+586 A band is marching in a rectangular formation with dimensions n-2 and n + 8. 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.
Solve for n:
(n - 2) (n + 8) = n (2 n - 3) + 4
Expand out terms of the left hand side:
n^2 + 6 n - 16 = n (2 n - 3) + 4
Expand out terms of the right hand side:
n^2 + 6 n - 16 = 2 n^2 - 3 n + 4
Subtract 2 n^2 - 3 n + 4 from both sides:
-n^2 + 9 n - 20 = 0
The left hand side factors into a product with three terms:
-(n - 5) (n - 4) = 0
Multiply both sides by -1:
(n - 5) (n - 4) = 0
Split into two equations:
n - 5 = 0 or n - 4 = 0
Add 5 to both sides:
n = 5 or n - 4 = 0
Add 4 to both sides:
n = 5 or n = 4
+129899 The total number of members is given by
(n - 2) ( n + 8) (1)
In the second formation, we have that the number of members, excluding the drummers is n (2n - 3) (2)
So...since there are at least 4 drummers, the difference between (1) and (2) is ≥ 4
So we have that
(n - 2) ( n + 8) - [ n (2n - 3) ] ≥ 4
n^2 + 6n - 16 - [2n^2 - 3n] - 4 ≥ 0 simplify
-n^2 + 9n -20 ≥ 0 multiply through by -1 and reverse the inequality sign
n^2 - 9n + 20 ≤ 0 (1) let's just set this to 0 and solve
n^2 - 9n + 20 = 0 factor this and we have that
(n - 5) ( n - 4) = 0 set each factor to 0 and solve for n and we have that
n = 5 or n = 4
The solution to the inequality (1) is 4 ≤ n ≤ 5
Since we assume n to be an integer, the sum of all possible n = 4 + 5 = 9
