Final answer:
Using Nagele's rule, the estimated date of birth for the client is July 27, 2013.
Step-by-step explanation:
Nagele's rule is a method used by healthcare providers to estimate the due date of a pregnant client. The rule states that you count back three months from the first day of the last menstrual period (LMP), and then add seven days. Applying Nagele's rule to the client's situation, the estimated date of birth would be July 27, 2013. This is calculated by counting back three months from October 20, which brings us to July 20, and then adding seven days.