Question

Solve the following problem and verify the data by drawing Venn diagrams. In a medical examination of $150$ people, it was found that $90$ had eye problem, $50$ had heart problem and $30$ had both complaints. What percentage of people had either eye problem or heart problem?

Answer 1

Let the number of people who had eye problem be $n\left(E\right)$ and number of people who had heart problembe $n\left(H\right)$.

Then, the number of people who had both eye and heart problem is $n(E\cap H)$ and the number of people having either of the problem is $n(E\cup H)$.

We know that $n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$ where $A$ and $B$ are the respective sets.

Here, it is given that in a medical examination of $150$ people, $90$ had eye problem, $50$ had heart problem and $30$ had both complaints, that is $n\left(E\right)=90,n\left(H\right)=50$ and $n(E\cap H)=30$ and therefore,

$n(E\cup H)=n\left(E\right)+n\left(H\right)-n(E\cap H)=90+50-30=140-30=110$

Thus, the number of people having either of the problem is $110$.

The percentage of people who had either eye problem or heart problem is:

$\frac{110}{150}\times 100=73.33$%

Hence, $73.33$% of people were there who had either eye problem or heart problem.