Multiplying and dividing decimals by 10

We've already learned that when we multiply by ten, let's say we took the number 53 and we were to multiply it by ten, it has the effect of shifting all the digits one place to the left. So this should be a review for you, but this was going to be 530. We could see that what used to be in the tens place has been shifted to the left to the hundreds place, and what used to be in the ones place has been shifted to the left to the tens place now.

And we saw if you divide by ten, you have the opposite effect. So let's say if I had 120, I could say that let's divide by ten. I could also say this is the same thing as 120 times 1/10. What is going to happen there? Well, in this situation, all the digits are going to shift one place to the right. So what used to be in the tens place will now be in the ones place, and what used to be in the hundreds place will now be in the tens place. So this is just going to be equal to 12.

So that was all review, but now we're going to extend this a little bit by thinking about things that have place values representing less than 1. I guess you could say, or we're gonna deal with decimals. So just to get ourselves warmed up, let's see if we could figure out what 3.015 times 10 is. Pause this video and see if you can figure that out.

Well, the exact same thing is going to happen. All of our digits are going to shift one place to the left. So right now we have a 3 in the ones place, we have a 0 in the tenths place, we have a 1 in the hundredths place, and we have a 5 in the thousands place. But now they're all going to shift one place to the left. So the 3 is now going to go into the tens place. It's going to shift one place to the left.

So we're going to have in the tens place; we're now going to have our three. And now this zero, which was in the tenths place, is now going to shift into the ones place. So zero is going to go right over. The zero is going to go right over there; that is now in the ones place. And then we'll put our decimal.

Now, what's going to go into the tenths place and the hundredths place? And actually, I'll rewrite the thousands place as well. This one is going to shift one place to the left into the tenths place. So into the tenths place. And then the five that was in the thousands place is now going to shift one place to the left into the hundredths place. So into the hundreds place, just like that.

Now we could put a trailing zero over here, but that's not going to change the value of this number, so I'll just leave it like that. And so there you have it; we see that every digit has shifted one place to the left. So this is equal to 30.15. I'll just put that zero there for kicks.

And so we could think about the other way around. What if I were to take 67.5 and if I were to divide it by ten or five? Another way of thinking about it is if I were to multiply this by one-tenth. Pause the video and see if you can figure out what that's going to be.

Well, now every digit is going to shift one place to the right. So the six is going to be in the ones place, the seven is going to go into the tenths place, and then the five is going to go into the hundredths place. So let's write that out. So our six is going to go into the ones place, and then we're going to have our decimal point. Our seven is going to go into the tenths place, and then our five is going to go into the hundredths place.

So there you have it; we get 6.75.