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To solve this problem, we need to understand how gear trains work, particularly compound gear trains like the one shown. In this setup:

1. **Gears B and C** are on the same shaft, so they rotate at the same speed.
2. **Gears A and C** are the driving gears, while **Gears B and D** are driven gears.
3. **Speed and teeth ratios** between gears determine the speeds of the driven gears.

### Step-by-Step Solution Approach

1. **Gear Ratio Formula**: 
   The relationship between the speeds of two meshing gears (A and B, or C and D) is given by the gear ratio:
   \[
   \frac{\text{Speed of Gear 1}}{\text{Speed of Gear 2}} = \frac{\text{Teeth of Gear 2}}{\text{Teeth of Gear 1}}
   \]
   
2. **Using Compound Gear Ratios**:
   Since gears B and C rotate at the same speed (they're keyed together), we can set up compound ratios. For example:
   - From Gear A to Gear B: \(\frac{\text{Speed of Gear A}}{\text{Speed of Gear B}} = \frac{\text{Teeth of Gear B}}{\text{Teeth of Gear A}}\)
   - From Gear C to Gear D: \(\frac{\text{Speed of Gear C}}{\text{Speed of Gear D}} = \frac{\text{Teeth of Gear D}}{\text{Teeth of Gear C}}\)

3. **Solving for Unknowns**:
   For each row, we use the known teeth and speed values to calculate the unknowns based on these proportions.

I'll solve each case from the table and provide the rounded answers. 

Let me proceed with this calculation.It appears there was an issue in calculating one of the values for part (c). Let me correct that and ensure all values are properly computed. I'll revise the calculations and present the answers for each part.Here are the calculated values for each part in the table:

- **Part (a)**:
  - \( x = 320.0 \) rpm (Speed of Gear B)
  - \( y = 320.0 \) rpm (Speed of Gear C, same as Gear B)
  - \( z = 800.0 \) rpm (Speed of Gear D)

- **Part (b)**:
  - \( x = 20.0 \) teeth (Teeth of Gear B)
  - \( y = 30.0 \) teeth (Teeth of Gear D)

- **Part (c)**:
  - \( x = 12.3 \) teeth (Teeth of Gear A)
  - \( y = 144 \) rpm (Speed of Gear B, given)
  - \( z = 280 \) rpm (Speed of Gear D, given)

- **Part (d)**:
  - \( x = 12.5 \) teeth (Teeth of Gear C)
  - \( y = 44.0 \) rpm (Speed of Gear B)
  - \( z = 350 \) rpm (Speed of Gear D, given)

Would you like further details on any specific calculation?

### Related Questions:
1. How does the speed of a driven gear change if the number of driving gear teeth increases?
2. What happens to the output speed if the driven gear’s number of teeth doubles?
3. How are compound gears typically used in gear train applications?
4. Why do Gears B and C have the same speed in this compound gear train?
5. How would the results change if the arrangement or shaft connections were different?

### Tip:
When working with compound gears, remember that gears on the same shaft will always rotate at the same speed, which simplifies calculations in complex gear systems.

The figure shows a compound gear train. Gears B and C are keyed to the same shaft; therefore, they turn at the same speed. Gear A and gear C are driving gears. Gear B and gear D are driven gears. Set up a proportion for each problem and determine the unknown values, x, y, and z in the table. Round the answers to 1 decimal place where necessary.

Explore how to solve compound gear train problems using gear ratios and proportionality. This guide includes a step-by-step approach to determining unknown speeds and teeth counts in a compound gear setup, suitable for high school students.

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