Jewish Time Line
Timeline for the History of Judaism

Arpachshad born to Shem 2 years after the flood when Shem was 100 years old (Gen 11:10)
The flood started when Noah was 600 years old (Gen 7:6, 11)
The flood ended 380 days later (maybe) (Gen 7:12, 17, Gen 7:24, Gen 8:3, Gen 8:5) 40 + 150 + 150 + 40 days
Therefore the flood ended when Noah was 601 years old
Therefore Noah was 503 years old when Shem was born (601 + 2 - 100)