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Decimal multiplication place value

When we rewrite decimal multiplication using fractions, we can use the commutative and associative properties of multiplication to justify how we place the decimal point in the standard multiplication algorithm. Created by Sal Khan.

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Video transcript

- [Instructor] This is an exercise from Khan Academy. It tells us that the product 75 times 61 is equal to 4,575. Use a previous fact to evaluate as a decimal. This right over here, 7.5 times 0.061. Pause this video and see if you can have a go at it. All right, now let's do this together. So the first thing that you might realize is that 7.5 is the same thing as 75 divided by 10. And 0.061, this is 61 thousandths. This right over here is the same thing as 61 divided by 1,000 and we're gonna take the product of these two things. Another way we could write this, 75 divided by 10, this is the same thing as 75 over 10 and I'm gonna take the product of that, and 61 thousandths, 61 divided by 1,000. So that would be 61/1,000. Now, when we look at it, either of these ways, well actually, I'll do both of them at the same time. You could change the order of the multiplication and the division here. So you could start with 75 times 61, 75 times 61, and then divide that by 10, and then divide that by 1,000. You could do it that way or you could look right over here and say, all right, if I'm taking this product, my numerator is going to be 75 times 61, 75 times 61. And then, my denominator is going to be 10 times 1,000 which is essentially the same thing as dividing by 10, and then dividing by 1,000. And of course, that is going to be 10,000. Now on the left-hand side, right over here, they told us what this is, it's 4,575. So it's 4,575 divided by 10, and then divided by 1,000. Well, if I divide by 10, and then I divide by 1,000, that's equivalent to dividing by 10,000. This is dividing by 10,000 and you could see that over here. We're dividing by 10,000 as well right over here. And the 75 times 61, this is 4,575. Now they want us to evaluate it as a decimal. We've now expressed it as a fraction and I still haven't fully evaluated this yet. So we really wanna think about this as 4,575 ten thousandths and you can see that very explicitly here. There's 4,575 ten thousandths. So how do we write that? Well, if I have a decimal right over here, that's the tenths place. This is the hundredths, thousandths, ten-thousandths place. So we have this many ten thousandths, 4,575 ten thousandths and we're done. So this is gonna be 0.4575. Now I know what some of you might be thinking. Hey, I learned a technique where if I'm taking the product of two numbers, I could take the product of those two numbers, or if I'm taking the product of two numbers that are decimals, I could remove the decimals from them essentially, take their product which they actually gave us right over here. And then, count how many digits to the right of the decimal there were in our original numbers. So we have one, two, three, four digits to the right of the decimal. And so what I do is I then move, I then make sure that there's four digits to the right of the decimal in the product. And so I would say, okay, one, two, three, four, that looks good and I've gotten the same answer a lot faster than we just did it. Well, whole reason why I just did it the way I did is to show you why that works. When we take the product of the two numbers without the decimals, we're essentially ignoring the fact that the original product was dividing by 10 and dividing by 1,000, and that's because we had one digit behind to the right of the decimal here, and we had three digits to the right of the decimal there. And so later after we take the product, we have to go and then actually take that product and divide by 10, and divide by 1,000 or divide by 10,000. So that's why you can then just say, all right, well now, we originally had four digits to the right, so we still have to have four digits to the right of the decimal point.