At the time the ship was as old as the captain is now, the captain was twice as old as the ship.
Together they are 56 years old.
How old is the captain and ship?
I've figured out the answer, but I did this through trial and error basically. An equation or way of thinking on how to solve would be much appreciated!
On
So let us call their current ages $x$,$y$ and the age difference since they were "double aged": $k$.
$x+y = 56$ : sum of current ages is 56.
$y = x-k = 2(y-k)$ which can be rewritten as 2 separate equations:
$y = x-k$
$y = 2(y-k)$
Now rewriting this as a linear equation system (x,y,k) $$\bf Ax = b$$ where $${\bf A} = \left[\begin{array}{rrr}1&1&0\\1&-1&-1\\0&-1&2\end{array}\right], {\bf b} = \left[\begin{array}{r}56\\0\\0\end{array}\right] $$
we find the solution:
$${\bf x} = \left[\begin{array}{c}33.6\\22.4\\11.2\end{array}\right]$$
$$33.6+22.4 = 56$$ and $$22.4 = 2 \cdot 11.2$$ which was their ages $11.2$ years ago.
So either I got some equation wrong or it is actually solvable despite the confusing grammar.
On
The ship is $56 - x$ years old,
The captain was the age of the ship $x - (56 - x)$ years ago.
So, $2x - 56$ years ago was when the captain was the age of the ship.
This indicates the ship was $(56 - x) - (2x - 56)$ years of age.
So $2(112 - 3x) = x$
So, $224 = 7x$
So $x = 32$
It was $32$ years ago.
Putting the facts into equations shows that the question is a paradox. Let's assume the captains age is $C$ and the age of the ship is $S$ now. Then we have $C+S = 56$.
At some other time $t$ from now the captain is $C+t$ old and the ship $S+t$ old. When the ship "was" as old as the captain is now we have $S+t = C$ and then the captain was $C+t$ old and twice as old as the ship that is $C+t$ = $2(S+t)$.
That is the three equations become:
$$C+S = 56$$ $$S+t = C$$ $$C+t = 2(S+t)$$
Substitution in of $C=S+t$ in the last gives:
$$S+t+t = S+2t = 2(S+t) = 2S+2t$$
which gives that $S=0$, substituting this back into the second gives that $C=t$ and that into the first gives $C+S=t+S=t+0=56$.
That is that the time mentioned as "was" is indeed in the future. The captain is now 56 and he "was" 104 when the boat "was" as old as he is now (56).
If on the other hand the question is as mrprottolo pointed out:
the equations become:
$$C+S=56$$ $$S-t = C$$ $$S = 2(C-t)$$
which has the solution $C=24$ and $S=32$ and $t=8$: a ship is twice as old (32y) as the captain was (16y, 8yrs ago) when the ship was as old as the captain is(24y, 8yrs ago).