This is a common type of problem that appear on algebra tests to test problem solving abilities (this is not from a real test or homework). I am curious on how to best approach these kind of problems since they can often overwhelm you (like this one has for me).
The final test was written by $U = 190$ people.
Of the $190$, $P = 75$ people passed the final test.
Of the $T = 161$ with all three previously completed exams, $74$ passed the final test.
Of the $B = 107$ with full bonus points, $63$ passed the final test.
There was $C = 99$ people who had made all previous exams and had full bonus.
Of those who lacked bonus and had also failed a previous exam there was 21 who did not pass the final test.
How many with full bonus and all three completed exams $C$ passed the final test?
We know that there was $C = 99$ people with full bonus and three completed exams.
Our problem is then to find the relation of $T$ and $B$ in $C$. Since then we can use that factor to multiply with the number of people who has passed the test in both $T$ and $B$.
That relation can be written as: $Tx + By = C \leftrightarrow 161x + 107y =99$.
Now we need another equation for $C$ expressed in $x$ and $y$ to have an equation system. And here I am stuck. What method would you use to find another equation?