Relating decimal multiplication to fractions and place value

0:00- [Instructor] What we're gonna do in this video is review what we know already about how we can go between fractions and decimals. And then we'll think about all the different ways that you can represent multiplying fractions in decimals. So for example, if I were to write the decimal 0.56, how could I write that as a fraction?

0:21Well, this is the same thing as 56 hundredths, so we could write this as 56/100. Similarly, if I had the fraction 5/100, this is 5 hundredths, so I could write it as a decimal as 0., I'm gonna have 0 tenths and 5 hundredths. And if we want to go back to the basics of how we first learned fractions.

0:49If we say 16, just the number 16, that's 16 wholes, or 16 ones, I guess you could say. So this is 16/1. You might already know this, but I'm just reviewing it because now we're going to start multiplying these types of numbers in different representations and think about all the different ways that we can represent that multiplication.

1:11So for example, if we were to write, actually let's write 0.05 times and let's multiply it times 14/100. What are all of the ways that we can represent this? Well one way we can represent this, we know that this first 0.05, we already said that could be written as 5 hundredths.

1:40So this is the same thing as 5/100 times 14/100, times 14/100. And, well that's going to be the same thing as, if we actually just wanted to multiply it out, that is going to be equal to our numerator. Let me do that in a neutral color. I'll do it in a, well, I don't know what color I haven't used yet.

2:03Let me see. I'll use this light yellow. So this would be 5 times 14 in the numerator and 100 times 100 in the denominator. We could calculate this if we like, but here I just want to think about the different ways of representing this. Now what's another way that we could represent this?

2:22Instead of making this 0.05 represented as a fraction, we could represent these 14 hundredths as a decimal. So we could rewrite it as. I lost track of my colors here. I had a little color. We could rewrite this as the original 0.05, 0.05, times 14 hundredths is the same thing as 0.14.

2:51Another way of representing the same multiplication, the same expression. Now of course, we could convert the first one into a fraction and the second one into a decimal. So then that one would be, and I'm just going through all of the different scenarios here, 5/100 times, times 0.14.

3:13And there's even other ways of doing it. For example, we could take what we had up here. We could say, "Hey, this is the same thing as." Well, I could write it as 5/100, 5/100, maybe times 14, times 14, times 1/100, because 14 hundredths, 14 hundredths. I could also write it as, I could also write it this way.

3:40I could say that this is equal to, let me say, 1/100 times 5 times 14 times 1/100. These are all equivalent expressions. Let's do a different example now, and I'll do it in a different color. Let's say we were to write 12 times. Let's do 12 times 6/10, 6/10. Well what are all the ways that we could represent that?

4:14Well we could represent the 6/10. I could represent the 6/10 as a decimal, so that'd be 12 times 6/10 is 0.6. I could write both of them as fractions. This is the same thing as 12/1 times 6/10. That's probably the direction I would actually go in if I was actually trying to solve it.

4:37And then of course, this would be the same thing as 12 times 6 in the numerator. Let me write this. 12 times 6 in the numerator. And then in the denominator, I'd have 1 times 10, or I could just write 10 there. I could write, of course, 72/10 if I actually wanted to figure out what this actually is.

4:59And then similarly to what we did over here, if I wanted to, I could also write this as, let me write it this way, 12 times 6 times 1/10. So this would be how many tenths do we have? Well it's going to be 12 times 6. In this case, it's going to be 72 tenths. The whole point of this video is to just see all of the different ways that you can represent these multiplication expressions involving decimals and fractions.

5:29And that's useful because sometimes it's easier to do things where both are decimals, both are fractions, or every now and then some mix of the two.