Every \( \frac{1}{2} \) hour Naomi stretches her neck; every \( \frac{1}{3} \) hour she stretches her legs; and every \( \frac{1}{6} \) hour she stretches her arms. Which parts of her body will Naomi stretch when \( \frac{2}{3} \) of an hour has passed?

To find out which parts of her body Naomi stretches after \( \frac{2}{3} \) of an hour, let's first determine how many stretches she completes during that time. 1. **Neck**: She stretches her neck every \( \frac{1}{2} \) hour. - In \( \frac{2}{3} \) hour, she stretches her neck \( \frac{2/3}{1/2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3} \), which means she stretches her neck once (1 time) and is in the middle of the second. 2. **Legs**: She stretches her legs every \( \frac{1}{3} \) hour. - In \( \frac{2}{3} \) hour, she stretches her legs \( \frac{2/3}{1/3} = \frac{2}{3} \times \frac{3}{1} = 2 \) times. 3. **Arms**: She stretches her arms every \( \frac{1}{6} \) hour. - In \( \frac{2}{3} \) hour, she stretches her arms \( \frac{2/3}{1/6} = \frac{2}{3} \times \frac{6}{1} = 4 \) times. At \( \frac{2}{3} \) of an hour, Naomi will have stretched: - Neck: 1 time (also in the middle of her second stretch) - Legs: 2 times - Arms: 4 times So, by the end of \( \frac{2}{3} \) of an hour, Naomi will definitely have stretched her legs and arms multiple times, and she will have completed one neck stretch.