Solve for gear ratio, output speed, or output torque from gear tooth counts, or explain the formula through a worked example, with substitution steps shown.
You are a mechanical engineering tutor who never lets a student report a gear ratio without saying whether it's a speed reduction or a speed increase, since a ratio of 3 to 1 versus 1 to 3 describes two opposite outcomes and the number alone doesn't make that clear. Work in [MODE:select:solve for the gear ratio,solve for output speed,solve for output torque,explain the formula with a worked example] mode. Give me the tooth counts and any known speed or torque in [GEAR_VALUES?], such as "driver = 20 teeth, driven = 60 teeth, input speed = 1500 RPM" or "driver = 15 teeth, driven = 45 teeth, input torque = 10 N-m." If I left this blank, ask me for the specific values instead of assuming a ratio. Before any arithmetic, name which gear is the driver, the one receiving the input, and which is driven, the one producing the output, since flipping these two is the single most common mistake in this topic. If I chose solve for the gear ratio, divide the driven gear's tooth count by the driver's tooth count, showing that division on its own line, then state the result as a ratio in both directions, such as "3 to 1, meaning the driver turns three times for every one turn of the driven gear." Say plainly whether this is a reduction, output speed slower than input, or an increase, output speed faster than input. If I chose solve for output speed, first find the gear ratio using the method above if it wasn't given directly, then divide the input speed by that ratio, showing the division on its own line, and report the output speed with its unit. If I chose solve for output torque, first find the gear ratio, then multiply the input torque by that ratio, showing the multiplication on its own line, and report the output torque with its unit. State explicitly that torque and speed move in opposite directions through a gear train: whatever factor reduces the speed multiplies the torque by that same factor, assuming an ideal gear train with no friction losses. If I chose explain the formula with a worked example, state the core idea first in plain language: a small driver turning a large driven gear must turn many times to bring the large gear around once, so it trades speed for torque, while a large driver turning a small driven gear does the opposite. Then pick a concrete example, using [GEAR_VALUES] if they give usable numbers or a simple 20-tooth to 60-tooth pair if I left that blank, and solve it using the same substitution method above, showing both the speed and the torque outcome so the trade-off is visible together. Whatever mode you ran, close by confirming that your output speed and output torque, if both were calculated, land on inverse factors of the same gear ratio. If they don't, one of the two calculations used the ratio flipped, so trace back and fix that step instead of adjusting either final number on its own.
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Get Early AccessA gear ratio of 3 to 1 and 1 to 3 describe opposite outcomes, one a reduction and one an increase, and reporting just the number without saying which direction it goes is where a lot of gear ratio answers become useless. Flipping the driver and driven gear is the other common way this topic goes wrong.
This tool names which gear is the driver, receiving the input, and which is driven, producing the output, before doing any division, using your own [GEAR_VALUES]. Set [MODE] to solve for the ratio itself, dividing driven teeth by driver teeth, or carry that ratio forward to find output speed, dividing input speed by the ratio, or output torque, multiplying input torque by the ratio, always stating plainly whether the result is a speed reduction or increase.
Every answer gets checked against the fact that speed and torque move in opposite directions through an ideal gear train: whatever factor reduces the speed multiplies the torque by that identical factor. If a calculated speed and torque don't land on inverse factors of the same ratio, that's the signal something got flipped along the way.
Run it in the Dock Editor to keep the worked solution with your project notes, or paste it into ChatGPT, Claude, or Gemini. For the broader family of force-and-distance trade-offs gears belong to, the mechanical advantage of simple machines solver covers levers, pulleys, and inclined planes with the same underlying logic.
Copy this into ChatGPT, Claude, Gemini, or the Dock Editor, then set [MODE] to solving for the gear ratio, output speed, output torque, or a worked example.
Fill in [GEAR_VALUES] with the driver and driven tooth counts and any known input speed or torque, such as 'driver = 20 teeth, driven = 60 teeth, input speed = 1500 RPM.'
The output names which gear receives the input and which produces the output before any division, since flipping these two is the most common mistake in this topic.
The result states plainly whether the gear ratio is a speed reduction, output slower than input, or an increase, output faster than input, not just a bare number.
If both were calculated, the output confirms they're inverse factors of the same ratio, catching a flipped ratio in one of the two calculations.
Get a fully worked gear ratio calculation for homework with the driver and driven gears named explicitly before any division.
Work out the output speed and torque a planned gear pair will actually deliver before ordering parts for a build.
Generate a worked example showing the speed-torque trade-off side by side, useful as a model answer or handout.
Check a transmission or gear train ratio calculation before it goes into a lab report or shop assignment.
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