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Multiplying decimals and whole numbers with visuals

Multiplying whole numbers and decimals can be fun. It starts with simple examples, such as multiplying 3/10 by 4, and then moves on to more complex examples, such as multiplying 52/100 by 3. In each case, we demonstrate how to use a number line or model to visualize the process and find the product. Created by Sal Khan.

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Video transcript

- [Instructor] So what we have here on this number line that we've now marked off with the tenths and you can see that this is three tenths here. We can think about this as a multiplication of a decimal. And so what is this representing? And I'll give you a hint. It's representing something times three tenths. So pause the video and try to think about that. Well, let's see. We are going one times three tenths, two times three tenths, three times three tenths, and then four times three tenths. So what's represented here is four times three tenths, and so what is this going to be equal to? Well you can see you go from three tenths, to six tenths, to nine tenths, and then you could view this as twelve tenths, but twelve tenths is the same thing as one, one and two tenths. So you could view this as 1.2. One and two tenths. Let's do another example. No, actually I'll do it on the same number line. If we wanted to represent three times 0.2 What would that look like on this number line? And what would this be equal to? So I'll put a little equal sign here. Pause this video and see if you an figure that out. All right, so let's think about where two tenths is this is one tenth, two tenths is right over there. This is 0.2, and we're gonna multiply it times three. So, we're gonna multiply it times one, then we're gonna multiply it times two, that takes us to four tenths and then we're gonna multiply it times three to get us to six tenths, 0.6. So it's six tenths just like that. Now you could also visualize two tenths as parts of a whole. So for example, this represents two tenths. I have this whole, this square is a whole it's split into ten equal columns here and we have two of them filled in. So this represents two tenths. So if you have three times two tenths, Well this is one times two tenths, this is two times two tenths, and this is three times two tenths. And so how many tenths do we now have? Well we have one, two, three, four, five, six tenths. Which is exactly what we have here, six tenths. Let's do one more example, that gets a little bit more involved. So here we're told to multiply. It says you many use the models shown to help find the product. And this is a screen shot from the exercise on Khan Academy. So pause this video and see if you can figure out what this is. All right, so they're saying 52 hundredths times three, and they have 52 hundredths depicted right over here and then they have it depicted three times. So the total number of hundredths depicted here that is 52 hundredths times three, because we have 52 hundredths here, another 52 hundredths, and then another 52 hundredths. So how many hundredths is that going to be? Well, you could view this as 52 times three and that will give you the number of hundredths we have. So let's think about this. So if we were to just say 52 times three, well this is going to be two times three is equal to six and then five tens times three is 15 tens, which is the same thing. We either just write it as 15 tens, or that's 100 and five tens. But either way if I have 52 of something and I multiply that by three, I now have 156 of that something. And here the something is hundredths. So if I say 52 hundredths times three that's going to be 156 hundredths. And how do we represent 156 hundredths. Well there is a couple of ways to think about it if this is the ones place, this is the tenths place, this is the hundredths place. Well we would write the six there, the five there, and the one there. So you could recognize this as hey look, a hundred hundredths, let me color code it, a hundred hundredths is the same thing as a whole and I'll circle that in red, and fifty hundredths is the same thing as five tenths, and of course six hundredths is the same thing as six hundredths. So this is going to be equal to 1.56, or you could view this as 156 hundredths, or you could view this as a whole, which is a hundred hundredths, and five tenths, which is fifty hundredths, and six hundredths.