Name the driver and driven gear before dividing teeth
The driver is the gear receiving the input motion, and the driven gear is the gear whose output the worksheet reports. This calculator defines ratio as driven tooth count divided by driver tooth count. Because engineering books and manufacturers may display the reciprocal or use labels such as speed ratio, reduction ratio, or transmission ratio, always state the convention beside a bare number. The fraction calculator can simplify an entered tooth-count pair, but it cannot decide which gear is the input or what a domain-specific ratio label means.
Tooth counts must be positive whole numbers. Input rotational speed and torque may be zero or positive, while the optional efficiency entry must be greater than zero and no more than 100 percent. Units also need discipline: input and output speed share the same revolutions-per-time unit, and the torque result stays in the entered torque unit. A ratio is dimensionless, so writing rpm or newton-metres beside it would confuse the relationship with one of its consequences.
Follow speed and torque through a 15-to-45-tooth mesh
A fully labelled example exposes both the arithmetic and the assumptions. Let the 15-tooth gear drive the 45-tooth gear at 3,000 rpm with 2 newton-metres of input torque.
- Calculate the declared reduction
Divide 45 driven teeth by 15 driver teeth. The declared ratio is 3, often written 3:1 when the convention is made explicit. Three driver revolutions correspond to one driven revolution for this pair.
- Divide input speed by the ratio
The driven speed is 3,000 ÷ 3 = 1,000 rpm in magnitude. One external mesh reverses rotational direction, so a clockwise driver produces a counter-clockwise driven gear when viewed consistently along parallel shafts.
- Multiply torque under an ideal model
Ideal output torque is 2 × 3 = 6 newton-metres. This is the loss-free relationship represented by the calculator; it is not a rating for the teeth, shaft, bearing, key, housing, or motor.
- Apply only the entered efficiency assumption
If the user explicitly enters 90 percent, the displayed adjusted torque is 6 × 0.90 = 5.4 newton-metres. The worksheet did not derive 90 percent from materials, lubrication, speed, temperature, alignment, or load.
Reduction, one-to-one, and overdrive have different outcomes
The ratio's position relative to one describes the direction of the speed-and-torque trade within this declared convention.
Ratio greater than one
The driven gear has more teeth, so output speed is lower in magnitude and ideal output torque is higher. This is a reduction in speed, not evidence that the mechanism can withstand the calculated torque.
Ratio equal to one
Equal tooth counts preserve speed and ideal torque magnitude through the pair while reversing direction. Equal counts do not guarantee equal pitch diameter unless the gears also share the required tooth system and compatible geometry.
Ratio less than one
The driven gear has fewer teeth, so output speed is higher and ideal output torque is lower. The word overdrive may describe that numeric outcome in some contexts, but the calculator does not assess permissible pitch-line velocity or structural capacity.
Efficiency is a declared scenario, not a solved loss model
Real transmitted torque is affected by tooth friction, bearing and seal losses, lubrication regime, alignment, manufacturing accuracy, deflection, speed, temperature, and load. A single efficiency percentage compresses all of those effects into a user assumption. It is useful for showing how a documented assumed loss changes the ideal result, but it cannot replace measured data, a manufacturer's efficiency map, or a power-loss analysis for the operating point.
The gap between ideal and adjusted output can be expressed as a percentage with the scientific calculator, provided the ideal value is explicitly the reference. That arithmetic check still says nothing about where energy is dissipated. Also remember that applying an efficiency factor to output torque while leaving speed fixed is a simplified accounting convention; it does not diagnose heat generation or transient behavior.
Read the vehicle result as an overall engine-to-wheel estimate
The vehicle path estimates wheel RPM from road speed divided by tire circumference, then divides engine RPM by that wheel RPM. The tire diameter and speed selectors are normalized before calculation, so equivalent inch-and-mph or metric entries return the same dimensionless ratio. Use the loaded rolling diameter measured or documented for the operating condition, not the wheel rim diameter.
This result represents the combined engine-to-wheel ratio at the entered operating point. It does not identify an individual transmission gear or axle ratio, and it assumes no clutch, torque-converter, or tire slip. Instrument error, tire growth, wear, inflation, load, and road conditions can all move a real observation away from the ideal circumference relationship.
Keep each calculation path inside its limits
The basic driver-and-driven relationship and unit conversions are reproducible, but neither mode designs or validates a working mechanism.
- Do not apply the result directly to compound trains, planetary sets, worm gears, bevel gears, belts, chains, or idler arrangements. Trace every stage with its own compatible convention, direction, and efficiency model.
- Verify module or diametral pitch, pressure angle, center distance, face width, interference and undercut risk, backlash, contact ratio, lubrication, material, tooth stress, shaft and bearing loads, critical speed, guarding, duty cycle, and applicable standards with a qualified designer.
- The vehicle mode reports the combined ratio implied by RPM, road speed, and loaded tire diameter; it cannot separate transmission and final-drive ratios or diagnose slip. The tire dimensions calculator supplies only nominal passenger-tire geometry, not the loaded rolling diameter at a specific operating point.
- Treat calculated torque as a relationship, never as an allowable load or component rating. Consult drawings, manufacturer limits, validated simulation, and physical testing before building or operating machinery, and include suitable guarding and safe test procedures.