Ana, Bia, Cátia, Diana, and Elaine work as street vendors selling sandwiches. Every day, they stop by Mr. Manoel's snack bar and take the same number of sandwiches to sell. One day, Mr. Manoel was sick and left a note explaining why he wasn't there, but asking each of them to take $\frac{1}{5}$ of the sandwiches. Ana arrived first, followed the note's instructions, and left to sell her sandwiches. Bia arrived next, but thought she was the first to arrive, taking $\frac{1}{5}$ of what was left and left. Cátia, Diana, and Elaine arrived together and divided what was left equally, as Cátia knew that Ana and Bia had already passed by.

a) What fraction of the total sandwiches did Bia get?

b) Who ended up with the smallest number of sandwiches? Who got the most?

c) At the end of the division, none of the vendors had more than 20 sandwiches. How many sandwiches did Mr. Manoel leave for them?
Solution

a) $\frac{4}{25}$. Since Bia took $\frac{1}{5}$ of $\frac{4}{5}$, she ended up with $\frac{1}{5} \cdot \frac{4}{5}=\frac{4}{25}$ of the sandwiches.

b) Ana took $\frac{1}{5}=\frac{5}{25}$, and Cátia, Diana, and Elaine divided the remaining fraction, which was $1-\frac{9}{25}=\frac{16}{25}$, among the three, meaning each took $\frac{16}{75}$ of the sandwiches, which is greater than $\frac{12}{75}$ and $\frac{15}{75}$. Therefore, Cátia, Diana, and Elaine ended up with the largest share, and Bia with the smallest.
c) 75. The number of sandwiches must be a common multiple of 5, 25, and 75, with 75 being the smallest. For 75 sandwiches, the quantities for Ana, Bia, Cátia, Diana, and Elaine are, respectively, $15, 12, 16, 16, 16$, all less than 20, but for the next common multiple of 5, 25, and 75, which is 150, all would end up with more than 20 sandwiches. Therefore, this quantity must be 75.