Henry Ernest Dudeney/Modern Puzzles/24 - Simple Arithmetic

by : $24$

 * Simple Arithmetic
 * Two gentlemen with an eccentric approach to philosophy were pinned down by your investigative reporter. 


 * They wished to riddle my mathematical understanding.


 * "Our two ages combined," said the first, "is $44$."


 * "Don't be silly," said the other, "it's $1280$."


 * They looked at me and said, "You see, we didn't tell you how we were combining them."


 * It was clear to me that the first number was their difference and the second was their product.


 * Now, how old were these two gentlemen?

Note that the above is a contemporary paraphrase of the original statement of this problem, which used language and concepts which are likely to cause offence in today's society.

Solution

 * $20$ and $64$.

Proof
Let their ages be $a$ and $b$.

Then:

You can either factorise $1280$ into its two factors which differ by $44$, which brings us no further than where we started from, or we use the Quadratic Formula to smash this exquisite gem with a mallet:


 * $b = \dfrac {-44 \pm \sqrt {44^2 + 4 \times 1280} } 2 = -22 \pm 42$

and we can pick out what we want from the debris.