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 goes in the numerator and which part in the denominator, always place the initial units opposite to the desired final units, so that they cancel out. For example, when converting miles to kilometers, set up the conversion factor to cancel the "mile" unit, leaving the "km" unit: \[ 5\,\text{miles} \times \frac{1.609\,\text{km}}{1\,\text{mile}} \] This ensures the correct arrangement of conversion factors and leads to the desired final units.

Step-by-step solution

Identify the units to be converted

To start with, determine the initial units that need to be converted and the desired final units. This information is vital in deciding which conversion factors to use and how to arrange them.

Find appropriate conversion factors

Look up the necessary conversion factors that you'll need for converting the initial unit to the final unit. Conversion factors relate two different units and give their equivalent quantities. For example, \( 1\,\text{mile} = 1.609\,\text{km}\).

Determine the proper arrangement of conversion factors

To decide which part of the conversion factor goes in the numerator and which part in the denominator, remember that you want to cancel out the initial units and obtain the final units. Therefore, the initial units should always be in the opposite part of the fraction (numerator or denominator) as the desired final units. For instance, if you want to convert miles to kilometers, place the conversion factor in a way that it cancels the "mile" unit and leaves you with the "km" unit.

Setting up the conversion factor ratio

Set up the conversion factor ratio with initial units and conversion factors. Make sure that the initial units are opposite to the units in the conversion factor, so they cancel each other out. Example: Convert \ (5\,\text{miles}\) to kilometers. Given, \( 1\,\text{mile} = 1.609\,\text{km}\) Set up the ratio as follows: \[ 5\,\text{miles} \times \frac{1.609\,\text{km}}{1\,\text{mile}} \]

Perform the calculation and cancel the units

Do the calculation and ensure the initial units are canceled out. In the example above, the "mile" unit will cancel out, leaving you with the "km" unit: \[ 5\,\cancel{\text{miles}} \times \frac{1.609\,\text{km}}{1\,\cancel{\text{mile}}} = 8.045\,\text{km} \] The result of converting \(5\) miles to kilometers is \(8.045\) kilometers.

Key concepts

Conversion Factors

When converting units, conversion factors play a crucial role. They are essentially the bridge between the initial units you have and the final units you wish to achieve.

To use conversion factors effectively, you first need to identify the relationship between the two different units. For example, if you're converting from miles to kilometers, you need to know that 1 mile is equivalent to 1.609 kilometers. This conversion factor provides the mathematical ratio that you'll use to convert one unit to another.

Conversion factors can be seen as fractions. They provide a way to multiply your original value such that the units "transform" while the numerical value gets updated based on the new unit measure. By understanding the relationship between units, you can ensure precise conversions every time.

• Choose the correct conversion factor.
• Understand the relationship it represents.
• Use it as a fraction to adjust your units.

Numerator and Denominator

When you set up your fraction with the conversion factor, it's essential to know which part of the conversion factor goes in the numerator and which goes in the denominator. This is crucial for ensuring that the units cancel out correctly.

The rule of thumb is to place the unit you want to get rid of in the opposite part of the fraction from where it currently stands. This means if your current unit is in the numerator, it should be in the denominator of the conversion factor.

For example, if you're converting miles to kilometers and you have your distance in miles in the numerator, your conversion factor's mile unit should be in the denominator:\[ 5\,\text{miles} \times \frac{1.609\,\text{km}}{1\,\text{mile}} \]This placement ensures that the miles cancel each other out.

• Identify where your current unit is (numerator or denominator).
• Place it in the opposite position in the conversion factor.
• This ensures the unit cancels during multiplication.

Cancellation of Units

One of the most satisfying parts of unit conversion is the cancellation of units. This is the process that ensures the correctness of your conversion.

Once you have your conversion factor set up correctly (with the appropriate numerator and denominator arrangement), you'll multiply your original quantity by the conversion factor. During this multiplication, the units you wish to convert will "cancel out."

In practice, this looks like striking through the units to indicate they are leaving the equation, allowing only the desired unit to remain. For instance:\[ 5\,\cancel{\text{miles}} \times \frac{1.609\,\text{km}}{1\,\cancel{\text{mile}}} = 8.045\,\text{km} \]Notice how the mile units are crossed out, leaving kilometers as the sole remaining unit, which is the final desired unit.

• Set your equation up with the correct conversion factor.
• Ensure units are placed to cancel each other.
• Cross out the units as they cancel, simplifying to your desired unit.