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.

mathtoo Aug 25, 2018

#1**+2 **

Number of band members in first fromation is

(n -2)(n + 8)

Number of band members in second formation = n (2n - 3)

But...the difference in the numbers will be ≥ 4...since at least 4 drummers are excluded in the second grouping....so we have this inequality

(n - 2) ( n + 8) - n (2n - 3) ≥ 4 simplify

n^2 + 6n - 16 - 2n^2 + 3n - 4 ≥ 0

-n^2 + 9n - 20 ≥ 0 multiply through by -1....change the inequality sign direction

n^2 - 9n + 20 ≤ 0 (1)

The easiest way to solve this is to change it to an equality

n^2 - 9n + 20 = 0 factor

(n - 5) ( n - 4) = 0

Setting each factor to 0 and solving for n produces n = 5 or n = 4

We have 3 possible intervals that will solve (1)

(-inf, 4) U [4 ,5 ] U (5 ,inf)

If we let n = 4.5...then note that

(4.5)^2 - 9(4.5) + 20 = -.25 which makes (1) true

So since n can only assume integer values...... n = 4 or n = 5 will make (1) true

And the sum of these values = 9

Here's a graph of the solution : https://www.wolframalpha.com/input/?i=n%5E2+-+9n+%2B+20%C2%A0%E2%89%A4+0

CPhill Aug 26, 2018