Question

When you convert units, how do you decide which part of the conversion factor is in the numerator and which is in the denominator? [Section 1.6]

Short answer

To decide which part of the conversion factor is in the numerator and which is in the denominator, follow these steps: 1. Identify the original and desired units. 2. Find the known relationship between the units, which will be the conversion factor. 3. Set up the conversion factor as a fraction to cancel out the original units when multiplied with the given value. 4. Multiply the given value by the conversion factor, keeping track of units. 5. Simplify the result and write the final answer with the desired units. For example, when converting meters to centimeters, arrange the conversion factor \(\frac{100\: \text{cm}}{1\: \text{m}}\) to cancel out the original unit (m) when multiplied.

Step-by-step solution

Identify the original and desired units

First, find the original units of the given value and the desired units you want to convert it to. This step helps to correctly set up the conversion factor.

Use known relationships between the units

Find the relationship between the original unit and the desired unit. This relationship will be the conversion factor we'll use to convert the original units to the desired units. Conversion factors can be found in textbooks, online resources, or other references.

Set up the conversion factor to cancel out the original units

Write the original value as a fraction with the original units in the numerator and a 1 in the denominator. Then, multiply this value by the conversion factor, arranging the conversion factor as a fraction so that the original units are set to cancel out. For example, if you are converting from meters (m) to centimeters (cm) and the given value is \(x\) meters, you will write it as \(\frac{x\: \text{m}}{1}\). The conversion factor is \(100\: \text{cm}=1\: \text{m}\). Arrange the factor so that it cancels the meters in the original value: \(\frac{100\: \text{cm}}{1\: \text{m}}\).

Multiply the given value by the conversion factor

Multiply the original value by the conversion factor, keeping track of units. In our example, that would be \(\frac{x\: \text{m}}{1} \times \frac{100\: \text{cm}}{1\: \text{m}}\). Notice that after multiplying these fractions, the meters will cancel out, leaving only centimeters as the unit.

Simplify and write the final result

After multiplying the given value by the conversion factor, simplify the result and write the final answer with the desired units. In our example, we get \(100x\: \text{cm}\) after m is cancelled out, leaving centimeters as the desired unit.