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?

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To determine which parts of Naomi's body she will stretch after $$ \frac{2}{3} $$ of an hour, we need to find out how many times she stretches each part of her body within that time frame.

1. **Stretching her neck**: 
   - Naomi stretches her neck every $$ \frac{1}{2} $$ hour.
   - To find out how many times she stretches her neck in $$ \frac{2}{3} $$ of an hour, we calculate:
     $$
     \frac{2/3}{1/2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3}
     $$
   - This means she stretches her neck $$ 1 $$ full time (since $$ \frac{4}{3} $$ is more than $$ 1 $$) and is in the process of stretching it again.

2. **Stretching her legs**: 
   - Naomi stretches her legs every $$ \frac{1}{3} $$ hour.
   - To find out how many times she stretches her legs in $$ \frac{2}{3} $$ of an hour, we calculate:
     $$
     \frac{2/3}{1/3} = \frac{2}{3} \times \frac{3}{1} = 2
     $$
   - This means she stretches her legs $$ 2 $$ times.

3. **Stretching her arms**: 
   - Naomi stretches her arms every $$ \frac{1}{6} $$ hour.
   - To find out how many times she stretches her arms in $$ \frac{2}{3} $$ of an hour, we calculate:
     $$
     \frac{2/3}{1/6} = \frac{2}{3} \times \frac{6}{1} = 4
     $$
   - This means she stretches her arms $$ 4 $$ times.

Now, we summarize the results:
- **Neck**: Stretched $$ 1 $$ time (in the process of stretching again).
- **Legs**: Stretched $$ 2 $$ times.
- **Arms**: Stretched $$ 4 $$ times.

Since $$ \frac{2}{3} $$ of an hour has passed, Naomi will have completed her stretches for her legs and arms, and she will be in the process of stretching her neck.

Thus, at $$ \frac{2}{3} $$ of an hour, Naomi will have stretched her **legs** and **arms**, and she will be in the process of stretching her **neck**.

In conclusion, the parts of her body that Naomi will stretch when $$ \frac{2}{3} $$ of an hour has passed are:
- **Neck** (in the process of stretching)
- **Legs** (stretched)
- **Arms** (stretched)
Naomi will have stretched her neck, legs, and arms when $$ \frac{2}{3} $$ of an hour has passed.

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?

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