Developing strategies for multiplying two digit decimals

Let's say I want to multiply 3 point 1, or 3 and 1/10, times 2.4, which can also be described as 2 and 4/10. So pause the video and see if you can do this.

Once again, I'll give you a hint: see if you can express these as fractions. There are a couple of ways you can express it as a fraction. You could express this as 3 and 1/10 times 2 and 4/10, the same color, 2 and 4/10.

Now, whatever your multiple, these are mixed numbers right over here, and mixed numbers are not super straightforward to multiply. It's easier if they were written as what's often known as improper fractions, but essentially not as mixed numbers.

So, 3 is the same thing as 30 tenths, so 30 tenths plus 1/10 is 31 tenths. Times 2 is the same thing as 20 tenths, so 20 tenths plus 4 is 24 tenths. Hopefully, this makes sense. To the 3.1, this 3 right over here is 30 tenths, or I could write all over at 30 tenths, and this is 1/10.

So this total is going to be 31 tenths. Likewise, this 2 is 20 tenths plus 4 tenths, giving us 24 tenths. Now we can multiply.

So, this is going to give us our denominator, which is pretty straightforward: 10 times 10 is 100. Then 31 times 24—we can multiply it in the traditional way that we're used to multiplying two-digit numbers.

31 times 24 is going to be equal to 4 times 1, which is 4. 4 times 3 is 12. Now we're going to be multiplying in the tens place; we're going to put a 0 here. So, 2 times 1 is 2.

We're really saying 20 times 1 is 20, but you get the idea. 2 times 1 is 2. 2 times 3 is 6—really 600, because it's times 30, but I'm just following the standard method for multiplication.

Then you add these, and you're going to get 4, 4, 7. So when you multiply these two things together in the numerator, you get seven hundred and forty-four hundredths, which can also be expressed as this: this is the same thing as seven hundred hundredths, I should say, plus forty-four hundredths.

And seven hundred hundredths, well, that's just going to be equal to seven. So this is seven plus forty-four hundredths, which we could write as 0.44. That's our seven and forty-four hundredths, and we would be done.

You might already be seeing a pattern: if you just took 31 and multiplied by 24, you get seven hundred and forty-four. Notice I have one and two digits behind the decimal point; and so think about whether that always works.

Think about why that might work if you just multiply the numbers as if they didn't have decimals.

So, you have gotten seven hundred forty-four, and you say, "Hey, I got two numbers behind the decimal, so my product is going to have to have two numbers behind the decimal." Why does that work, or does it always work? How does it relate to what we did here, which is converting these things to improper fractions and then multiplying that way?