To determine the earliest time all 120 parts can be finished, we must analyze the production rates, setup times, and the constraint that partial parts do not count (integer completions).

### 1. Define Parameters

**Machine A:**
*   Rate ($R_A$): 8 parts/hour
*   Setup Time ($S_A$): 30 minutes = $0.5$ hours
*   Production Start Time: $0.5$ hours

**Machine B:**
*   Rate ($R_B$): 5 parts/hour
*   Setup Time ($S_B$): 12 minutes = $12/60$ hours = $0.2$ hours
*   Production Start Time: $0.2$ hours

**Total Parts Required:** 120

### 2. Analyze Production Logic

Since both machines run simultaneously, we need to find the time $T$ (in hours) when the cumulative integer number of parts produced by Machine A ($N_A$) and Machine B ($N_B$) sum to at least 120.

The number of parts produced by a machine is determined by the time available for production ($T - S$) multiplied by the rate, floored to the nearest integer (since partial parts don't count).

$$ N_A(T) = \lfloor 8 \times (T - 0.5) \rfloor \quad \text{for } T \ge 0.5 $$
$$ N_B(T) = \lfloor 5 \times (T - 0.2) \rfloor \quad \text{for } T \ge 0.2 $$

We need to find the smallest $T$ such that:
$$ N_A(T) + N_B(T) \ge 120 $$

### 3. Step-by-Step Calculation

Let's estimate the time required.
If we ignore setup times and integer constraints, the combined rate is $8 + 5 = 13$ parts/hour.
$120 / 13 \approx 9.23$ hours.
Adding the maximum setup time ($0.5$ hours), the answer should be around $9.73$ hours. Let's test integer hours or half-hours near this estimate.

**Test $T = 9.5$ hours:**
*   **Machine A:**
    *   Production time: $9.5 - 0.5 = 9.0$ hours
    *   Parts: $\lfloor 8 \times 9.0 \rfloor = 72$ parts
*   **Machine B:**
    *   Production time: $9.5 - 0.2 = 9.3$ hours
    *   Parts: $\lfloor 5 \times 9.3 \rfloor = \lfloor 46.5 \rfloor = 46$ parts
*   **Total:** $72 + 46 = 118$ parts.
*   **Result:** Not enough (Need 120).

**Test $T = 9.6$ hours:**
*   **Machine A:**
    *   Production time: $9.6 - 0.5 = 9.1$ hours
    *   Parts: $\lfloor 8 \times 9.1 \rfloor = \lfloor 72.8 \rfloor = 72$ parts
*   **Machine B:**
    *   Production time: $9.6 - 0.2 = 9.4$ hours
    *   Parts: $\lfloor 5 \times 9.4 \rfloor = \lfloor 47.0 \rfloor = 47$ parts
*   **Total:** $72 + 47 = 119$ parts.
*   **Result:** Not enough (Need 120).

**Test $T = 9.7$ hours:**
*   **Machine A:**
    *   Production time: $9.7 - 0.5 = 9.2$ hours
    *   Parts: $\lfloor 8 \times 9.2 \rfloor = \lfloor 73.6 \rfloor = 73$ parts
*   **Machine B:**
    *   Production time: $9.7 - 0.2 = 9.5$ hours
    *   Parts: $\lfloor 5 \times 9.5 \rfloor = \lfloor 47.5 \rfloor = 47$ parts
*   **Total:** $73 + 47 = 120$ parts.
*   **Result:** **Success.**

### 4. Verification of "Immediately Preceding Completion Time"

The problem asks to check the immediately preceding completion time. This implies checking if the count reached 120 earlier or if there is a gap between integer completions.

Let's look closely at the transition between $T=9.6$ and $T=9.7$.

At **$T = 9.6$ hours**:
*   $N_A = 72$
*   $N_B = 47$
*   Total = 119

At **$T = 9.7$ hours**:
*   $N_A$ jumps from 72 to 73.
    *   Machine A produces parts every $1/8$ hour ($7.5$ minutes).
    *   Previous completion for A was at $T$ where $8(T-0.5) = 72 \Rightarrow T-0.5=9 \Rightarrow T=9.5$.
    *   Next completion for A is at $T$ where $8(T-0.5) = 73 \Rightarrow T-0.5=9.125 \Rightarrow T=9.625$.
    *   So, Machine A produces its 73rd part at **$T = 9.625$ hours**.
*   $N_B$ stays at 47.
    *   Machine B produces parts every $1/5$ hour ($12$ minutes).
    *   Previous completion for B was at $T$ where $5(T-0.2) = 47 \Rightarrow T-0.2=9.4 \Rightarrow T=9.6$.
    *   Next completion for B is at $T$ where $5(T-0.2) = 48 \Rightarrow T-0.2=9.6 \Rightarrow T=9.8$.
    *   So, Machine B produces its 48th part at **$T = 9.8$ hours**.

Let's check the total count at the specific completion events:

1.  **At $T = 9.6$ hours:**
    *   Machine B just finished its 47th part (since $5 \times 9.4 = 47$).
    *   Machine A has finished 72 parts (since $8 \times 9.1 = 72.8$, floor is 72).
    *   Total = $72 + 47 = 119$.

2.  **At $T = 9.625$ hours ($9$ hours $37.5$ minutes):**
    *   Machine A finishes its 73rd part ($8 \times 9.125 = 73$).
    *   Machine B still has 47 parts (next one is at 9.8).
    *   Total = $73 + 47 = 120$.

Therefore, the 120th part is completed exactly when Machine A finishes its 73rd part.

### 5. Final Conclusion

The earliest time all 120 parts can be finished is **9.625 hours**.

**Arithmetic Summary:**
*   **Machine A:**
    *   Setup: 0.5 h
    *   Time to produce 73 parts: $73 / 8 = 9.125$ h
    *   Total time for A to have 73 parts: $0.5 + 9.125 = 9.625$ h
*   **Machine B:**
    *   Setup: 0.2 h
    *   Time to produce 47 parts: $47 / 5 = 9.4$ h
    *   Total time for B to have 47 parts: $0.2 + 9.4 = 9.6$ h
    *   Time to produce 48 parts: $48 / 5 = 9.6$ h
    *   Total time for B to have 48 parts: $0.2 + 9.6 = 9.8$ h

At $T = 9.625$ h:
*   Machine A has produced $\lfloor 8(9.625 - 0.5) \rfloor = \lfloor 8(9.125) \rfloor = 73$ parts.
*   Machine B has produced $\lfloor 5(9.625 - 0.2) \rfloor = \lfloor 5(9.425) \rfloor = \lfloor 47.125 \rfloor = 47$ parts.
*   Total = $73 + 47 = 120$ parts.

At the immediately preceding completion time ($T = 9.6$ h):
*   Machine A has produced $\lfloor 8(9.6 - 0.5) \rfloor = \lfloor 8(9.1) \rfloor = 72$ parts.
*   Machine B has produced $\lfloor 5(9.6 - 0.2) \rfloor = \lfloor 5(9.4) \rfloor = 47$ parts.
*   Total = $72 + 47 = 119$ parts.

**Answer:** The earliest time all 120 parts can be finished is **9.625 hours** (or 9 hours and 37.5 minutes).