Step 1: Let the two-digit number be represented as: Original number: $10x + y$ Reversed number: $10y + x$ Here, $x$ is the tens digit, and $y$ is the unit digit.

Step 2: According to the problem, the ratio of the original number to the reversed number is 4:7. This gives us the equation: $\frac{(10x + y)}{(10y + x)} = \frac{4}{7}$

Step 3: Cross-multiply to get rid of the denominators: $7 \cdot (10x + y) = 4 \cdot (10y + x)$

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

Step 5: Move all terms involving $x$ to one side and all terms involving $y$ to the other side: $70x - 4x = 40y - 7y$ $66x = 33y$

Step 6: Simplify the equation by dividing both sides by 33: $2x = y$

Step 7: Since $x$ and $y$ are digits of a two-digit number, they must be integers between 0 and 9. From the equation $2x = y$, the possible pairs $(x, y)$ are: $(1, 2)$ $(2, 4)$ $(3, 6)$ $(4, 8)$