Determine the ratio of the number of people having characteristic X to the number of people having characteristic Y in a population of 100 subjects from the following table : | | | | :--- | :--- | | Having X and Y | 10 | | Having X but not Y | 30 | | Having Y but not X | 20 | | Having neither X nor Y | 40 |

### Concept & Set Theory Basics This problem is solved using the principles of Set Theory and Venn diagrams. To find the total number of people possessing a specific characteristic (like X), we must sum everyone who has that characteristic exclusively AND everyone who has that characteristic in conjunction with another. $$ n(X) = n(\text{X only}) + n(X \cap Y) $$ ### Step-by-Step Solution 1. Identify the goal: Find the ratio of total people with characteristic X to total people with characteristic Y, which means we need $\frac{n(X)}{n(Y)}$. 2. Calculate the total number of people having characteristic X ($n(X)$): * People having X and Y = $10$ * People having X but not Y = $30$ * Total having X = $10 + 30 = 40$ 3. Calculate the total number of people having characteristic Y ($n(Y)$): * People having X and Y = $10$ * People having Y but not X = $20$ * Total having Y = $10 + 20 = 30$ 4. Calculate the final ratio: $$ \text{Ratio} = \frac{\text{Total with X}}{\text{Total with Y}} = \frac{40}{30} = \frac{4}{3} $$ 5. This corresponds to the ratio $4 : 3$. ### Exam Strategy & Shortcut In a simple tabular set breakdown like this, ignore the "Having neither X nor Y" row entirely, as it does not contribute to the totals for X or Y. Quickly sum the relevant rows for X ($10+30$) and Y ($10+20$) and form the fraction $\frac{40}{30}$. ### Common Pitfall A very common mistake is to assume "Having X but not Y" represents all people with characteristic X. Students often incorrectly use the values $30$ and $20$ to form a ratio of $30:20$ or $3:2$. Remember that "Having X" is an inclusive term encompassing anyone in the X circle of a Venn diagram. ### Final Answer Therefore, the correct answer is **4 : 3**.