The ratio of a two-digit natural number to a number formed by reversing its digits is 4 : 7. The number of such pairs is

The correct option is 4.

Let the two-digit natural number be represented as $\mathrm{10}x+y$, where $x$ is the tens digit and $y$ is the units digit.
Since it is a two-digit number, $x$ must be an integer from $1$ to $9$ ($x\in \{1,2,...,9\}$), and $y$ must be a digit from $0$ to $9$ ($y\in \{0,1,...,9\}$).

When the digits are reversed, the new number formed is $\mathrm{10}y+x$.
For this reversed number to be a valid number and to maintain a ratio, its first digit $y$ cannot be $0$. Thus, $y\in \{1,2,...,9\}$.

According to the given condition, the ratio of the original number to the reversed number is $4:7$. We can set up the following equation:
$\frac{10x+y}{10y+x}=\frac{4}{7}$

By cross-multiplying, we get:
$7(10x+y)=4(10y+x)$

Expanding both sides of the equation:
$70x+7y=40y+4x$

Rearranging the terms to group the $x$ terms on one side and the $y$ terms on the other:
$70x-4x=40y-7y$
$66x=33y$

Dividing both sides by $33$, we obtain a simple relationship between the digits:
$2x=y$

Since $x$ and $y$ must be single-digit positive integers (from $1$ to $9$), let us find the possible values for $x$ that keep $y$ as a single-digit integer:
1. If $x=1$, then $y=2\left(1\right)=2$. The pair of numbers is $(12,21)$.
2. If $x=2$, then $y=2\left(2\right)=4$. The pair of numbers is $(24,42)$.
3. If $x=3$, then $y=2\left(3\right)=6$. The pair of numbers is $(36,63)$.
4. If $x=4$, then $y=2\left(4\right)=8$. The pair of numbers is $(48,84)$.

For any value of $x\ge 5$, the value of $y$ would be $10$ or greater, which is not possible as $y$ must be a single digit.

Thus, there are exactly 4 such pairs of numbers: $(12,21)$, $(24,42)$, $(36,63)$, and $(48,84)$.