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To convert between measurements in different measuring systems, you can multiply by a conversion factor. A conversion factor is a ratio that, when you multiply it by the item you’re converting, cancels out the units you don’t want and leaves those that you do. The conversion factor must equal 1.

Here’s how it works: For every relation between units — for example, 24 hours = 1 day — you can make a fraction that has the value of 1. If, for example, you divide both sides of the equation 24 hours = 1 day by 1 day, you get

Suppose you want to convert 3 days to hours. You can just multiply your time by the preceding fraction. Doing so doesn’t change the value of the time because you’re multiplying by 1. You can see that the unit of days cancels out, leaving you with a number of hours:

Words such as days, seconds, and meters act like the variables x and y in that if they’re present in both the numerator and the denominator, they cancel each other out.

To convert the other way — hours into days, in this example — you simply use the same original relation, 24 hours = 1 day, but this time divide both sides by 24 hours to get

Then multiply by this fraction to cancel the units from the bottom, which leaves you with the units on the top.

Consider the following problem. Passing the state line, you note that you’ve gone 4,680 miles in exactly three days. Very impressive. If you went at a constant speed, how fast were you going? Speed is just as you may expect — distance divided by time. So you calculate your speed as follows:

Your answer, however, isn’t exactly in a standard unit of measure. You have a result in miles per day, which you write as miles/day. To calculate miles per hour, you need a conversion factor that knocks days out of the denominator and leaves hours in its place, so you multiply by days/hour and cancel out days:

Your conversion factor is days/hour. When you multiply by the conversion factor, your work looks like this:

Note that because there are 24 hours in a day, the conversion factor equals exactly 1, as all conversion factors must. So when you multiply 1,560 miles/day by this conversion factor, you’re not changing anything — all you’re doing is multiplying by 1.

When you cancel out days and multiply across the fractions, you get the answer you’ve been searching for: