jellybeans are green.” Reference to the larger group is often omitted if the larger group is understood from context, but this omission can be the source of errors.

8. 8 Determining Percentages Recreate this spreadsheet. = cell with a number in it = cell with a formula in it You should use SUM() to compute Total Memberships and to check your values. As a first step, determine the Percentage of the memberships that occurs in each of the organizations. Work in tens of dollars.

9. 9 Definitions: Quotas The standard quota for a claimant is obtained by multiplying the number of items to be distributed by the percentage of the claims belonging to the claimant. For example, if a group has 20% of the people and you are distributing 103 jellybeans, then this group receives a standard quota of 0.20 * 103 jellybeans = 20.6 jellybeans. Notice that the standard quota has a unit (jellybeans in this case). The lower quota for a claimant is obtained by rounding the standard quota down to the nearest integer. The upper quota for a claimant is obtained by rounding the standard quota up to the nearest integer. In the previous example, the lower quota is 20 jellybeans and the upper quota is 21 jellybeans. Like the standard quota, the lower and upper quotas have a unit (jellybeans in this case).

10. 10 Determining Lower Quotas Add Columns F and G to recreate this spreadsheet. = cell with a number in it = cell with a formula in it Use ROUNDDOWN() to obtain these numbers This is the extra money left over after everyone has been allotted their lower quota. Next determine the Standard Quota and the Lower Quota for each group.

11. 11 Fractional Part and Hamilton’s Method The integer part of a positive number is the result of rounding the number down to the nearest integer, this is exactly the same as the lower quota which we determined in the previous worksheet. The fractional part of a positive number is equal to the number minus the integer part of the number. For example, 3.21 has an integer part of 3 and a fractional part of 0.21. To allot the excess using Hamilton’s method, we give the first excess $10 to the claimant with the largest fractional part, the second excess $10 to the claimant with the next largest fractional part, and so forth until we have run out of excess $10s. However, we would like to make Excel do this work for us and the next slide describes an Excel function that can help us with this task.

12. 12 Using LARGE() and IF() We need to find the organizations with the two largest fractional parts because we have two excess $10s. We start by determining the 2 nd largest fractional part, which we will call the cutoff. To determine the cutoff, we will use the function LARGE(). This function takes an array and an integer n and returns the n th largest value from the array. In our case, the function call will be LARGE(H4:H7, G11). Any organization whose fractional part is greater than or equal to this value will be among those given an extra $10. To determine if an organization gets an extra $10, we will use the IF() function. This function takes a test, a value to be used if the test is true, and a value to be used if the test is false. For the case of the Booster Club, the function call will be IF(H4 >= H11, 1, 0).

13. 13 Determining Hamilton’s Allotment Add Columns H, I, and J to recreate this spreadsheet. Notice that I’ve compressed Columns D, E, and F so you can see the new columns better. You should leave these columns expanded in your worksheet. = cell with a number in it = cell with a formula in it Use IF() here. Sum up your allocations to check your work. Now determine how much everyone gets. Use LARGE() here.

14. 14 End of Module Assignment 1.A jar contains 35 green jellybeans, 20 red jellybeans, 12 yellow jellybeans and 15 black jellybeans. Compute the following without Excel. a.What percentage of jellybeans are green? b.What percentage of jellybeans are not black? c.In a collection of 10 jellybeans, how many would you expect to be red? 2.Use your spreadsheet to answer the following questions. In all cases, assume that you are starting with the scenario described at the start of this module and that we are using Hamilton’s Method. a.Assume that student memberships have not changed but there is one more dollar that can be spent on student activities. How does the distribution change. Who gets the extra dollar? Why is the Math Club upset by this increase in funding? Is the result fair? b.Assume that a Latin Club has been started with exactly the same membership size as the Math Club (which has 7 people and $12 of funding). To accommodate this club, there is an increase of $12 funding above the original $545. When money is allocated, why are the Math and Latin club members upset? Is the result fair? 3.Assume that we are distributing legislative seats among different districts. The three districts have populations of sizes 657,000, 237,000, and 106,000. There are 100 seats in the legislature and we are using Hamilton’s Method. a.How many seats should each group receive. Which district is over represented? Which district is under represented? b.Assume that the populations change with the first group growing to 660,000, the second growing to 245,100, and the third shrinking to 104,900. How many seats should each group receive? Is this new allocation surprising? Is it fair? 4.Who is featured on the $10 bill? What are some of his claims to fame?

15. 15 Pre- and Post-Assessment 1.If a group of people consists of 12 Republicans, 16 Democrats and 5 Libertarians, what percentage of these people are Republicans? 2.$120 is to be divided among three groups of people with sizes of 5, 3, 2. Roughly how much should be given to the group of 3 people. 3.Three groups are represented in a parliament. The groups have sizes of 34,505, 25,123, and 10,240 and have 3, 3, 1 representatives. Which of these groups is over- represented and which is under-represented? 4.Idaho (population 1,297,274 in the 2000 Census), Oregon (population 3,428,543 in the 2000 Census), and Washington (population 5,908,684 in the 2000 Census) have a total of 16 representatives in the House of Representatives. How should these representatives be divided up among these three states.