The answer is $\boxed{216}$.
The question is asking for the number that is to $64$ as $125$ is to $27$. In other words, we are looking for a number that, when multiplied by $64$, gives $125$.
We can solve this problem by using the following formula:
$x \times y = z \times w$
where $x$ and $y$ are the numbers in the first ratio, $z$ and $w$ are the numbers in the second ratio, and $x$ is the number we are looking for.
Substituting in the given values, we get:
$x \times 64 = 125 \times 27$
$x = \frac{125 \times 27}{64}$
$x = 216$
Therefore, the number that is to $64$ as $125$ is to $27$ is $216$.
Here is a brief explanation of each option:
- Option A: $162$. This is the product of $64$ and $2.5$. However, $2.5$ is not equal to $\frac{125}{27}$, so $162$ is not the answer.
- Option B: $216$. This is the product of $64$ and $\frac{125}{27}$. Therefore, $216$ is the answer.
- Option C: $517$. This is the product of $64$ and $\frac{517}{27}$. However, $\frac{517}{27}$ is not a whole number, so $517$ is not the answer.
- Option D: $273$. This is the product of $64$ and $\frac{273}{27}$. However, $\frac{273}{27}$ is not a whole number, so $273$ is not the answer.
