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This is a center of mass (CM) problem, where you need to find the center of mass of a system of objects placed on a meter stick, including the stick itself.

### Step-by-step solution:

We are given:
- Masses: \( m_1 = 50 \, \text{g}, m_2 = 100 \, \text{g}, m_3 = 200 \, \text{g}, m_{\text{stick}} = 300 \, \text{g} \).
- Positions of the masses: \( x_1 = 30 \, \text{cm}, x_2 = 60 \, \text{cm}, x_3 = 80 \, \text{cm}, x_{\text{stick}} = 50 \, \text{cm} \).

The formula for the center of mass (CM) of the system is:
\[
x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}
\]
Where:
- \( m_i \) is the mass of each object (including the meter stick),
- \( x_i \) is the position of the object.

### Step 1: Compute the total mass of the system
\[
M = m_1 + m_2 + m_3 + m_{\text{stick}} = 50 \, \text{g} + 100 \, \text{g} + 200 \, \text{g} + 300 \, \text{g} = 650 \, \text{g}
\]

### Step 2: Calculate the moment contributions from each mass
For each mass, the moment (product of mass and position) is:
\[
m_1 x_1 = 50 \times 30 = 1500 \, \text{g} \cdot \text{cm}
\]
\[
m_2 x_2 = 100 \times 60 = 6000 \, \text{g} \cdot \text{cm}
\]
\[
m_3 x_3 = 200 \times 80 = 16000 \, \text{g} \cdot \text{cm}
\]
\[
m_{\text{stick}} x_{\text{stick}} = 300 \times 50 = 15000 \, \text{g} \cdot \text{cm}
\]

### Step 3: Sum the moments
\[
\sum m_i x_i = 1500 + 6000 + 16000 + 15000 = 38500 \, \text{g} \cdot \text{cm}
\]

### Step 4: Calculate the center of mass
\[
x_{\text{cm}} = \frac{38500}{650} = 59.23 \, \text{cm}
\]

### Final answer:
The center of mass of the system is located at **59.23 cm** from the 0-cm mark of the meter stick.

---

Do you have any questions or would you like a more detailed explanation of any part? Here are some related questions to deepen your understanding:

1. How does the distribution of mass affect the location of the center of mass?
2. If we removed one of the masses, how would the center of mass change?
3. What would happen to the center of mass if the meter stick were not uniform?
4. How is the center of mass different from the center of gravity in this context?
5. Can the center of mass be located outside of the physical object in some cases?

**Tip:** When calculating the center of mass, it is important to include all objects in the system, including the support or base structure if it contributes to the total mass.

Masses of 50 g, 100 g, and 200 g are placed at the 30-cm, 60-cm, and 80-cm marks of a meter stick. If the meter stick's mass is 300 g, locate the center of mass of the system.

This problem demonstrates how to calculate the center of mass of a system, including three masses and a uniform meter stick. We calculate the center of mass using the positions and weights of each object placed at different points on the stick. The total mass and the combined moments of each object lead to a center of mass located at 59.23 cm from the 0-cm mark.

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