Question: If one person gets saved doesn’t the Bible say the whole family is saved?
Answer: No the Bible absolutely does not teach that at all. The proof text I have heard given for this is Acts 16:31, “And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.” If we take this verse and have it stand alone then we can get that out of it but if we look at it in context then we can see that he is not saying one person getting saved counts for everyone.
“And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.And they spake unto him the word of the Lord, and to all that were in his house.And he took them the same hour of the night, and washed their stripes; and was baptized, he and all his, straightway.And when he had brought them into his house, he set meat before them, and rejoiced, believing in God with all his house.” (Acts 16:31-34) Looking at the entire interaction the man asks what he must do to be saved. The apostles reply in verse 31 that he must believe on the Lord Jesus Christ. By saying “and thy house” they are not saying he can be saved for his household bu that they too must believe on the Lord Jesus Christ.
Look at what happens after this. In verse 32 it says they spoke the Word of God to him and all that were in his house. So they all heard the Gospel message. In verse 33 we see the apostles baptize him and all that were in his house. Why did they baptize them all? Well look at verse 34 it says he believed “with all his house.” So we have a consistent story where Paul says that he and his house must believe, the Gospel is preached to him and his house, he and his house believe, and he and his house are baptized.