Strategies for multiplying multi-digit decimals by whole numbers

- [Instructor] So in this video, we're gonna try to think of ways to compute what 31.2 times 19 is. And there's multiple ways to approach this, but like always, try to pause this video and see if you can work through this on your own. All right, now let's do this together. Now one way to think about this is you could view 31.2 as a certain number of tenths. And how many tenths would this be? Well, you could view this, 31.2, as 312 tenths. And so, this is the same thing as 312 tenths times 19. And so, what we could do is we could figure out what is 312 of something times 19, or what's 312 times 19. It's gonna be that many tenths, and then we could convert it back to a decimal. I'll also show another strategy, but let's just do that. So if we were to just multiply 312 times 19, times 19, this is going to be, let's see, two, obviously, another color. Two times nine is 18. One times nine is nine. Plus one is 10. Three times nine is 27, plus one is 28. If what I just did looks unfamiliar, we have videos that explain how this process works. And then, we go to the tens place, right over here. And so, we'd say one times two is two. One times one is one. One times is three is three. And then we add everything together. We get eight. Zero plus two is two. Eight plus one is nine. Two plus three is five. So we get five, nine, two, eight, or 5928. Now that's not going to be the answer here. The answer is going to be 5928 tenths. So this is going to be equal to 5928 tenths. Now, how can we express this as a decimal? Well, we could think of it this way. If that's the decimal, this is the tenths place, this is the ones place, which is the same, this is the same thing as 10 tenths place, which is the ones place. This is the tens place, and this is the hundreds place. Well, you have eight tenths. We could put that in the tenths place. You have these 20 tenths. That's the same thing as two ones. You have the 900 tenths. Do that in a different color. 900 tenths is the same thing as nine tens. And then, your 5000 tenths is the same thing as five hundreds. Another way to think about it is we wrote all the places out, and we wrote it in terms of tenths. So the eight went there, and then every place to the left of that went to the place to the left of that. So this is going to be 592 and eight tenths. So we could write it like that. 592 and eight tenths. Now another way to approach this is to just think about the digits, not the actual numbers, to figure out, well, the answer will have what digits in it. And then try to estimate to think about where the decimal place should go. So, for example, you could do 312 times 19. So since you remove the decimal, do the computation, and say, okay, the answer should have the digits five, nine, two, eight in that order. Now where should I put the decimal in order for that to be a reasonable answer? And that's where estimation comes in. You could say, hey, 31.2 times 19, that's going to be approximately equal to, in fact, if I were to estimate these with numbers that are easy to multiply, that's going to be roughly equal to 30 times 20, which is equal to 600. So that tells me that my product here should be roughly equal to 600. And so where would I put the decimal here for it to be roughly equal to 600? So I know the answer has the digits five, nine, two, eight. Where do I put a decimal for it to be roughly equal to 600? Well, if I were to put the decimal there, that's not roughly equal to 600. If I were to put the decimal there, that's not roughly equal to 600. That's close to 60. If I put the decimal there, that's close to six. If I wanna be close to 600, I'd have to put the decimal right over there. And so that's also a good way to test the reasonableness of what's going on. This should be roughly equal to 600, if we were to estimate it. And so, we like that our process got an answer that is roughly equal to 600.