In a given gear system, a driven gear with 30 teeth has a speed of 100rev/min, drives the driven gear with 40 teeth.calculate; A)the gear ratio B)the speed of the driven gear

Here's how to calculate the gear ratio and the speed of the driven gear:

Step 1: Identify the given values.
The driving gear has $N_1 = 30$ teeth.
The speed of the driving gear is $S_1 = 100 \text{ rev/min}$.
The driven gear has $N_2 = 40$ teeth.

A) Calculate the gear ratio.
The gear ratio (GR) is the ratio of the number of teeth on the driven gear to the number of teeth on the driving gear.

$$GR = \frac{N_2}{N_1}$$

Substitute the given values:

$$GR = \frac{40}{30}$$ $$GR = \frac{4}{3}$$ $$GR \approx 1.333$$

The gear ratio is $\boxed{\text{4/3 or 1.333}}$.

B) Calculate the speed of the driven gear.
The relationship between the number of teeth and the speeds of the gears is given by the formula:

$$N_1 S_1 = N_2 S_2$$

Where $S_2$ is the speed of the driven gear.
Rearrange the formula to solve for $S_2$:

$$S_2 = \frac{N_1 S_1}{N_2}$$

Substitute the given values:

$$S_2 = \frac{30 \text{ teeth} \times 100 \text{ rev/min}}{40 \text{ teeth}}$$ $$S_2 = \frac{3000}{40} \text{ rev/min}$$ $$S_2 = 75 \text{ rev/min}$$

The speed of the driven gear is $\boxed{\text{75 rev/min}}$.