Solution



To solve this problem you want the greatest possible value of x. It is clear from the diagram that x cannot be greater than 142 nor greater than 121, otherwise or would be negative. Hence, x must be less than or 142−x 121−x equal to 121. Since there is no information to exclude the correct x=121,

Answer is \(121\).
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Because the question asks for the greatest possible number of students. It is only possible if the set of students who took chemistry class is a subset of the set of students who took algebra as well. This basically means that all the students who took chemistry also took algebra, but all the students who took algebra didn't necessarily take chemistry.

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Question: 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?

The book gives the answer as 121, with the explanation that It is clear the number of students who took both cannot exceed 121 or 142, therefore the answer is 121.

How is this possible if there are a total of 236 students? The question did not state that there were students that didn't take either class, does anyone else feel that this question is a little unfair?

Another approach is to use the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
Here, we have a population of students, and the two characteristics are:
- took algebra or did NOT take algebra
- took chemistry or did NOT take chemistry

Note: the TOTAL population is 236 students.
If 142 students took algebra, then the remaining 94 students did NOT take algebra (236 - 142 = 94).
Also, If 121 students took chemistry, then the remaining 115 students did NOT take chemistry (236 - 121 = 115).

Let's add all of this to our diagram to get...



What is the greatest possible number of students that could have taken both algebra and chemistry?
So, what's the biggest number we can place in the starred box below?


Well, we can't place a number greater than 121 in that box, since the two boxes in the left-hand column (took Chem) must add to 121
So, the biggest number we can place in the starred box is 121
When we do this, we get the following diagram:



So, the answer to the question is 121

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