Multiplying and dividing decimals by 10, 100, 1000

In this video, we're gonna get a little bit of practice multiplying and dividing decimals by ten, hundred, and a thousand. So let's just start with a little bit of a warm-up. If I were to say, "What is two point zero five times ten?" Pause this video and see if you can figure that out.

Well, in previous videos, we've already said that when you multiply by ten, you shift each of the digits one place to the left. So this is going to be equal to, instead of two ones, we're now going to have two tens. So, two in the tens place, and then instead of zero tenths, we're now going to have zero ones. And instead of five hundredths, we're now going to have five tenths. So this is equal to twenty point five.

Now, what if we were to go the other way around? What if we were to say, "Two point zero five divided by 10"? Pause the video and see if you can figure that out.

Well, here, all of our digits are going to shift one place to the right because we're dividing by ten. We could also view that as multiplying by one tenth. And so our two ones are going to become two tenths, so this is going to be zero point. So we're gonna have two tenths. Now, our zero tenths are going to be zero hundredths, and then our five hundredths are going to be five thousandths. We've covered that in other videos, but now let's do this with, say, a hundred or a thousand.

So if I were to ask you, "What is 57 divided by 1,000?" Pause this video and see if you can work that out.

All right, now let's do this together. So when you divide by a thousand, that's the same thing as dividing by 10 three times, or it's the same thing as multiplying by one tenth three times. Or you could just say, "Hey, that means I'm going to shift each of these digits three places to the right."

Let me create some places here. So there's tens, ones, tenths, hundreds, thousands, and so five was in the tens place. It's five tens, so it was here, but we're going to shift three places to the right: one, two, three. So our five will go there. What was five tens is now five hundredths, and then our seven is similarly going to— it was in the ones place, but we're going to shift three places to the right: one, two, three.

And there you have it. What was just 57 is now fifty-seven thousandths. To make that very clear, I'll put a zero in the tenths place and a zero in the ones place. And that makes sense.

So it's another example. Let's say someone walks up to you on the street and says, "I started with one point zero three two and I multiplied it by something and I was able to get one hundred and three point— what did they multiply by to get one hundred and three point two?" Pause this video and try to work it out.

Well, to understand this, we just have to think: well, how much did each digit get shifted by? What was in the ones place got shifted, not just to the tens place, but it got shifted to the hundreds place. So it got shifted two places to the left. The zero similarly got shifted two places to the left, from the tenths to the tens.

If you look at it, every digit got shifted two places to the left. So, we must have multiplied by 10 twice. So you could say times 10 times 10, or you could just rewrite this as times 100.

Let's just do one more example for kicks. Let's say that someone were to say, "What is 0.015 times 100?" Pause this video and see if you can figure that out.

All right, well, like we've done before, we're just going to shift every digit two places to the left. So the one, which is in the hundredths place, is going to end up in the ones place, and then the five, which is in the thousandths place, is going to end up in the tenths place. So this is going to be equal to one point five, and we're done.