At the 'Shoot and Win' stall in a fair, $X$ fires $5$ shots to $Y$ 's $3$, but $X$ hits the target only once in three shots while $Y$ hits once in $2$ shots. When $Y$ has missed $27$ times, how many times has $X$ hit his target?

Let the common timeframe or rounds be considered.
The ratio of shots fired by $X$ to $Y$ is $5:3$. Let $X$ fire $5k$ shots and $Y$ fire $3k$ shots.

Given hitting efficiencies:

* For $X$: Hits $\frac{1}{3}$ of his shots, so misses $\frac{2}{3}$ of his shots.
* For $Y$: Hits $\frac{1}{2}$ of his shots, so misses $\frac{1}{2}$ of his shots.

The number of times $Y$ misses is given as $27$:

$$\text{Misses of }Y=\frac{1}{2}\times 3k=27⟹1.5k=27⟹k=\frac{27}{1.5}=18$$

Now, find the number of times $X$ hits his target:

$$\text{Hits of }X=\frac{1}{3}\times 5k=\frac{5}{3}\times 18=5\times 6=30\text{ times}$$