At Allegany College of Maryland freshman need to take a 100 level Math and English plus optional Philosophy course. In any quarter the college needs to make available 8 less English sections than Math sections. In any quarter student demand for the optional Philosophy course is half as many sections as English sections. Available classrooms limit the total sections of all three courses to 48 Given these constraints how many sections of each course should the college make available each quarte to meet demand? English sections Philosophy sections

To solve this problem, let's denote the number of Math sections as \( M \), English sections as \( E \), and Philosophy sections as \( P \). Based on the constraints given: 1. The college needs to make 8 less English sections than Math sections: \( E = M - 8 \) 2. The demand for Philosophy sections is half of the English sections: \( P = \frac{1}{2}E \) 3. The total sections available for all three courses are limited to 48: \( M + E + P \leq 48 \) Substituting the equations into the total sections constraint: \[ M + (M - 8) + \frac{1}{2}(M - 8) \leq 48 \] This simplifies to: \[ M + M - 8 + \frac{1}{2}M - 4 \leq 48 \] Combining terms yields: \[ \frac{5}{2}M - 12 \leq 48 \] Adding 12 to both sides results in: \[ \frac{5}{2}M \leq 60 \] Multiplying by \(\frac{2}{5}\): \[ M \leq 24 \] Using \( M = 24 \): - \( E = 24 - 8 = 16 \) - \( P = \frac{1}{2}(16) = 8 \) So, the college should offer **24 Math sections**, **16 English sections**, and **8 Philosophy sections** each quarter to meet the demand! Happy studying! 😊