jmc

Question

A band is marching in a rectangular formation with dimensions n-2 and n + 8 (members). In the second stage of their performance, they re-arrange to form a different rectangle with dimensions n and 2n - 3, excluding all the drummers. If there are at least 4 drummers, then find the sum of all possible values of n.

Accepted Answer

Originally, there are (n-2)(n+8) = n^2 + 6n - 16 members in the band. In the second formation, there are at least 4 more than (n)(2n-3) = 2n^2 - 3n members in the band. Thus, n^2 + 6n - 16 2n^2 - 3n + 4, or simplifying, 0 n^2 - 9n + 20. The quadratic expression factors as 0 (n-4)(n-5). Thus 4 n 5, and n = 4,5. We can verify that both values work, from which it follows that the answer is 4+5 = 9.