If a 38-tooth gear runs at 360 rpm, how many teeth does the driven gear have when it's running at 144 rpm?

To find the number of teeth on the driven gear, we can use the relationship between the speeds and the number of teeth of the gears, which is often represented in gear systems by the formula:

( \text{Speed of driver gear} \times \text{Teeth of driver gear} = \text{Speed of driven gear} \times \text{Teeth of driven gear} )

In this situation, we can substitute in the given values. The driver gear has 38 teeth and runs at 360 rpm, while the driven gear runs at 144 rpm. Let ( T_d ) represent the number of teeth on the driven gear.

Using the formula:

( 360 , \text{rpm} \times 38 = 144 , \text{rpm} \times T_d )

Now we can rearrange the equation to solve for ( T_d ):

1. Calculate the left side: ( 360 \times 38 = 13680 )

2. Set up the equation: ( 13680 = 144 \times T_d )

3. Now divide both sides by 144 to isolate ( T_d ): ( T_d = \frac{