How many teeth must the larger gear have if the smaller gear has 50 teeth and they turn at 250 and 300 rpm respectively?

To determine how many teeth the larger gear must have, we start by applying the relationship that governs gears in terms of their speeds and teeth count. This relationship states that the product of the speed (rpm) and the number of teeth for two meshing gears should be equal. Specifically, if we denote the number of teeth on the smaller gear as T1, the larger gear as T2, and their respective speeds as N1 and N2, we can express this relationship mathematically as:

N1 × T1 = N2 × T2.

In your case, the smaller gear has 50 teeth and operates at 250 rpm, while the larger gear operates at 300 rpm. Plugging these values into the equation gives:

250 × 50 = 300 × T2.

Calculating the left side, we have:

12500 = 300 × T2.

To find T2, we can rearrange the equation:

T2 = 12500 / 300.

After performing the division, we find that:

T2 = 41.67 teeth.

Since the number of teeth must be a whole number, we can round T2 to the nearest whole number, which is 42. Given this context of possible answers