SOLUTION: Aaron is three years older than Ben and Ben is three years older than Chris. The sum of their ages in years is between 12 and 21. How old are Aaron, Ben, and Chris if the product

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Question 1183919: Aaron is three years older than Ben and Ben is three years older than Chris. The sum of their ages in years is between 12 and 21. How old are Aaron, Ben, and Chris if the product of their ages is 80?
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
Aaron is three years older than Ben
A = B + 3

and Ben is three years older than Chris.
B = C + 3,

The sum of their ages in years is between 12 and 21. 
12 < A + B + C < 21

How old are Aaron, Ben, and Chris if the product of their ages is 80?
A = B + 3
C = B - 3

Substitute in 

ABC = 80
(B + 3)B(B - 3) = 80
B(B + 3)(B - 3) = 80
B(B2 - 9) = 80
B3 - 9B2 = 80
B3 - 9B2 - 80 = 0

12 < A + B + C < 21
12 < B + 3 + B + B - 3 < 21
12 < 3B < 21
 4 < B < 7

By the "P/Q" rule, if there are any rational roots to

B3 - 9B2 - 80 = 0  ,

they must be divisors of 80.  The only divisor of 80 between 4 and 7 is 5.

B3 - 9B2 - 80 = 0

5 | 1   0   -9   -80
  |     5   25    80
    1   5   16     0

(B - 5)(B2 + 5 + 16) = 0

So B - 5 = 0
       B = 5 is the only rational root.

So Ben is 5
Aaron is three years older than Ben
So Aaron is 8

Ben is three years older than Chris.
So Chris is 2.

Edwin
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