When converting from one unit to another, we multiply a quantity by a conversion factor. A conversion factor is a fraction equivalent to 1, so multiplying it by a quantity doesn't change its actual measure. However, the fraction certainly doesn't look like it's equal to one (at least not until you inspect it). The expressions in the numerator and denominator are equal quantities, but measured in different units. During unit conversion, units are also canceled out when they appear in both the numerator and the denominator. To illustrate this idea, let's convert 1.5 feet into inches, using the fact that 1 foot is equivalent to 12 inches.
A unit rate is a ratio between two quantities with different units. For examples, speed is a unit rate, because it is the ratio between distance traveled and the time taken to travel that distance. There are two different types of units that make up this ratio, distance and time. Another characteristic of a unit rate is that a unit rate has a denominator of 1. Let's think about speed again: we wouldn't normally say that a car travels 120 miles per 2 hours. Instead, we simplify the ratio to 1 hour, and say 60 miles per hour.
Converting Unit Rates
When converting unit rates, we need to make use of multiple conversion factors: as many conversion factors needed to convert from one quantity to another for every unit involved in our unit rate. Let's work more with miles per hour, and convert 60 miles per hour into feet per second. We'll need to convert miles to kilometers, but also hours to seconds. Let's first list our conversion factors:
When using a conversion factor to create a fraction equal to 1, think about what unit you wish to convert into. In most cases, this will be the numerator of the fraction, which places the unit you wish to cancel in the denominator of that fraction. This is because the unit you wish to cancel will likely be in the numerator of another fraction. Units are canceled when they appear in the numerator and denominator of the combined fraction.
Keep in mind that although we have listed some very helpful conversion factors, we may not need to use all of them. For example, as we will see when we are converting from miles per hour into feet per second, we will use the conversion factor that 1 mile = 5280 feet, instead of 1 foot = 0.0002 miles. This all depends on which unit you want to convert into and which unit you wish to cancel.