1. **State the problem:** We have a survey of traders selling tomatoes and onions with the following information: - One fourth of traders who sell tomatoes also sell onions. - Two fifths sell only tomatoes. - Eleven twentieths sell onions. - Five traders sell neither tomatoes nor onions. We need to find: I. The fraction of traders that sell tomatoes. II. The total number of traders involved in the survey. 2. **Define variables:** Let $T$ be the fraction of traders who sell tomatoes. Let $O$ be the fraction of traders who sell onions. Let $N$ be the total number of traders. 3. **Translate given information into equations:** - One fourth of traders who sell tomatoes sell onions means the fraction of traders who sell both tomatoes and onions is $\frac{1}{4}T$. - Two fifths sell only tomatoes means the fraction who sell tomatoes but not onions is $\frac{2}{5}$. - Eleven twentieths sell onions means $O = \frac{11}{20}$. - Five traders sell neither means the fraction who sell neither is $\frac{5}{N}$. 4. **Express the fraction of traders who sell tomatoes:** The total fraction selling tomatoes is those who sell only tomatoes plus those who sell both: $$T = \frac{2}{5} + \frac{1}{4}T$$ 5. **Solve for $T$:** $$T - \frac{1}{4}T = \frac{2}{5}$$ $$\frac{3}{4}T = \frac{2}{5}$$ $$T = \frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$$ So, the fraction of traders selling tomatoes is $\frac{8}{15}$. 6. **Find the fraction selling both tomatoes and onions:** $$\text{Both} = \frac{1}{4}T = \frac{1}{4} \times \frac{8}{15} = \frac{2}{15}$$ 7. **Find the fraction selling only onions:** Since $O = \frac{11}{20}$ and $\text{Both} = \frac{2}{15}$, $$\text{Only onions} = O - \text{Both} = \frac{11}{20} - \frac{2}{15} = \frac{33}{60} - \frac{8}{60} = \frac{25}{60} = \frac{5}{12}$$ 8. **Sum of all fractions:** $$\text{Only tomatoes} + \text{Only onions} + \text{Both} + \text{Neither} = 1$$ Substitute known values: $$\frac{2}{5} + \frac{5}{12} + \frac{2}{15} + \frac{5}{N} = 1$$ 9. **Find common denominator and sum fractions:** Common denominator for 5, 12, 15 is 60: $$\frac{2}{5} = \frac{24}{60}, \quad \frac{5}{12} = \frac{25}{60}, \quad \frac{2}{15} = \frac{8}{60}$$ Sum: $$\frac{24}{60} + \frac{25}{60} + \frac{8}{60} = \frac{57}{60}$$ 10. **Solve for $N$:** $$\frac{57}{60} + \frac{5}{N} = 1$$ $$\frac{5}{N} = 1 - \frac{57}{60} = \frac{3}{60} = \frac{1}{20}$$ $$N = 5 \times 20 = 100$$ **Final answers:** - Fraction of traders selling tomatoes: $\frac{8}{15}$. - Total number of traders surveyed: 100.