Comparing numbers represented different ways

- So we're asked to compare these two numbers right over here and I just took a picture from the Khan Academy exercise but this isn't it, itself, so I can't, I can't click on this, but if you did click on this on the exercise, you'd get to choose between one of these three, but I'll just write it down. So we're gonna compare 2.7 to two and 17 hundredths, so there's a bunch of ways you could do this, you could convert both of them to mixed numbers. So, for example, 2.7, 2.7, you could rewrite, this is the same thing as two and seven tenths, two and seven tenths, you could compare two and seven tenths to, to this right over here, which is two and 17 hundredths, two and 17 hundredths. So, they both have a two, so you want to compare the fraction part, so what's larger? Seven tenths or 17 hundredths? Well, if it doesn't jump out at you, you could convert this to hundredths. Seven tenths is the same thing if you want to convert it to hundredths, you can multiply the denominator by 10, which would make the 10 into 100, and then you'd also have to multiply the numerator by 10. So two and 7 tenths is the same thing as two and 70 hundredths. And so 70 hundredths is for sure bigger than 17 hundredths. So, this number, two and seven tenths, or two and 70 hundredths, or 2.7, is going to be greater than this number, so it's going to be greater than, we open the symbol to the larger number. Another way that we could have thought about this, is we could have converted both of them to decimals. So we have 2.7, which we can leave as a decimal. 2.7 but then two and 17 hundredths, we could write as 2.17. This is 17 hundredths over here. And there's a bunch of ways that you could make this comparison, you could just say, oh look, two and seven tenths is the same thing as two and 70 hundredths, over here I have 70 hundredths, over here I only have 17 hundredths, so this is going to be larger. This one's going to be larger right over here. Another way you could have thought about it is we'll just start in the ones place, they both have two ones, so that doesn't tell you much, but as soon as you go to the tenths place, this number as seven tenths and this one only has one tenth, so this number right over here is going to be greater than that number. Let's go to, let's do another example. Which of the following are less than two point zero three? Or two and three hundredths? Each big square represents one whole. So one way we could do it, and this is kind of the way that jumps out at me, is let's, let's write all of these as decimals. So this one over here, we have one whole, another whole, and then over here I don't have a full whole, let's see, I've taken a whole, I've divided it into tenths, and I have two tenths right over here, so this number right over here is equal to two and two tenths, you see that, two, one, two, and then two of the ten equal sections, so two and two tenths, which is the same thing as 2.2. And if we wanted to compare it to 2.03, well we just have to think about this, well there's a, there's a bunch of ways you could think about this, you could view this as two point two zero, which would mean this is two and 20 hundredths, and two and 20 hundredths is for sure greater than two and three hundredths. So, this is not less than 2.03, this one over here is greater than 2.03, so I'm gonna rule that one out. Another way you could have thought about this is this is two and two tenths which is the same thing as two and 20 hundredths, which this is another way of representing that, and you say, well this one over, the number you're comparing against is two and three hundredths. So 20 hundredths is greater than three hundredths. So once again this is not less than 2.03. Here we have two and 12 hundredths, and actually this is useful because I already have our original number represented as two and three hundredths, so the whole number part, the two, is the same, so when you look at the fraction part, 12 hundredths is greater than three hundredths, so this one is also not less than 2.03. Now what about 23 hundredths? 23 hundredths, we can write like this, 23 hundredths, or we could write it like this, 23 hundredths, and you might be tempted to say, oh well hey, 23 hundredths, that's bigger than three hundredths, but remember, we have two wholes over here, we have two wholes, we have no wholes over here, we have, over here, one would think, but you have nothing in the ones place, over here you have something in the ones place. So if you look at the ones place, you have two ones, you have zero ones over here, it doesn't matter how many hundredths you have, if you have less than a 100 hundredths, and so, this one is for sure less than 2.03. This one is, this is significantly less than even one. So this one is the only, of the three, that is less than 2.03. Let's do one more of these. Order the following values from least to greatest. And once again I took a picture from the Khan Academy exercise, so I'm not gonna be able to drag 'em around, if you're doing the exercise, you would be able to drag them around, but I'm just gonna, I'm just gonna think about the numbers and then write them in order. So, I want to write them all, let's write them all in, as decimals, let's write them all as decimals, I guess is the easiest thing to do. So here, this one's already written as a decimal, 2.59. 24 tenths, 24 tenths, so let's think about this. That's 24 over 10. Well, 24 over 10 is the same thing, this is equal to 20 over 10, plus four over 10, 20 tenths, this is two wholes right over here, so this is the same thing as two and four tenths, two and four tenths, or, the same thing as 2.4, two and four tenths, now if we wanna make a comparison to two and 59 hundredths, we could throw another zero right over here, and say OK, we can view this as two and 40 hundredths. So these are all different ways of representing this, some of you all might have been able to go immediately, hey look, 10 tenths is one, 20 tenths is two, so 24 tenths is going to be two and four tenths, or 2.4, or 2.40, now let's try this number, two and three tenths, represented as 2.3, if we want to think of it in terms of if we want to think of it in terms of hundredths, you could say, well this is the same thing as two and 30 hundredths, which you could also represent as 2.30, and the reason why I'm adding the hundredth there is so we can more easily make the comparison. So you're comparing, you're comparing two and 59 hundredths, to two and 40 hundredths, to two and 30 hundredths. Well, they all have two in the ones place, they all have two in the ones place. So then we just have to go to the tenths place. And you see, this has the most tenths, five tenths, this is the second most, four tenths, and this has the least tenths, three tenths, so, ordering from least to greatest: the least is two and three tenths, the least and three tenths, which is equal to 2.3. The next, the one larger, next larger than that, is going to be 24 tenths, 24 tenths, which is equal to 2.4, and then the largest is 2.59. And once again, this only, they all have two in the ones place and then you go to the tenths, this has three tenths, four tenths, five tenths.