Question

(two points)
given
•pinion (driver): ( n_1 = 8 ) teeth
•driven gear: ( n_2 = 12 ) teeth
compute the gear ratio:
& tell which of the two gears rotates slower or the driven? (three points)
(the __ gear rotates times slower than the __.)

Explanation:

Step1: Calculate gear ratio

Gear ratio is defined as $\frac{\text{Number of teeth on driven gear}}{\text{Number of teeth on driver gear}}$.
$\text{Gear Ratio} = \frac{N_2}{N_1} = \frac{12}{8} = \frac{3}{2} = 1.5$

Step2: Relate gear ratio to speed

For gears, speed is inversely proportional to the number of teeth. So $\frac{\text{Speed of driven gear}}{\text{Speed of driver gear}} = \frac{N_1}{N_2}$.
$\frac{\text{Speed of driven gear}}{\text{Speed of driver gear}} = \frac{8}{12} = \frac{2}{3}$
This means the driven gear speed is $\frac{2}{3}$ of the driver gear speed, so the driver gear is slower? No, rearrange: $\text{Speed of driver gear} = \frac{3}{2} \times \text{Speed of driven gear}$. Wait, correction: Wait, driver is pinion ($N_1=8$), driven is $N_2=12$. Since more teeth mean slower speed, driven gear is slower. The factor is $\frac{N_2}{N_1} = 1.5$, so driver is faster, driven is slower by $\frac{3}{2}$ times? Wait no: $\text{Speed}_1 \times N_1 = \text{Speed}_2 \times N_2$. So $\text{Speed}_2 = \text{Speed}_1 \times \frac{N_1}{N_2} = \text{Speed}_1 \times \frac{8}{12} = \text{Speed}_1 \times \frac{2}{3}$. So $\text{Speed}_1 = \text{Speed}_2 \times \frac{3}{2}$. So the driven gear (gear 2) rotates slower, and the driver gear is $\frac{3}{2}$ times faster, meaning the driven gear is $\frac{2}{3}$ as fast, so the driver is 1.5 times faster, so driven is slower than driver by 1.5? No, the question says "which rotates slower the driver or the driven? (The __ gear rotates times slower than the __.)" So driven is slower. Driven speed is $\frac{2}{3}$ driver speed, so driven is $\frac{3}{2}$ times slower than driver? Wait, if A is $\frac{2}{3}$ as fast as B, then A is $\frac{3}{2}$ times slower than B. Yes, because $\text{Speed}_B = \frac{3}{2} \text{Speed}_A$, so $\text{Speed}_A = \frac{2}{3} \text{Speed}_B$, so A is $\frac{3}{2}$ times slower than B.

Answer:

Gear ratio: $1.5$
The driven gear rotates $1.5$ times slower than the driver.