Side-by-side comparison of old and new gear systems illustrating gear ratios, velocity ratios, and RPM calculations.

Problem Analysis:

The task involves calculating the gear ratio and velocity ratio for two different gear systems. Additionally, we need to determine the RPM (revolutions per minute) of the driven gear based on the given driver gear RPM.

#### Key Concepts:
1. Gear Ratio: The gear ratio is the ratio of the number of teeth on the driver gear to the number of teeth on the driven gear.
\[
\text{Gear Ratio} = \frac{\text{Number of Teeth on Driver Gear}}{\text{Number of Teeth on Driven Gear}}
\]

2. Velocity Ratio: The velocity ratio is the inverse of the gear ratio because it represents the speed relationship between the driver and driven gears.
\[
\text{Velocity Ratio} = \frac{\text{RPM of Driven Gear}}{\text{RPM of Driver Gear}} = \frac{1}{\text{Gear Ratio}}
\]

3. RPM Calculation: Using the velocity ratio, we can calculate the RPM of the driven gear if the RPM of the driver gear is known.
\[
\text{RPM of Driven Gear} = \text{RPM of Driver Gear} \times \text{Velocity Ratio}
\]

---

Solution:

#### Old Gear System (1A):
- Driver Gear (A): 60 teeth
- Driven Gear (B): 30 teeth

##### Step 1: Calculate the Gear Ratio
\[
\text{Gear Ratio} = \frac{\text{Number of Teeth on Driver Gear}}{\text{Number of Teeth on Driven Gear}} = \frac{60}{30} = 2
\]

##### Step 2: Calculate the Velocity Ratio
\[
\text{Velocity Ratio} = \frac{1}{\text{Gear Ratio}} = \frac{1}{2}
\]

##### Step 3: RPM Calculation
Assume the RPM of the driver gear (A) is \( \text{RPM}_A \). Then:
\[
\text{RPM of Driven Gear (B)} = \text{RPM}_A \times \text{Velocity Ratio} = \text{RPM}_A \times \frac{1}{2}
\]

If the RPM of the driver gear is not specified, we leave the expression in terms of \( \text{RPM}_A \).

---

#### New Gear System (2A):
- Driver Gear (A): 25 teeth
- Driven Gear (B): 75 teeth

##### Step 1: Calculate the Gear Ratio
\[
\text{Gear Ratio} = \frac{\text{Number of Teeth on Driver Gear}}{\text{Number of Teeth on Driven Gear}} = \frac{25}{75} = \frac{1}{3}
\]

##### Step 2: Calculate the Velocity Ratio
\[
\text{Velocity Ratio} = \frac{1}{\text{Gear Ratio}} = \frac{1}{\frac{1}{3}} = 3
\]

##### Step 3: RPM Calculation
Assume the RPM of the driver gear (A) is \( \text{RPM}_A \). Then:
\[
\text{RPM of Driven Gear (B)} = \text{RPM}_A \times \text{Velocity Ratio} = \text{RPM}_A \times 3
\]

If the RPM of the driver gear is not specified, we leave the expression in terms of \( \text{RPM}_A \).

---

Final Answers:

#### Old Gear System (1A):
- Gear Ratio: \( 2 \)
- Velocity Ratio: \( \frac{1}{2} \)
- RPM of Driven Gear (B): \( \text{RPM}_A \times \frac{1}{2} \)

#### New Gear System (2A):
- Gear Ratio: \( \frac{1}{3} \)
- Velocity Ratio: \( 3 \)
- RPM of Driven Gear (B): \( \text{RPM}_A \times 3 \)

---

Boxed Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{Old Gear System (1A):} & \\
\text{Gear Ratio} = 2 & \text{Velocity Ratio} = \frac{1}{2} \\
\text{RPM of Driven Gear (B)} = \text{RPM}_A \times \frac{1}{2} & \\
\\
\text{New Gear System (2A):} & \\
\text{Gear Ratio} = \frac{1}{3} & \text{Velocity Ratio} = 3 \\
\text{RPM of Driven Gear (B)} = \text{RPM}_A \times 3 & \\
\end{array}
}
\]