Strategies for multiplying decimals and whole numbers

- [Instructor] In this video, we're going to further build our intuition for multiplying decimals. So let's say that we wanted to figure out what eight times seven tenths is. Pause this video and see if you can figure this out on your own. All right, now there's several ways that we could approach what eight times seven tenths is. We could view this as eight times, and we could write seven tenths as a fraction. So we can re-express this as seven tenths, seven over 10 is the same thing as 0.7. And we already know how to multiply fractions, you could view this as being equal to, eight is the same thing as eight over one or eight wholes, I guess you could say, times seven over 10, times seven tenths, which is going to be equal to, if we multiply our numerator, we're going to get 56. And if we multiply our denominators, we get tenths. And that makes sense. If I have eight times seven tenths, I end up with 56 tenths. Now 56 tenths can also be written as, this is the same thing as 50, plus six over 10, which is the same thing as 50 over 10, plus six over 10. And so this is the same thing as, this is five wholes, so five and six tenths, five and six tenths, which we can write as five and six tenths, or 5.6. And it's always good to do a little bit of a reality check, whenever you get an answer when you're multiplying decimals. Say, okay, seven tenths is a little bit less than one. So we would expect this product, if we're multiplying eight times something a little bit less than one, we would expect the product to be a little bit less than eight. So 5.6 makes sense. If for some reason we got, the we you computed something and you were to get 60, you say, well, that doesn't make sense, I should get a value less than eight. And similarly, if you somehow got a value or product of like one, you're like, well, that's a lot less than eight, I should get something that is seven tenths of eight. Now, another way that you could approach this is you could view this as the same thing as eight times, and once again, I'm just gonna write this in a different way, eight times seven, eight times seven tenths. So, if you have eight times seven of something, what is that going to be equal to? Well, eight times seven, that's 56. So you're going to be, this is going to be equal to 56 tenths, 56 tenths. And one way to think about 56 tenths, 56 tenths is the same thing as 50 tenths, 50, let me color code that differently. So this is going to be the same thing as 50 tenths, 50 tenths, plus six tenths, plus the six tenths, get right tenths, six tenths, and 50 tenths is the same thing as five ones. So five ones, and six tenths, which is exactly what we have here, five ones, and six tenths. Let's do another example, that's a little bit more involved. So let's say that we want to figure out, we want to figure out what is three times 0.87. Pause this video and try to figure that out. Well, once again, there's many ways to approach it. But we could just start with the way that we just looked at. We could say, hey, this is the same thing as three times, and we can re-express this as, this is the same thing as 87 hundredths. 87 hundredths, and so if I have three times 87 of something, what am I going to be left with? Well, this is going to be equal to some number of hundredths, and to figure out that, we just figure out what's three times 87? So 87 times three, seven times three is 21, we regroup that two, becomes two 10s. And then eight times three is 24. And that's really 24 10s plus those other two 10s, so we get 26 10s, which is the same thing as 206 10s, but it's gonna be 261. So the three times 87 of something is going to be 261 of that something, and this case something is hundredths. So this is 261 hundredths. So how do we express this as a decimal? Well, there's a couple of ways that you can approach it. You can think about is this is the ones place, this is the tenths place, this is the hundredths place. And so very clearly, 100th here would be one in the hundredths place. If you have 60 hundredths, which is what the six represents, 60 hundredths is the same thing as six tenths. And then last but not least, if you have 200 hundredths, that's the same thing as two wholes. Another way to think about it is, you go to the hundredths place, and then you start from there, but you write out 261, one, the 60 hundredths, and then the 200 hundredths, and you get 2.61. Now another way that you could have approached this, and we saw this in the last example, is you could say hey, this is going to be the same thing as three times at 87 hundredths. (mumbles) These are all equivalent, but hopefully one of these, or more than one of these register with you of what's really going on. Well, this is going to be the same thing as three wholes, times 87 hundredths, 87 hundredths. And so this is going to be equal to, in the numerator, we have three times 87. Three times 87 hundredths, one times 100 is 100. Three times 87 hundredths, well, we already know what three times 87 is, this is equal to 261 hundredths. And you can see 100 goes into 261, two times and you're left with 61 hundredths. So these are all equivalent representations. And just reminder, so it's always good to estimate. And so what you have is you have three times something that's a little bit less than one. So you would expect a value, a little bit less than three. And so 2.61 also meets that sniff test, that this seems about right. If for some reason you got 26 or 261, that would be way off or even if you got 0.261, that would also feel way off. So hopefully this is helpful.