Gear Ratios & Velocity Ratio Calculations

This lesson focuses on the mathematical calculations associated with gears and pulleys. Being able to calculate gear ratios, output speeds and velocity ratios is essential for AQA GCSE Design and Technology (8552), Section 3.1.5, and these calculations frequently appear on Paper 1.

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Gear Ratio (GR)

The gear ratio tells you how the speed and torque change between the driver (input) gear and the driven (output) gear.

Formula

$\text{Gear Ratio} = \frac{\text{Number of teeth on driven gear}}{\text{Number of teeth on driver gear}}$Gear Ratio=Number of teeth on driver gearNumber of teeth on driven gear​

Interpreting the Gear Ratio

Gear Ratio	Meaning	Effect
GR > 1	The driven gear has MORE teeth than the driver	Speed decreases, torque increases (gearing down)
GR < 1	The driven gear has FEWER teeth than the driver	Speed increases, torque decreases (gearing up)
GR = 1	Both gears have the same number of teeth	Speed and torque are unchanged (direction changes)

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Calculating Output Speed

Once you know the gear ratio, you can calculate the output speed:

$\text{Output speed (RPM)} = \frac{\text{Input speed (RPM)}}{\text{Gear Ratio}}$Output speed (RPM)=Gear RatioInput speed (RPM)​

Worked Example 1: Simple Gear Train

A motor drives a gear with 20 teeth at 600 RPM. This meshes with a gear with 60 teeth. Calculate the output speed.

Step 1: Calculate gear ratio

$GR = \frac{60}{20} = 3$GR=2060​=3

Step 2: Calculate output speed

$\text{Output speed} = \frac{600}{3} = 200 \text{ RPM}$Output speed=3600​=200 RPM

Answer: The output gear rotates at 200 RPM. The system has geared down — slower speed but higher torque.

Worked Example 2: Gearing Up

A motor drives a gear with 40 teeth at 300 RPM. This meshes with a gear with 10 teeth. Calculate the output speed.

$GR = \frac{10}{40} = 0.25$GR=4010​=0.25

$\text{Output speed} = \frac{300}{0.25} = 1200 \text{ RPM}$Output speed=0.25300​=1200 RPM

Answer: The output gear rotates at 1200 RPM. The system has geared up — faster speed but lower torque.

AQA Exam Tip: Always show each step of your working clearly. Even if you get the final answer wrong, you can gain marks for correct method (gear ratio formula, output speed formula).

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Compound Gear Ratios

For a compound gear train, multiply the individual gear ratios together:

$\text{Overall GR} = GR_1 \times GR_2 \times GR_3 \times \ldots$Overall GR=GR1​×GR2​×GR3​×…

Worked Example 3: Compound Gear Train

Gear	Teeth	Notes
A (driver)	10	Input at 1000 RPM
B	40	Meshes with A
C (same shaft as B)	10	Meshes with D
D (output)	50	Output

Step 1: Individual gear ratios

$GR_1 = \frac{40}{10} = 4$GR1​=1040​=4

$GR_2 = \frac{50}{10} = 5$GR2​=1050​=5

Step 2: Overall gear ratio

$\text{Overall GR} = 4 \times 5 = 20$Overall GR=4×5=20

Step 3: Output speed

$\text{Output speed} = \frac{1000}{20} = 50 \text{ RPM}$Output speed=201000​=50 RPM

Answer: The output speed is 50 RPM — a significant reduction from the 1000 RPM input, with a corresponding increase in torque.

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Velocity Ratio (VR)