Trying to find the error in my attempt at basic probability set complement problem I am working through a textbook on probability for actuaries and I am having trouble with this problem:

In a universe $U$ of $100$, let $A$ and $B$ be subsets of $U$ such that $|A \cup B| = 70$ and $|A \cup B'| = 90$. Then what is $|A|$?

So here is my thinking: I know $|B|+|B'|=100$, so I would think that I can just subtract $100$ from the sum $|A \cup B| + |A \cup B'| = 160$. Then we have $|A| + |A| = 60$, so $|A| = 30$.
However, the right answer seems to be $60$. Was my method close, or is it just a coincidence that I arrived at exactly half that number?

When subtracting $|U|$ from $|A\cup B|+|A\cup B'|$, you also subtract one copy of the elements that lie in $A$. Thus what you have left is not $2|A|$, but $|A|$ itself. The last step is unnecessary and incorrect.
