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# Band Problem!!!!

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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.

Aug 25, 2018

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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

Aug 26, 2018