# How to find the number of a students using sets when different clauses are stated using a Venn diagram or a table?

The problem is as follows:

From a group of $$90$$ students it is known that: $$12$$ prefer mathematics, but not literature; $$27$$ prefer literature, but are not $$18$$ years old; $$18$$ who do not prefer literature do not prefer mathematics and are $$18$$ years old; $$7$$ prefer literature and are $$18$$ years old, but do not prefer mathematics, $$4$$ prefer mathematics and literature and are $$18$$ years old. How many students who are not $$18$$ years old, do not prefer mathematics and not literature?

The alternatives given in my book were:

$$\begin{array}{ll} 1.&22\\ 2.&24\\ 3.&25\\ 4.&21\\ 5.&20\\ \end{array}$$

I'm confused exactly how to arrange this information in a table or in a Venn diagram to find the requested information. Can someone help me?.

What I've attempted so far was to make up a table as this:

$$\begin{array}{|c|c|c|c|} \hline \, &\textrm{Maths} &\textrm{Literature} &\textrm{Neither maths nor literature} \\ \hline \textrm{More than 18 years} &x &y &z\\ \hline \textrm{Less than 18 years} &a &b &c\\ \hline \end{array}\\$$

But that's how far I went. Does it exist a way to solve this using this approach or a Venn diagram?.

I really would appreciate someone could include a drawing or table so I can better understand this and avoid getting confused with the equations.

Can someone help me here please?.

Here is one way to lay out the information in the problem statement.

Explanation: For example, "FT" (think False-True) under Math-Literature means that in that column students do not prefer math and they do prefer literature. The "T" to the right of 18 means that in that row students are 18. So the $$7$$ in the column with "FT" and the row with "T" means that $$7$$ students fall in the category of being 18 years old, do not like math, and do like literature. The column to the right of the 2 by 4 grid is for row totals (but none are given in this problem), and the row below the grid is for column totals; so we see $$12$$ is the sum of the two numbers in the Math-Literature "TF" column. The cell to the lower right of the grid is for a grand total; so the sum of all the cells in the 2 by 4 grid is $$90$$. You could also say that $$90$$ is the sum of the row totals or the sum of the column totals.

The only number given in the problem statement that is awkward in the sense of not having a well-designated cell is the $$27$$ in the 18 "F" row which is placed between the Math-Literature "FT" and "TT" columns. This is meant to show that the sum of the two cells in the "FT" and "TT" columns is $$27$$.

The problem asks us to find the $$x$$ in the upper left corner of the 2 by 4 grid, in the intersection of the 18 "F" row and the Math-Literature "FF" column, i.e. the number of students who are not 18, do not like math, and do not like literature.

Hint: You might think about what you can do about filling out the missing data in the column subtotals--the row below the 2 by 4 grid.

• Is that the only way to solve this problem?. Why did you split the table with true-false columns?. Is this any sort of arrangement for these kinds of situations which I'm unaware of?. Do we have an intersection in this whole story?. Assuming all is right and if I add up with $x$ does it mean $x=22$?. you put 27 in between FT and TT, why? what is the interpretation?. Again why did you split the table in this sort of way and why the method which I attempted did not worked? Did I missed to take account for something?.Can you explain this part please. – Chris Steinbeck Bell Oct 28 '20 at 0:06
• I attempted to fill out the subtotals as you mentioned but I'm stuck with it, can it be done?. I think there isn't enough information for filling out the table?. – Chris Steinbeck Bell Oct 28 '20 at 0:08
• How do you account for the age factor in this problem?. There's false true in the columns and also in the rows. This cross is confusing. – Chris Steinbeck Bell Oct 28 '20 at 0:11
• @ChrisSteinbeckBell With regard to age: There are two rows in the table. The first row consists of 4 cells in which "student is 18" is False (F). The second row consists of 4 cells in which "student is 18" is True (T). The intersection of one column and one row uniquely determines the combination of age/math/literature for that cell. The reason 27 is placed between two cells is because it is the sum of the two logical combinations: age/math/literature FFT and FTT. If you find this approach confusing, feel free to use a different method--but this is the way I would do it. – awkward Oct 28 '20 at 12:40
• @ChrisSteinbeckBell With regard to filling in the column subtotals: maybe there isn't enough information to determine each subtotal uniquely, but maybe it's enough if you can find the sum of a couple of subtotals. – awkward Oct 28 '20 at 12:43