Strategies for multiplying multi digit decimals

So in this video, we're gonna try to think of ways to compute what 31.2 times 19 is. There are 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. One way to think about this is you could view 31.2 as a certain number of tenths. How many tenths would this be? Well, you could view this 31.2 as three hundred and twelve tenths. So this is the same thing as three hundred and twelve tenths times nineteen.

What we could do is we could figure out what is three hundred twelve of something times 19, or what's three hundred twelve times 19. It's going to 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.

If we were to just multiply 312 times 19, let's see. Two obvious in another color: two times nine is 18. One times nine is 9 plus 1 is 10. Three times nine is 27 plus 1 is 28. If what I just did looks unfamiliar, we have videos that explain why and how this process works.

Then we go to the tens place right over here, and so we would say 1 times 2 is 2. 1 times 1 is 1. 1 times 3 is 3. Then we add everything together: we get 8. 0 plus 2 is 2. 8 plus 1 is 9. 2 plus 3 is 5. So we get 5,928. Now that's not going to be the answer here. The answer is going to be 5 thousand nine hundred and twenty-eight tenths.

This is going to be equal to five thousand nine hundred and twenty-eight 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 as ten tenths place, this is the tens place, and this is the hundreds place.

Well, you have 8/10; we could put that in the tenths place. You have these twenty tenths; that's the same thing as two ones. You have the nine hundred tenths in a different color; nine hundred tenths is the same thing as nine tens, and then your five thousand 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 five hundred ninety-two and eight tenths. So we could write it like that: five hundred ninety-two 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. Then try to estimate to think about where the decimal place should go. For example, you could do 312 times nineteen. Essentially, remove the decimal, do the computation, and say okay, the answer should have the digits five, nine, two, and eight in that order.

Now where should I put the decimal in order for that to be a reasonable answer? That's where estimation comes in. You could say, "Hey, thirty-one point two times nineteen, that's going to be approximately equal to..." If I try 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. 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 want to be close to 600, I'd have to put the decimal right over there.

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.