Need help figuring out my date of conception

Patient: Date of conception? I possibly had sex on July 4th and for sure on July 20th. On July 29th, a home pregnancy test, tested positive. On the 31st, I had an ultrasound and the sac measured 4weeks 6 days. Do those measurements even add up for a conception date of the 20th?

Doctor: Hello,Thanks for contacting askthedoctor.comIf your scan on 31st july showed that the pregnancy measured 4 weeks and 6 days then it is unlikely that you conceived due to sex on july 20th . The conception arrived by early july or late june. However the calculations are based on your LAST MENSTRUAL PERIOD for calculating the gestational age for all practical purposes and ultrasonological age is not accurate.I hope i have answered your query.regards