if the turns of the output gear over the turns of the input gear is equal to the number of teeth on the input gear over the number of teeth on the output gear, the input gear has 36 teeth and the output has 10, the output gear has turned twice as much as the input gear, how many times has the output and input gear moved?

this question really confused me... and the rest of the class..

It confuses me. The ratio of turns is 3.6 whatever the input turns, the output is 3.6 times that.

I can understand why this question might be confusing. Let me break it down for you.

First, let's establish the given information:
- The input gear has 36 teeth.
- The output gear has 10 teeth.
- The output gear has turned twice as much as the input gear.

To solve this problem, we can use the ratio of the turns of the output gear to the turns of the input gear. According to the given condition, this ratio should be equal to the ratio of the number of teeth on the input gear to the number of teeth on the output gear.

So, let's set up the equation using the gears' turns:
(Turns of output gear) / (Turns of input gear) = (Teeth on input gear) / (Teeth on output gear)

Plugging in the given values:
(Turns of output gear) / (Turns of input gear) = 36 / 10

Since it is stated that the output gear has turned twice as much as the input gear, we can set up another equation:
(Turns of output gear) = 2 * (Turns of input gear)

Now we can substitute the second equation into the first equation:
2 * (Turns of input gear) / (Turns of input gear) = 36 / 10

Simplifying the equation, we get:
2 = 36 / 10

To find the number of times the gears have moved, we need to find the value of "2" in the equation. We can solve this by cross-multiplying:
2 * 10 = 36
20 = 36

Since 20 is not equal to 36, the given information in the question is contradictory, and there is no solution to the problem as presented.

It's important to note that contradictory information or errors can sometimes be present in problems, which can cause confusion. If you encounter similar situations, always double-check the given information to ensure its accuracy.