To estimate the sum of $45.6$ and $78.9$, we first round each number to the nearest whole number before performing the addition.

Rounding each number simplifies the calculation. For $45.6$, the nearest whole number is $46$, since $0.6$ is closer to $1$ than to $0$. Similarly, for $78.9$, the nearest whole number is $79$, as $0.9$ is also closer to $1$ than to $0$.

Next, we add the rounded numbers:

To make this addition easier, we can break it down into smaller steps. First, we add $40$ and $70$, which gives us $110$. Then, we add the remaining $6$ and $9$, resulting in $15$. Finally, we combine these two results:

Thus, the estimated sum of $45.6$ and $78.9$ is $125$. This estimation method is particularly useful for quick calculations, providing a rough idea of the total without the need for a calculator. Remember, the essential step is to round each number to the nearest whole number before adding them together.