Gear Ratios

So recently a topic has come up on our robot reveal post on Instagram about the way gear ratios are written. It seems that another user thinks that we are writing them the wrong way around, and have sources to back that up. However, we also have sources to back up how we are writing them.

If we have a 12 tooth gear (powered by a motor) turning an 84 tooth gear, we would write that as 1:7 (a gear ratio for higher torque). However, this user is of the opinion that this should be written as 7:1 (The user agrees that this is for torque, but thinks it should be written like this).

Similarly, if we were to have a 60 tooth gear (powered by a motor) turning a 36 tooth gear, we would write that as 5:3 (A ratio for speed). This person would write it as 3:5.

My source is the VEX Robotics curriculum website, which states:

The full web page can be found here:

However, the other person’s source is PLTW Principles of Engineering, which states:

My question is, which is correct, and why are there these inconsistencies between two sources?

@DRow (Tagged in case it turns out the problem is with the VEX page)

It doesn’t really matter, but I always thought of it as driving:driven.

1 Like

This discussion:
Gear Ratios How to show properly…

As long as the math is correct at the end, you did it right! If you are computing angular velocities of the shafts, slower is less than one, faster is a ratio more than one. And if you are doing torque calculations, more power is a ratio more than one, less torque for greater speed is a ratio less than one.

This discussion just spins in circles with no conclusive answer. I also have this question…for the exact same reason. What I have been using for years is in agreement with the OP’s convention. 3:1 is torque. 1:3 is speed. I am being “challenged” with an exactly inverse convention. I would appreciate any feedback that you all might have on an actual answer.

As suggested by others in the referenced thread, I have a hard time believing that this is just up to preference. What it really seems to me is that people (possibly including me) are merging vocabulary and not using proper terminology in communicating Gear Ratio…hope that was a clear question.

Bottom line…what is the proper way to communicate gear ratio? Not trying to beat a dead horse, but I kinda need an answer here… so I can be right. I love to be right. :slight_smile:


  1. I think you are actually using the opposite of the OP’s convention.
  2. The terms have inherent ambiguity.

If you want to avoid ambiguity you can also use the terms, torque ratio and speed ratio. It is always expressed
But the speed ratio is output:input angular velocity. And the torque ratio is output:input torque.

You are correct. I looked back at the post just before reading this and realized we were actually “enemies”! :slight_smile:

So, this confusion and “argument” are unavoidable by the nature of the vocabulary?

@jonathans your expression of ‘gear ratio’ is correct. The term ‘gear ratio’ does have a clear definition, its just that the term is commonly misused. (At least, to my knowledge)

According to the SAE and numerous other sources…

That quote can be found on this PDF from the SAE, page 3 line 2

Just a side note:
Gear ratio is basically the same as the velocity ratio
And the way the OP expresses the ratio is known as the torque ratio or teeth ratio

@Amadeus Funny you use that document. I am using the same one as one of my documents of support. However, the VEX page referenced by the OP is just the opposite…and he is quoting that.

So, is the consensus that (in the example from OP) that VEX is using confused terminology in its EDR Curriculum?

Well. according to The SAE, US Bureau of Transport, MIT, Carnegie, NYU, Brooklyn College, and a whole list of other reputable sources, (just google search “gear ratio” and .gov) yes apparently the VEX curriculum got ‘gear ratio’ backwards.

@5YNAP5E That is my thought as well, but I wanted to get some feedback. Thanks for your input. It was a great help.

I had posted something, but I need to go back and clarify things a bit. I’ll repost soon.

Reading and rereading definitions over and over, I think I can make some clearer statements now on what things are an where the confusion comes from, and I know I made some incorrect (or at least unconventional) statements not so long ago. I think I should also make some statements about some better and worse terminology.

First, I’ll highlight the reasons that this gets really confusing:

  1. From the get-go, you could reasonably define things any way you want. It’s ultimately arbitrary, as long as you apply your rules consistently. Ideally, you define things in a consistent way for clarity, but that has not actually been done.

  2. The origins of all this are in simple machines and mechanical advantage, which is defined as output force:input force. However, for some reason along the way, people have decided to sometimes name things as output:input and sometimes as input:output. (Or driving:driven and driven:driving.) Switching orders in definitions (not in calculations, as those are dictated by the math) causes confusion. So much of the blame for the confusion should really fall on those who were inconsistent in the naming process.

  3. As you’ll see in a comment below, there is absolutely no consistency with gear ratios. There is significant consistency within subsets of speech, though. Of course, there is more consistency in the non-technical subset than in the technical subset.

  4. A lot more confusion shows up when people just write driving:driven or driven:driving instead of that they’re actually talking about. Those ratios are nonsensical on their own because there is no statement about what’s actually going into the ratios. Notice about how I specified output force:input force for mechanical advantage.

Now, where does this come from. With machines we have mechanical advantage being output force:input force. However, that is really tricky to work with outside of making measurements. The reason is that mechanical advantage also includes things like friction, the weights of parts of the machine themselves, etc. But we can get a good idea of what’s going on with ideal mechanical advantage, the mechanical advantage if friction, the machine’s weights, etc. can be ignored. Then, from conservation of energy, we get

IMA = ideal output force:input force = input distance:output distance

So we can see that, as IMA only depends on distances, it can be used quite readily with different set-ups to identify what is happening in the design. Sure, some force is lost, but as long as that is understood, the design can be communicated fairly well. Also, it means we can use it to examine the motion of parts of the machine since it’s based distances. Consider a machine that ideally multiplies the force by 4 (IMA), but due to friction it’s actually 3.5 (MA). That means the machine multiplies the distance or speed by 0.25; the output moves only a quarter as much as the input.

However, someone in what I consider questionable wisdom, decided to define speed (or velocity or movement) ratio as input distance:output distance. That’s the same as IMA. Why do I consider this questionable wisdom? Mechanical advantage is how a machine affects force, and theoretically (and for consistency) speed ratio is how a machine affects speed. But speed ratio isn’t how the machine affects the speed, but rather it’s the multiplicative inverse (reciprocal). With gears, this speed ratio is known as the gear ratio. Gear ratio could have been set to IMA with the speed ratio still being chosen the other way. But, alas, no. And now we have yet more confusion: a low-gear’s ratio is a high gear ratio and a high-gear’s ratio is a low gear ratio. I expect this is why I can find these ratios listed the other way, and not just on the VEX site.

But there is even more bad terminology. “Torque ratio” is labeled as the same thing. This sounds good and in some ways better, as this ratio is the idea torque ratio. But, due to friction and related losses, the actual ratio of the torques is less than this. It would have been less confusing if the torque ratio were more carefully defined like the force ratio (MA) instead of the ideal it never is (IMA).

So what we the majority in academia say is

idea mechanical advantage = speed ratio = gear ratio (for gears) = torque ratio (for gears and similar) = input distance:output distance = output (follower, driven) teeth:input (driver) teeth (for gears)

Before ending, I think there is one other important point to address. I’ve seen a lot of comments about mechanical advantage of a rack and pinion being undefined or nonsensical. That really isn’t true. The problem only arises when people decide to create a force:torque ratio. However, we know that if we multiply the pinion’s diameter by some factor, the ideal mechanical advantage gets divided by the same factor. What you really want to do is look at the torque from the motor and divide by its shaft’s diameter. We can then compare the force there to the force along the rack, and we can compare the diameter of the shaft to the diameter of the pinion to get the ideal mechanical advantage.

Hey everyone…

I emailed VEX support the other day to get to the bottom of this on their end concerning the referenced lesson in the EDR curriculum. Below is their response…looks like we (@callen, @5NAP5E, @tabor473).

Thank you for the email.
I have asked one of our Engineers and he said that you are correct! Most books on Gear Ratio have it reversed vs what the
EDR curriculum says. Thanks for bringing it to our attention.

I assume this means they will correct their curriculum content…waiting to confirm that.