Developing strategies for multiplying decimals

So right over here, we want to compute what three times 0.25, or 3 times 25 hundredths, is. I encourage you to pause the video and see if you can figure this out.

All right, now let's work through this together. In this video, we're going to explore multiple strategies. In the future, we're going to show you what's called the standard strategy, which you might use a lot. But the strategies we're going to look at in this video are actually very helpful for understanding what multiplying decimals actually means and how it relates to multiplying fractions. This is often the way that people— even people who have a lot of math behind them— multiply decimals.

So here, 3 times 25 hundredths. There are a couple of ways to think about it. One way is to say, "Hey, this is the same thing as three times,” and I'm just going to write it a different way: 25 hundredths. So if I have 3 times 25 of something, what is it going to be?

Well, what's 3 times 25? Let's see: 2 times 25 is 50, 3 times 25 is 75. So it's going to be 75. I'm multiplying not just 3 times 25; I'm multiplying 3 times 25 hundredths. So instead of 2500, I'm going to have 75, 75 hundredths. Written out in words, this would be 75 hundredths.

Now, how would we write that as a decimal? Well, that is the same thing as this: 75 hundreds.

Now, another way to conceptualize this, to think about what this is, is if we were to write 3 times. We could write it as a fraction; we could write 25 over 100. This is another way of writing 25 hundredths. These are all equivalent.

So what is 3 times 25 over 100? Well, same idea. You could say this is 25 over 100 plus 25 over 100 plus 25 over 100. This is going to be 75, 75 hundredths, which once again is 0.75. If you wanted to view it more formally as fraction multiplication, you could view it as 3 over 1 times 25 over 100.

You multiply the numerators; you get 75. You multiply the denominators; you get 100. Either way, in all of these situations, you're going to get 75 hundreds. Or another way to think about it is, "Hey look, this thing right over here, this 25 over 100, this is the same thing as 1/4." So you could view this as 3 times 1/4. In fact, this is a decimal that it's good to recognize: this is the same thing as 1/4.

So you could view this as 3 times 1/4 or 3/4. This is 1/4 right here. 1/4 could be viewed as 1/4. So this is going to be equal to 3/4, 3 over 4, 3/4. All of these are equivalent.

Now, if someone wanted it written out as a decimal, you could say, if you might know that 3/4 can be expressed as 75 hundredths, what's in general is a good thing to know.

Now, let's tackle slightly more complicated examples. So let's say we wanted to figure out what 0.4 times— just a new color— times 0.3 is going to be equal to. Pause the video and see if you can compute this, and I'll give you a hint: see if you can express these as fractions.

All right, so what we have here in red, we could read this as 4/10, 4/10. We could write it as a fraction as 4 over 10, and we're going to multiply that by what we have over here. This is 3/10, 3/10, which we could write as a fraction as 3/10.

So you could view this as 4/10 of 3 tenths, or 3 times 4 tenths. But we're multiplying these fractions, which we've seen before in other videos. So what's going to happen? Well, if we multiply the numerator, we get 12. We multiply the denominators; you get 100. So you get 12 hundredths.

If you wanted to write that as a decimal, it would be 0.12, 12 hundredths. Now, you might notice something interesting here, and you'll see this more and more as you learn the standard method. Well, 12 is 4 times 3, which is 12, but now I have 2 digits behind the decimal.

But notice I have one digit behind the decimal here and one digit behind the decimal here, for a total of two digits behind the decimal. So I'm giving you a little bit of a hint about where we're going. But the important thing for this video is to recognize that you can express each of these as fractions and then multiply the fractions to get something expressed in terms of hundreds and then express that as a decimal.