1. **State the problem:** We are given fractions of traders selling tomatoes and onions and need to find the fraction of traders that sell tomatoes. 2. **Define variables:** Let the total number of traders be 1 (considering fractions). 3. **Given:** - One fourth of traders who sell tomatoes also sell onions: $\frac{1}{4}$ of tomato sellers sell onions. - Two fifths sell only tomatoes: $\frac{2}{5}$ of all traders sell only tomatoes. - Eleven twentieths sell onions: $\frac{11}{20}$ of all traders sell onions. 4. **Let $T$ be the fraction of traders that sell tomatoes.** 5. **From the problem:** - Traders who sell both tomatoes and onions = $\frac{1}{4}T$ - Traders who sell only tomatoes = $\frac{2}{5}$ - Traders who sell onions = $\frac{11}{20}$ 6. **Traders who sell onions include those who sell both tomatoes and onions and those who sell only onions.** Let $O_{only}$ be the fraction selling only onions. So, $$O_{only} + \frac{1}{4}T = \frac{11}{20}$$ 7. **Total tomato sellers $T$ is sum of only tomato sellers and both sellers:** $$T = \frac{2}{5} + \frac{1}{4}T$$ 8. **Solve for $T$:** $$T - \frac{1}{4}T = \frac{2}{5}$$ $$\frac{3}{4}T = \frac{2}{5}$$ $$T = \frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$$ 9. **Final answer:** The fraction of traders that sell tomatoes is $\boxed{\frac{8}{15}}$. This means $\frac{8}{15}$ of all traders sell tomatoes either only tomatoes or tomatoes with onions.