Formulas and units: Comparing rates

- [Instructor] We're told that Hannah and Martine each got a plant for their home. Hannah measured that her plant grows, on average, 2 centimeters per week. Martine measured that her plant grows, on average, 3 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 at first when you look at it, you might just compare 3 to 2 and say, oh, 3 is larger than 2, therefore maybe Martine's plant grows faster. And 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, 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 write H, grows at 2 centimeters per week. And then you have Martine, grows at an average of 3 millimeters per day. Now how do we convert 3 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 7 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 7 days. So let me do that. So times 7 days in a week. That would get us, 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. And that makes sense, 3 millimeters a day, you're able to do 7 times that in a week, 21 millimeters a week. And then when you think about 21 millimeters is how many centimeters? Well, we just have to remember that 1 centimeter is equal to 10 millimeters, so if we wanna covert this into centimeters, we essentially have to divide by 10. We could just say 1/10 of a centimeter per millimeter and then that gets us, we could write it 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. Another way you could have just thought about it is we could say 1 centimeter is equal to 10 millimeters, or if you divide both sides by 10, 1/10 of a centimeter is equal to 1 millimeter, and if 1 millimeter is equal to 1/10 of a centimeter then 21 millimeters is 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.