Strategies for multiplying decimals

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.

Alright, 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 ten is the same thing as zero point seven, 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 holes, I guess you could say, times seven over ten.

Multiplying our numerator, we're going to get 56, and if we multiply our denominators, we get ten. 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 6 over 10, which is the same thing as 50 over 10 plus 6 over 10. So this is the same thing as this is 5 holes, so 5 and 6 tenths.

5 and 6 tenths, which we can write as 5 and 6 tenths or 5.6. 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 1, so we would expect this product—if we're multiplying eight times something a little bit less than 1—we would expect the product to be a little bit less than eight.

So 5.6 makes sense. If for some reason we computed something and you were to get 60, you'd say, "Wait, that doesn't make sense. I should get a value less than eight." Similarly, if you somehow got a product of like 1, 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, and once again, I'm just gonna write this in a different way: eight times seven tenths. 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.

One way to think about 56 tenths is the same thing as 50 tenths. Let me color code that differently. So this is going to be the same thing as 50 tenths plus 6 tenths. Can't write 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 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.

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 do have to figure out what three times 87 is. So 87 times three: seven times three is twenty-one. We regroup that two, it becomes two tens, and then eight times three is 24, and that's really 24 tens plus those other two tens.

So we get 26 tens, which is the same thing as two hundred. 206 tens. There's going to be 261. So the three times 87 of something is going to be 261 of that something, and in this case, something is hundreds. So this is two hundred and sixty-one hundredths.

So how do we express this as a decimal? Well, there are a couple of ways that you can approach it. You can think about it: this is the ones place, this is the tenths place, this is the hundredths place. And so very clearly, one hundredths here would be one in the hundreds 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 two hundred 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 two hundred sixty-one, one in the 60 hundredths, and then the two hundred hundredths. And you get to point six one.

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 87 hundredths." See, these are all equivalent. But hopefully, one of these—or more than one of these—registers with you of what's really going on.

Well, this is going to be the same thing as three wholes times 87 hundredths. So this is going to be equal to—in the numerator, we have three times 87, three times 87—and in the denominator, one times 100 is a hundred.

Three times 87 hundredths, well, we already know what three times 87 is. This is equal to 261 hundredths, and you can see a hundred goes into 261 two times, and you're left with 61 hundredths. So these are all equivalent representations.

And just a reminder: so it's always good to estimate. And so what you have here 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—this seems about right.

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.