What does unit analysis mean in math?

Answer 1

It means that you determine what units the answer must have, look at what units you are starting with, and figure out what unit ratios must be multiplied by the one you are starting with in order to end up with the right units in the answer. For example, you have 389 minutes, and you want to know how many seconds that is. You know that there are 60 seconds in 1 minute, and this can be written either 60 sec. / min. or 1 min. / 60 sec. Since you are starting with minutes, you want to multiply it by something that will cancel out the minutes and leave you with just seconds in the answer. You always put the first thing you start with (389 minutes) over a 1 (389 min./1), so that it is in the numerator. Then you choose which ratio (either 60sec./min. or 1 min./60 sec.) that you would want to multiply this first thing by in order to cancel out the minutes units. If the minutes units are in the denominator, they will cancel out the minutes units in the numerator, so you choose to multiply by 60 sec./1 min., where the minutes are in the denominator. Here is how it looks:

389 min./1 x 60 sec./min. = 23340 sec. This only required one ratio. Unit analysis really helps when there are several ratios to use:

Let's say you have 2340 seconds, and you want to know how many years that is. You know the following: 1 year = 365 days, 1 day = 24 hours, 1 hour = 60 min., and 1 min = 60 sec. All of these can be expressed as ratios (fractions), such as

1 year/365 days. Unit analysis lets you know that if you are starting with seconds, and you want to end up with years, you will have to flip these fractions (when you multiply them) so that seconds will cancel out and years will stay at the end. If you multiply 2340 sec./1 x 1 min./60 sec., then you will have minutes in the numerator, and that will have to cancel out with the next ratio. Here's what has to be multiplied:

2340 sec./1 x 1 min./60 sec. x 1 hr./60 min. x 1 day/24 hrs. x 1 year/365 days

Notice how each ratio is chosen so that the previous units will cancel (anything the same above the fraction line as below it will cancel out) and you'll end up with new units.