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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.

 Jul 12, 2018
 #1
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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

 Jul 12, 2018
 #2
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+2

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

 

 

cool cool cool

 Jul 12, 2018

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