My Math Forum How to Distribute scholarships among students

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 November 12th, 2007, 04:39 AM #1 Newbie   Joined: Nov 2007 Posts: 3 Thanks: 0 How to Distribute scholarships among students Hello, We are trying to reward the intelligent students with scholarships. However, the amount is not the same rather it varies proportionally with the Grade Point Average (GPA). 1. We reserve 15% of the semester tuition fee paid by students for awarding scholarships. The total amount of scholarships shall not exceed this 15% money and no (or little) money shall be left unawarded. 2. We award scholarships for students having GPA >= 3.5 with a max. limit of 4.0. 3. We award scholarships that are in the limit of min. 15% or max 75% of the tuition fee that a student has paid, depending on his GPA. We do not want a distribution where student with a GPA of 3.9 and one with a GPA of 3.5 have a very small difference in the scholarship amount. We want that the higher the GPA the more shall be the scholarship amount. However, the scholarship amount for lower GAP level shall also not be too small. Can you please help me in figuring how the formula would look like of if there are some existing formulas that cater for such distributions? If I was to award the scholarships with an equal amount to all the students then it would have been very simple. But since the amount depends on the GPA and we want to see a significant difference in what is offered to a student with the highest GPA versus lowest GPA so I am having problem in figuring it out. Till now we have been doing it manually. We start with a an amount for the student with highest GPA and then gradually go down. But some times by using this method we exceed the rules 1 & 3 (discussed above) so we have to start again till the time that we have a distribution that satisfied the rules mentioned above. This is more or less hit and trial method. Then we came up with this formula Max. Scholarship=1000+(GPA-3.5) x 8000 . It didnâ€™t work out as it gives a linear distribution with the total amount of scholarship exceeding the 15% amount we have. Then we checked with some variations in this formula but they also had some shortcomings that gives me an idea that it won't work this way.
 November 13th, 2007, 06:59 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Choose a scholarship amount on the low end (I'll use $1000 as in your above example) and you can find a linear amount that uses up all of your money. Let there be n students with GPA >= 3.5. Then let s = sum (GPA - 3.5) for all students with GPA >= 3.5. Further, let A = 0.15T be the amount of money reserved for scholarships. Then A -$1000n is the money you can distribute for the higher scorers, so let k = (A - $1000n) / s. Then each student with GPA at least 3.5 gets$1000 + k * (GPA - 3.5).
November 14th, 2007, 04:28 AM   #3
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Quote:
 Originally Posted by CRGreathouse Choose a scholarship amount on the low end (I'll use $1000 as in your above example) and you can find a linear amount that uses up all of your money. Let there be n students with GPA >= 3.5. Then let s = sum (GPA - 3.5) for all students with GPA >= 3.5. Further, let A = 0.15T be the amount of money reserved for scholarships. Then A -$1000n is the money you can distribute for the higher scorers, so let k = (A - $1000n) / s. Then each student with GPA at least 3.5 gets$1000 + k * (GPA - 3.5).
Dear GRGreathouse,

Thank you for the response. I have tried the formula as you have explained and it works fine except for one things.

When I fix the minimum as $2000 then the amount to be paid to the student with 4 GPA ($95K) becomes larger then 75% of what that student has paid in 6 months ($36K) (Condition 3 mentioned in my question). Now if I keep increasing the minimum then the max. also drops but the overall gap between the ones with low GPA and others with high GPA becomes too narrow. I may be able to live with a narrow gap but I need a smart way of finding the minimum value which will restrict the max. value below 75% of tuition paid by the student instead of doing it on hit and trial basis. Given below is the results for a class according to your formula T = Tution fee generated for one semester =$ 1,692,000.00
A =Scholarship allocation (15% of T) = $253,800.00 N = No. of students GPA>3.5 = 12 Min = Minimum Scholarship =$ 2000
Max. that can paid to a student=75% of 6000 x 6 months = $27,000 Money for higher scorers =$ 241,800.00
K = (A - Min) / S = 186,829.27
S = Sum(GPA-3.5)=1.23

S. No.-GPA--P=(GPA-3.5)--I=Min + k * P
1------ 4.00-----0.50-----------95,414.63
2------ 3.75-----0.25-----------48,707.32
3------ 3.67-----0.17-----------33,760.98
4------ 3.67-----0.17-----------33,760.98
5------ 3.58-----0.08-----------16,946.34
6------ 3.53-----0.03-----------7,604.88
7------ 3.52-----0.02-----------5,736.59
8------ 3.51-----0.01-----------3,868.29
9------ 3.50-----0.00-----------2,000.00
10-----3.50-----0.00-----------2,000.00
11-----3.50-----0.00-----------2,000.00
12-----3.50-----0.00-----------2,000.00

---------------------------Total--253,800.00

I also did not understand the Money for high scorers since its never used in the overall calculation.

Thank you for the help.[/code]

 November 14th, 2007, 06:14 AM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms OK, that helps a lot. A few more questions, if I may: how much do the numbers need to vary? For example, if the 4.0 got the full $27,000, would it be acceptable for the 3.75 to get$27,000 as well, or would it have to be a smaller amount? How much are you comfortable giving to the 3.5s? You have a lot of money to distribute amongst a small number of students. In fact, you have so much that even if you gave everyone above a 3.5 the full $27,000 you would need to give the 3.5s$9,450 each to exhaust the scholarship fund. This is enough that I wonder about other possibilities -- setting aside some money for the next year, lowering tuition, or giving out an additional scholarship, this one not based on GPA. If you stick to this scheme of scholarship funding, you need to choose an amount between $9,450 and$21,150 to give to those with a 3.5 GPA, and then distribute the remainder as fairly as possible. It's hard to make suggestions without a notion of what you consider 'fair', but here's an attempt: $10,000 to the 3.50 GPAs$27,000 to everyone over 3.55 $26,341 to 3.53$26,267 to 3.52 \$26,192 to 3.51 This comes from a linear function on (GPA - 3) for everyone over 3.5 but not getting the maximum.
 November 15th, 2007, 02:04 AM #5 Newbie   Joined: Nov 2007 Posts: 3 Thanks: 0 How to Distribute scholarships among students You are right, I also observed that if available amount is huge then the difference between award scholarships between varying GPA would not be significant. We certainly do not want to keep money with ourselves unless it is evident that we cannot grant all of it to the students because of the 75% constraint. I have done the exercise for another class where conditions are more favorable. I have setup the minimum amount as 1.7% of the available amount and the overall distribution seems nice. It looks like that the Min. has to vary according to the funds available so it will be something like an if-then-else statement. Here is the results for another exercise T = Tution fee generated "2,124,000.00" A =Scholarship allocation "318,600.00" N = No. of students GPA>3.5 25 "Min = Minimum amount ,< 1.7% >" 5416.2 1.7 S. No. 1 4 0.5 "26,767.60" 2 4 0.5 "26,767.60" 3 3.94 0.44 "24,205.43" 4 3.85 0.35 "20,362.18" 5 3.82 0.32 "19,081.10" 6 3.79 0.29 "17,800.01" 7 3.71 0.21 "14,383.79" 8 3.68 0.18 "13,102.70" 9 3.68 0.18 "13,102.70" 10 3.65 0.15 "11,821.62" 11 3.65 0.15 "11,821.62" 12 3.65 0.15 "11,821.62" 13 3.65 0.15 "11,821.62" 14 3.65 0.15 "11,821.62" 15 3.62 0.12 "10,540.54" 16 3.59 0.09 "9,259.45" 17 3.59 0.09 "9,259.45" 18 3.59 0.09 "9,259.45" 19 3.56 0.06 "7,978.37" 20 3.53 0.03 "6,697.28" 21 3.53 0.03 "6,697.28" 22 3.53 0.03 "6,697.28" 23 3.53 0.03 "6,697.28" 24 3.5 0 "5,416.20" 25 3.5 0 "5,416.20" S 4.29 "318,600.00" Money for higher scorers "293,600.00" K "42,702.80"

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