Solving for $(x,y): 2+\frac1{x+\frac1{y+\frac15}}=\frac{478}{221}$ Solving for $x,y\in\mathbb{N}$:
$$2+\dfrac1{x+\dfrac1{y+\dfrac15}}=\frac{478}{221}$$
This doesn't make any sense; I made $y+\frac15=\frac{5y+1}5$, and so on, but it turns out to be a very complicated fraction on the left hand side, and I don't even know if what I got is correct.
Is there a more mathematical way to approach this problem?

Let's see,
$$\frac{478}{221}=2+\frac{36}{221}=2+\dfrac1{\frac{221}{36}}=2+\frac1{6+\frac5{36}}=2+\dfrac1{6+\dfrac1{\frac{36}{5}}}=2+\dfrac1{6+\dfrac1{7+\frac15}}$$
Thus, $x=6,y=7$.
