Let’s address the math problems one at a time.

4 (a)

In the diagram, $O$ is the center of the circle $XYZ$. Given $\angle ZXO = 34^\circ$ and $\angle XOY = 146^\circ$, find the value of $\angle OYZ$.

1. Explanation:

   • In a circle, angles subtended by the same arc are related. Specifically, the central angle $\angle XOY$ (146°) subtends the same arc $XY$ as the angle at the circumference $\angle OYZ$.
   • The angle at the circumference is half the central angle subtending the same arc.

   $\angle OYZ = \frac{1}{2} \angle XOY$

2. Calculation: $\angle OYZ = \frac{1}{2} \times 146^\circ = 73^\circ$

Final Answer: $\angle OYZ = 73^\circ$

4 (b)

If $\frac{3m}{2} + \frac{n}{2} = 4$, find $m$ and $n$.

1. Combine the terms: $\frac{3m + n}{2} = 4$

2. Multiply through by 2 to eliminate the denominator: $3m + n = 8$

3. To find the specific values of $m$ and $n$, we would need either additional equations or conditions. (The problem might implicitly assume $m$ and $n$ are integers, leading to possible pairs that satisfy this equation.)

Let me know which specific part you want explained in further detail or solved next! Here are some related questions:

1. What is the relationship between central and inscribed angles in a circle?
2. How do you solve systems of equations involving two variables?
3. What are the key properties of chords, radii, and arcs in a circle?
4. Can you explain why $146^\circ$ leads to $73^\circ$ in the arc problem?
5. How do you approach ratio-based money-sharing problems like in 4(b)?

Tip: For geometry problems, always try sketching additional auxiliary lines to visualize relationships better.