Gear ratio:
what two meshed gears do to speed and force

Teeth tell you the ratio

When two gears mesh, each tooth of one fits between teeth of the other, so the same number of teeth pass the contact point on both gears. A small gear therefore has to turn more times than a large one to move the same number of teeth.

The driving gear is the one connected to the power source, such as a motor shaft or the pedals' chainring. The driven gear is the one that receives motion and is attached to the load, such as the wheel or a roller.

What the ratio does to speed and torque

A ratio above 1 is a reduction: the output turns slower than the input, but with more twisting force. A ratio below 1 is an overdrive: the output turns faster and delivers less torque.

The trade is exact: power stays about the same (minus losses), so what you gain in torque you give up in speed. A reduction does not create energy.

Chains of gears multiply

When a gear pair feeds another pair on a shared shaft, the ratios multiply. A 3:1 stage followed by a 2.5:1 stage produces a total reduction of 3 × 2.5 = 7.5:1. Input speed of 1,500 rpm falls to 200 rpm at the final output.

An idler gear placed between two others changes the direction of rotation but not the ratio, because its tooth count cancels out. Only the first driving gear and the last driven gear matter for the overall ratio of a simple train with idlers.

Where you meet gear ratios

• A bicycle: a 48-tooth chainring driving a 16-tooth rear sprocket is a 1:3 ratio, so the wheel turns three times per pedal turn, giving speed at the cost of effort.
• A car gearbox: first gear has a high reduction for pulling away, top gear has a ratio near or below 1 for cruising.
• A drill or screwdriver: a reduction gearbox gives the torque needed to drive screws.
• Clocks: gear trains step motion down so that one shaft turns once every hour while another turns once every minute.

For comparing different quantities that follow ratios and proportions in physics problems, a general magnitude tool can be useful, but gear ratio itself needs only the two tooth counts.

Watch the convention

Some texts write the ratio as driving ÷ driven, which gives the inverse number, 1:3 instead of 3:1. Cycling and automotive writing often say a gear is 'higher' when the ratio is closer to 1 or below, whichever way it is written. Always state which gear is on top of the fraction.

A bicycle worked through

A bicycle makes the idea concrete. Take a 48-tooth chainring at the pedals and a 16-tooth sprocket on the rear wheel. The driven gear (the sprocket) has fewer teeth, so the ratio, driven ÷ driving, is 16 ÷ 48 = 0.33, or 1:3. The wheel turns three times for each pedal revolution.

With a 2.1 m wheel circumference, one pedal turn moves the bike 3 × 2.1 = 6.3 metres. At a cadence of 80 pedal revolutions per minute, that is 504 m per minute, or about 30 km/h. Switch to a 32-tooth sprocket and each pedal turn moves the bike only 48 ÷ 32 × 2.1 = 3.15 m, which is why low gears feel easy and slow.

The same relation links any motor, gearbox and wheel: road speed is wheel circumference times wheel rpm, and wheel rpm is motor rpm divided by the total reduction.

Picking a ratio for a job

Choosing a ratio is a matter of what you want to maximise. A winch lifting a heavy load wants a large reduction, so a small motor can produce a big force at a slow speed. A fan driven from a motor may want a ratio close to 1. A bicycle wants a range of ratios so the rider's legs can work at a comfortable cadence on flat ground and hills.

A practical method is to start from the output you need, speed and torque, and divide by the motor's available speed and torque. The two requirements usually give slightly different ratios, and the final choice is a compromise that also considers gear size, noise and cost.

Why tooth counts are whole numbers

Because teeth are counted, ratios come out as fractions of whole numbers, such as 60:20 or 47:13. Designers often pick tooth counts that share no common factor, so that every tooth on one gear meets every tooth on the other over time. This spreads wear evenly and avoids the same pair of teeth meeting on every revolution.

Real-world limits

The simple formula assumes perfect meshing. In practice, gear pairs lose about 1% to 3% of power per stage in friction, and there is a small amount of backlash. Helical and spur gears differ slightly in efficiency, and lubrication matters. The ratio itself stays exact, since it comes from counting teeth, but the torque and power figures are always a touch lower than the ideal calculation.