1. **Stating the problem:** We have a market survey with traders selling tomatoes and onions. - One fourth of traders who sell tomatoes also sell onions. - Two fifths of all traders sell only tomatoes. - Eleven twentieths of all traders sell onions. We need to find: - The fraction of traders that sell tomatoes. - The fraction of traders who sell only one commodity. 2. **Define variables:** Let $T$ be the fraction of traders who sell tomatoes. Let $O$ be the fraction of traders who sell onions. Let $B$ be the fraction who sell both tomatoes and onions. Let $T_{only}$ be the fraction who sell only tomatoes. Let $O_{only}$ be the fraction who sell only onions. 3. **Translate given information into equations:** - One fourth of traders who sell tomatoes sell onions: $$B = \frac{1}{4}T$$ - Two fifths sell only tomatoes: $$T_{only} = \frac{2}{5}$$ - Eleven twentieths sell onions: $$O = \frac{11}{20}$$ 4. **Express $T$ and $O$ in terms of only and both:** $$T = T_{only} + B$$ $$O = O_{only} + B$$ 5. **Use given values:** $$T = \frac{2}{5} + B$$ $$O = O_{only} + B = \frac{11}{20}$$ 6. **Substitute $B = \frac{1}{4}T$ into $T$ equation:** $$T = \frac{2}{5} + \frac{1}{4}T$$ 7. **Solve for $T$:** $$T - \frac{1}{4}T = \frac{2}{5}$$ $$\frac{3}{4}T = \frac{2}{5}$$ $$T = \frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$$ 8. **Find $B$ using $B = \frac{1}{4}T$:** $$B = \frac{1}{4} \times \frac{8}{15} = \frac{2}{15}$$ 9. **Find $O_{only}$ using $O = O_{only} + B$:** $$O_{only} = O - B = \frac{11}{20} - \frac{2}{15} = \frac{33}{60} - \frac{8}{60} = \frac{25}{60} = \frac{5}{12}$$ 10. **Find fraction of traders who sell only one commodity:** $$T_{only} + O_{only} = \frac{2}{5} + \frac{5}{12} = \frac{24}{60} + \frac{25}{60} = \frac{49}{60}$$ **Final answers:** - Fraction of traders that sell tomatoes: $$\boxed{\frac{8}{15}}$$ - Fraction of traders who sell only one commodity: $$\boxed{\frac{49}{60}}$$