# Scholarship Allocation Define variables: $x=$ number of $1000 scholarships $y=$ number of $2500 scholarships $z=$ number of $4000 scholarships Note that $1000x+2500y+4000z=500,000$, so we can solve for $z = -\frac{1}{4}x - \frac{5}{8}y + 125$ Next we examine how to calculate the expected values. A capital $E$ is used to denote expected value. We want to maximize: E[Graduations] = E[# graduates 3.25-3.5] + E[# graduates 3.5-3.75] + E[# graduates 3.75-4]. To estimate those expected values, you need to know some probabilities (denoted with a capital $P$). We define the following quantities. * $p_{3.25}=$P(a 3.25-3.5 student graduates without a scholarship) * $q_{3.25}=$P(a 3.25-3.5 student graduates with a scholarship) * $p_{3.5}=$P(a 3.5-3.75 student graduates without a scholarship) * $q_{3.5}=$P(a 3.5-3.75 student graduates with a scholarship) * $p_{3.75}=$P(a 3.75-4.0 student graduates without a scholarship) * $q_{3.75}=$P(a 3.75-4.0 student graduates with a scholarship) * $N_{3.25}=$ # of students with a GPA from 3.25-3.5 * $N_{3.5}=$ # of students with a GPA from 3.5-3.75 * $N_{3.75}=$ # of students with a GPA from 3.75-4.0 E[# graduates 3.25-3.5]= $(N_{3.25}-x)p_{3.25}+xq_{3.25}$ E[# graduates 3.5-3.75]= $(N_{3.5}-y)p_{3.5}+yq_{3.5}$ E[# graduates 3.75-3.5]= $(N_{3.75}-\left(-\frac{1}{4}x - \frac{5}{8}y + 125\right))p_{3.75}+\left(-\frac{1}{4}x - \frac{5}{8}y + 125\right)q_{3.75}$ We need to maximize a 2-variable function after making those 9 assumptions. I created a calculator at https://www.desmos.com/calculator/uywvkgggyy where you can just enter those 9 numbers and it will give you the values you are after.