Henry Ernest Dudeney/Modern Puzzles/24 - Simple Arithmetic/Solution

From ProofWiki
Jump to navigation Jump to search

Modern Puzzles by Henry Ernest Dudeney: $24$

"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?


Solution

$20$ and $64$.


Proof

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

Then:

\(\ds a - b\) \(=\) \(\ds 44\)
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds b + 44\)
\(\ds a b\) \(=\) \(\ds 1280\)
\(\ds \leadsto \ \ \) \(\ds \paren {b + 44} b\) \(=\) \(\ds 1280\)
\(\ds \leadsto \ \ \) \(\ds b^2 + 44 b - 1280\) \(=\) \(\ds 0\)


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.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Henry_Ernest_Dudeney/Modern_Puzzles/24_-_Simple_Arithmetic/Solution&oldid=559096"