10. Kate had some stickers. She kept \( \frac{2}{5} \) of them for herself and gave \( \frac{5}{6} \) of the remaining stickers to 6 of her friends equally. (a) What fraction of her stickers did Kate give to her 6 friends? (b) What fraction of Kate's stickers did each of her 6 friends receive?

To solve the problem, let’s start by determining how many stickers Kate initially had, which we can call \( S \). First, Kate kept \( \frac{2}{5}S \) for herself. This means she gave away \( S - \frac{2}{5}S = \frac{3}{5}S \) stickers. Next, she gave \( \frac{5}{6} \) of the remaining stickers to her friends. Hence, the stickers given away to her friends are: \[ \frac{5}{6} \times \frac{3}{5}S = \frac{1}{2}S. \] Now, let’s answer the questions: (a) The fraction of her stickers that Kate gave to her 6 friends is \( \frac{1}{2} \). (b) Since she gave \( \frac{1}{2}S \) to her friends, and there are 6 friends, each friend received: \[ \frac{1/2 S}{6} = \frac{1}{12}S. \] So, each of her 6 friends received \( \frac{1}{12} \) of her original stickers. Enjoy sharing the sticker math!