In the context of multiplication and division, uncertainties are combined by adding the percentage uncertainties of the involved quantities.

When working with the multiplication or division of physical quantities, it is crucial to accurately account for the uncertainties associated with these quantities. The process for combining uncertainties involves adding the percentage uncertainties of each quantity. The percentage uncertainty for a given quantity is calculated as the absolute uncertainty divided by the value of the quantity, multiplied by $100\%$.

For instance, consider two quantities, $A$ and $B$. Let $A$ have a value of $10$ with an uncertainty of $\pm 1$, and $B$ have a value of $5$ with an uncertainty of $\pm 0.5$. To find the percentage uncertainty of $A$, we compute:

Similarly, for $B$:

When we multiply $A$ and $B$, the percentage uncertainty of the resulting product is the sum of the percentage uncertainties of $A$ and $B$:

The same principle applies when dividing two quantities. If we divide $A$ by $B$, the percentage uncertainty of the result is again the sum of the percentage uncertainties of $A$ and $B$.

This method is based on the assumption that the uncertainties are relatively small compared to the values of the quantities, and that they are random and independent of one another. If these assumptions do not hold, a more complex method for combining uncertainties may be required.

In summary, when multiplying or dividing quantities in physics, the uncertainties are combined by adding their percentage uncertainties. This involves computing each quantity’s percentage uncertainty (the absolute uncertainty divided by the quantity’s value, multiplied by $100\%$) and summing these percentages. It is important to note that this approach assumes the uncertainties are small, random, and independent. If these conditions are not satisfied, a more sophisticated approach to uncertainty propagation should be considered.