Question: In a class 16 students read Mathematics, 17 read General science and 6 read both (of these). The num...

Question

In a class 16 students read Mathematics, 17 read General science and 6 read both (of these). The number of students in class which read either Mathematics or general science is
A. 6
B. 10
C. 11
D. 27

Answer 1

Solution

These types of questions are solved with the help of Venn diagrams which help us in explaining the relation between various sets. We calculate the union of students reading Mathematics and General science using the formula for union of two sets i.e. $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
If $n(A),n(B)$ denotes number of elements in each set A and B respectively, then their union i.e. $n(A \cup B)$ and their intersection i.e. $n(A \cap B)$ gives us the formula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .

Complete step by step answer:
Let us denote Mathematics by A and General science by B
Number of students who read Mathematics $= 16$
$n(A) = 16$
Number of students who read General science $= 17$
$n(B) = 17$
Then, the set of students reading both Mathematics and General science will be the intersection of students reading mathematics and students reading General science i.e. $A \cap B$
$n(A \cap B) = 6$
The students in the class which are enrolled in either Mathematics or General science means the students those who are enrolled in Mathematics but not in General science and students who are enrolled in General science but not in Mathematics.
So the set of students in the class which are enrolled in either Mathematics or General science is denoted by $A \cup B$ .

The formula for Union of two sets $A$ and $B$ is given by $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ .
Substitute the values in the formula.
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
$\Rightarrow n(A \cup B) = 16 + 17 - 6$
$\Rightarrow n(A \cup B) = 33 - 6$
$\Rightarrow n(A \cup B) = 27$
So, the number of students reading either of the subjects is 27

$\therefore$ The option D is correct.

Note: Students are very likely to get confused with either/or statement and not either/or statement. If we have to calculate ‘either this or that’ kind of value we always calculate the union, but if we have to calculate ‘not either this or that’ which means ‘neither this nor that’ then we calculate the union and subtract it from the universal set. Also, many students make mistakes in calculation if they don’t follow BODMAS rule which states we add the positive terms first and then subtract the number from the sum.