Comparing decimals in different representations

So what we're going to do in this video is build our muscles at comparing numbers that are represented in different ways. So, for example, right over here on the left we have 0.37; you could also view this as 37 hundredths. And on the right we have 307 thousandths.

So what I want you to do is pause this video and figure out: are these equal to each other, or is one of them larger than the other? And if one of them is, which one is larger and which one is smaller? Pause this video and try to figure that out.

Alright, now let's try to do this together. The way that my brain works is I try to put them into a common representation. So one way we could do it is we could try to rewrite this one on the right as a decimal. So let's do that. We could rewrite this as it's expressing a certain number of thousandths.

Let me just make some blanks for our various places. So let's say that's the ones place, and that's our decimal; that's going to be our tenths place, that's going to be our hundredths place, and that's going to be our thousandths place.

So one way to view three hundred and seven thousandths is that we have three hundred and seven of this place right over there. So we could just write the seven there, the zero there, and the three over there. This over here would be three hundred and seven thousandths, and so we would have no ones.

When you look at it this way, it's a little bit easier to compare. You can say, "Alright, we have the same number of ones; we have the same number of tenths." Let me compare the like ones to like ones. Our tenths are equal, but what happens when we get to the hundredths? Here we have seven hundredths, and here we have zero hundredths.

So this number on the left is going to be larger. So 37 hundredths is greater than three hundred and seven thousandths. Another way that we could have done this is we could have re-expressed this left number in terms of thousandths. We could have rewritten it as instead of 37 hundredths, we could have just said 0.37 and just put another zero on the right. This is 370 thousandths.

I'll write it out: 370 thousandths. When you look at it this way, once again, it's clear that 370 of something is more than 307 of that something. So this quantity on the left is larger.

Let's do another example, but I'll use different formats. So let's say on the left I'll use decimal format. I'll have 0.6 or six tenths, and then on the right I'm going to have six times one over a hundred. Pause this video and tell me which of these quantities, if either, are greater or are they equal to each other?

Alright, so once again, in order to tackle this, you really just have to think about what are different ways to represent them and really just try to get to a common representation. So I could rewrite six tenths as six times 1/10. Six times 1/10. This might be enough to be able to compare the two because six times 1/10— is that going to be greater than, less than, or equal to six times one hundredths?

Well, a tenth is ten times larger than a hundredth. So because this is ten times larger than that, if you multiply it by six, well, this is going to be a larger quantity. So we could go and say, "Hey, this is greater than that."

Another way that you might have realized that is if you were to express this right quantity as a decimal like this. So this is six times a hundred or six hundredths. So we could write: that's our ones, that's our tenths, and then in our hundredths place you would have six.

If it isn't obvious that this is less than that, you could add a zero here, and this we would read as 60 hundredths. And 60 hundredths is for sure larger than six hundredths.

So these are all very reasonable ways of re-representing these numbers and putting them in the same format so we can make the comparison and realize the one on the left actually, in both scenarios, is larger than the one on the right.