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 the problem, we can set up a system of equations based on the constraints provided. Let: - \( M \) = number of Math sections - \( E \) = number of English sections - \( P \) = number of Philosophy sections From the problem, we have the following constraints: 1. The college needs to make available 8 less English sections than Math sections: \[ E = M - 8 \] 2. The student demand for the optional Philosophy course is half as many sections as English sections: \[ P = \frac{1}{2}E \] 3. The total sections of all three courses must not exceed 48: \[ M + E + P \leq 48 \] Now, we can substitute the expressions for \( E \) and \( P \) into the total sections equation. Substituting \( E \) from the first equation into the second equation: \[ P = \frac{1}{2}(M - 8) \] Now substituting \( E \) and \( P \) into the total sections equation: \[ M + (M - 8) + \frac{1}{2}(M - 8) \leq 48 \] Now simplify this equation: \[ M + M - 8 + \frac{1}{2}M - 4 \leq 48 \] \[ 2M + \frac{1}{2}M - 12 \leq 48 \] \[ \frac{5}{2}M - 12 \leq 48 \] \[ \frac{5}{2}M \leq 60 \] \[ M \leq 24 \] Now, since \( M \) must be a whole number, the maximum number of Math sections is 24. Now we can find the corresponding values for \( E \) and \( P \): 1. If \( M = 24 \): \[ E = M - 8 = 24 - 8 = 16 \] \[ P = \frac{1}{2}E = \frac{1}{2}(16) = 8 \] Now we check the total: \[ M + E + P = 24 + 16 + 8 = 48 \] This satisfies the total sections constraint. Thus, the college should make available: - **Math sections (M)**: 24 - **English sections (E)**: 16 - **Philosophy sections (P)**: 8 These values meet all the constraints provided in the problem.