In mechanical engineering, the gear ratio is a measure of the rotational speed of two or more interconnected sprockets. As a general rule, when it comes to two sprockets, if the driving wheel (the one that directly receives the rotational force from the motor) is greater than the driven wheel, the latter will rotate faster and vice versa. This basic concept can be expressed by the formula **Gear ratio = T2 / T1**, where T1 is the number of teeth of the first sprocket and T2 is the number of teeth of the second.

## Steps

### Method 1 of 2: Finding the Gear Ratio of a Gear Train

#### Gear with two sprockets

#### Step 1. Start with a two-sprocket train

To determine the gear ratio, you must have at least two sprockets interconnected with each other; This is called a "gear train." Normally, the first sprocket is the "driving" or "driving" and is coupled to the motor shaft, and the second is the "driven" or "driven", which is connected to the first. Between these two wheels there may be many more that help to transmit the movement, called intermediate or "crazy" wheels.

### For now, a gear train with only two sprockets will be considered. To find the gear ratio, these wheels must be interconnected and one of them must be turning the other. For example, consider that a small driving wheel (wheel 1) moves a larger one (wheel 2)

#### Step 2. Count the number of teeth on the drive wheel

An easy way to find the gear ratio between two interconnected wheels is to compare the number of teeth on the two (these are the bumps on the edges of the wheels). First, determine how many teeth are on the drive wheel. You can count manually or sometimes just check the information on the label.

- As an example, consider that the smallest wheel in the system has
**20 teeth**.

#### Step 3. Count the number of teeth on the driven wheel

Determine how many teeth the wheel has driven in the same way that you did with the other.

- As an example, consider that the driven wheel has
**30 teeth**.

#### Step 4. Divide the values between them

Now that you know how many teeth each wheel has, you can find the gear ratio pretty easily. Divide the number of teeth on the driven wheel by the number of teeth on the driving wheel. Depending on your need, the result can be expressed with a decimal number, a fraction or in the form of a relation, that is, x: y.

- In our case, if you divide the 30 teeth of the driven wheel by the 20 teeth of the driving wheel, we get 30/20 =
**1, 5**. You can also write it as**3/2**or**1, 5: 1**, etc. - This value indicates that the small drive wheel has to go one and a half turns for the larger one to complete one turn. It makes sense since the driven wheel is larger and will spin slower.

#### More than two wheels

#### Step 1. Consider a gear train with more than two sprockets

As the name implies, a "gear train" can also be made up of a long sequence of sprockets, and not just two. In these cases, the first sprocket remains the driving wheel and the last the driven one. The wheels in the middle are called "idlers" or "intermediate" wheels. These are often used to change the direction of rotation or to connect two wheels that could not be connected directly.

### For example, consider that the two sprockets in the previous section are driven by a small wheel with 7 teeth. In this case, the 30 tooth wheel is still driven, but the 20 tooth wheel, which was previously the driving wheel, is now an idler wheel

#### Step 2. Divide the number of teeth on the driving wheel and the driven wheel

It is very important to know that when it comes to gear trains with more than two sprockets, only the driving wheel and the driven wheel (usually the first and last) matter. In other words, the idler does not affect the gear train gear ratio at all. When you have identified which is the driving wheel and which is the driven one, you can calculate the gear ratio in the same way as you did before.

- In the example, the 30 teeth of the driven wheel are divided by the 7 teeth of the new driving wheel: 30/7 = approximately
**4, 3**(or 4, 3: 1, etc.). This means that the driving wheel must turn 4.3 times so that the driven wheel turns once.

#### Step 3. If you want, you can find the gear ratio of the middle wheels

In some situations, it may be necessary to find the gear ratio for the idlers as well. In these cases, start from the driving wheel and move towards the driven wheel. In other words, consider the first driving wheel and the second driving. Divide the number of teeth on each "driven" wheel by the number of teeth on each "driven" wheel, for each set of gears, in order to calculate intermediate gear ratios.

- In our case, the intermediate gear ratios are 20/7 =
**2, 9**and 30/20 =**1, 5**. Note that none of these is equal to the gear ratio of the entire train, which is 4.3. **Nevertheless**, note that (20/7) × (30/20) = 4, 3. In general, the intermediate gear ratios, when multiplied together, will result in the overall gear ratio.

### Method 2 of 2: Do Ratio / Speed Calculations

#### Step 1. Find the rotational speed of the driving wheel

Using the gear ratio concept, it is easy to calculate how fast the driven wheel is turning based on the input speed of the driving wheel. To start, find the rotational speed of the driving wheel. In most gear calculations, this is calculated in rotations per minute (rpm), although other units of measure can also be used.

### For example, consider that in the gear train above you have a 7 tooth wheel spinning at 130 rpm. With this information, you can find the speed of the driven wheel in a few steps

#### Step 2. Substitute this information into the formula S1 × T1 = S2 × T2

In this formula, S1 is the speed of rotation of the driving wheel, T1 is the number of teeth of the driving wheel, and S2 and T2 represent the speed and number of teeth of the driven sprocket, respectively. Enter the values you have, leaving the unknown aside.

- Often in these types of problems, you will have to find the value of S2, although you can get the value of any other unknown factor. Enter the data in this formula and you will get:
**130 rpm × 7 = S2 × 30**

#### Step 3. Solve the problem

To find the value of the unknown, you will have to apply your basic algebra knowledge. You simply have to simplify the rest of the equation and isolate the unknown value on one side of the equals sign and you will have the answer. You must not forget to express the result with the correct unit of measurement; you can lose points on a high school job, for example, if you don't.

- In our example, you can solve as follows:
- 130 rpm × 7 = S2 × 30
- 910 = S2 × 30
- 910/30 = S2
**30, 33 rpm**= S2- In other words, if the driving wheel turns at 130 rpm, the driven wheel will turn at 30.33 rpm. This makes sense as the driven wheel is larger and will therefore turn more slowly.

## Advice

- To see the principles of the gear ratio in action, take a bike ride! Note that it is easier to climb a hill when you have a small wheel at the front and a large wheel at the rear. In this case, despite being much easier to turn the small gear with the force of the pedals, it takes more revolutions to turn the rear wheel that has a larger sprocket. That is, you will go slower but it will be easier to pedal. Having a larger wheel at the front will reverse this process.
- In a speed reduction system (where the speed of the driven wheel is lower than that of the driving wheel), you will need a motor that provides optimal power at high rotational speeds.
- The power required to move the driven wheel is amplified or reduced by the gear ratio. Once the gear ratio is taken into account, the engine size is determined according to the power required to drive the load. In a speed multiplication system (where the speed of the driven wheel is higher than that of the driving wheel), you will need a motor that provides optimal power at low rotational speeds.