Formulas and units: Comparing rates | Working with units | Algebra I | Khan Academy

We're told that Hannah and Martine each got a plant for their home. Hannah measured that her plant grows on average two centimeters per week. Martine measured that her plant grows on average three millimeters per day.

Which plant grows faster? Pause this video and see if you can figure that out on your own.

All right, now let's go through this together. So first, when you look at it, you might just compare three to two and say, "Oh, three is larger than two, therefore maybe Martine's plant grows faster." You would think that until you look at the units. This is millimeters per day, while for Hannah's plant it's in centimeters per week.

So in order to really compare them, we have to convert them to the same units in both length and time. So let's convert both of them. Let's convert them both to centimeters per week. You could just try to convert both of them to millimeters per day, or I guess you could try to convert both of them to meters per year with a third set of units. But centimeters per week seems reasonable since we already have Hannah's plant rate at.

So let me write this down. So Hannah, I'll just say she grows at two centimeters per week. And then you have Martine. Martine grows at an average of three millimeters per day.

Now, how do we convert three millimeters per day to centimeters per week? Well, first we could convert the millimeters. Actually, first, let's convert the days into weeks. So how many days are there in a week? Well, there's seven days in a week. So if we have how many millimeters per day, if we wanted to know millimeters per week, we would multiply times seven days.

So let me do that. So times seven days, days in a week, that would get us this. This would be equal to 3 times 7, which is equal to 21 millimeters in a week. And you can see actually that the units cancel out nicely like that, so you're left with millimeters a week. That makes sense; three millimeters a day, you're able to do seven times that in a week, 21 millimeters a week.

And then when you think about 21 millimeters, how many centimeters is that? Well, we just have to remember that one centimeter is equal to 10 millimeters. So if we want to convert this into centimeters, we essentially have to divide by 10. We could just say one-tenth of a centimeter per millimeter.

And then that gets us, we could write in different ways, but we could write this, and even here the units cancel out nicely: 21 divided by 10 is 2.1 centimeters per week.

Centimeters per week. Another way you could have just thought about it is, we could say one centimeter is equal to 10 millimeters, or if you divide both sides by 10, one-tenth of a centimeter is equal to one millimeter.

And if one millimeter is equal to one-tenth of a centimeter, then 21 millimeters it's just going to be 21 times this. 21 times 1/10 is the same thing as 21 divided by 10. It would be 2.1 centimeters.

And so now we can compare 2.1 centimeters per week compared to 2 centimeters a week. Well, it turns out that when you actually compare the appropriate units, it still turns out that Martine’s plant is growing just a little bit faster.