Problem statement,
In a graduating class of $236$ students, $142$ took algebra and $121$ took chemistry. What is the greatest possible number of students that could have taken both algebra and chemistry?
What I have so far,
$A \cup B = 263-x$ where ($x=A \cap B)$ where $A,B$ are students with algebra and chemistry respectively.
$y = $students with neither $A$ or $B$.
$$263 - x + y = 236$$
$$x = y - 27$$
Not sure how to proceed further.
On
Exactly 121 students took chemistry. It is possible that all of them also took algebra, in which case 121 students would have taken both chemistry and algebra. It is not possible that more than 121 students took both chemistry and algebra, since more than 121 students did not take chemistry.
The total class size is irrelevant, as there is no requirement that every student took at least one of chemistry or algebra. The answer would still be 121 no matter how large the graduating class is.
On
Assuming students in this question either took algebra or took Chemistry, let $A$ be the number of students who took Algebra, and $B$ be the number of students who took Chemistry.
We know that:
Therefore, in order to compute the greatest possible number of students that could have taken both algebra and chemistry ($A \cap B$):
$$A \cap B = A + B - A \cup B = 142 + 121 - 236 = 27$$
On
The question does not specify that each student took either Chemistry or Algebra,
so rather obviously, the maximum number that could have taken both is $121$
On
Since it is not specified in the question, not necessarily every student has taken one of the algebra or chemistry courses. Therefore, the maximum number of students who took both algebra and chemistry will be 121. In fact, assume that A is the set of students who have taken chemistry (with size 121) and B is the set of students who have taken algebra (with size 142). The maximum number of students who have taken algebra And chemistry is achieved when A is a subset of B or B is a subset of A (Here, as $|A|<|B|$ the first one should occur). So, the answer to this question is 121.
Other students have not taken any of the algebra and chemistry courses.
We can define some value $a$ for students that didn't take algebra. We can see that $a=236-142=95$. Now this means that there are $94$ students who have only taken chemistry. The rest of those who've taken chemistry also took algebra, and let that value be $b$. So we can see that $b=121-94=27$.
This is the value you're looking for under the assumption that students have to take one at least one of these two classes. If they don't that then there is the answer Nuclear Hoagie already wrote.