Question 1183919
Aaron is three years older than Ben<pre>
A = B + 3</pre>
and Ben is three years older than Chris.<pre>
B = C + 3, </pre>
The sum of their ages in years is between 12 and 21. <pre>
12 < A + B + C < 21</pre>
How old are Aaron, Ben, and Chris if the product of their ages is 80?<pre>
A = B + 3
C = B - 3

Substitute in 

ABC = 80
(B + 3)B(B - 3) = 80
B(B + 3)(B - 3) = 80
B(B<sup>2</sup> - 9) = 80
B<sup>3</sup> - 9B<sup>2</sup> = 80
B<sup>3</sup> - 9B<sup>2</sup> - 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

B<sup>3</sup> - 9B<sup>2</sup> - 80 = 0  ,

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

B<sup>3</sup> - 9B<sup>2</sup> - 80 = 0

5 | 1   0   -9   -80
  |<u>     5   25    80</u>
    1   5   16     0

(B - 5)(B<sup>2</sup> + 5 + 16) = 0

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

So Ben is 5</pre>Aaron is three years older than Ben<pre>
So Aaron is 8</pre>
Ben is three years older than Chris.<pre>
So Chris is 2.

Edwin</pre>