Multiplying challenging decimals

Let's multiply 1.21, or 1 and 21 hundredths, times 43 thousandths, or 0.043. And I encourage you to pause this video and try it on your own. So let's just think about a very similar problem but one where essentially we don't write the decimals. Let's just think about multiplying 121 times 43, which we know how to do. So let's just think about this problem first as kind of a simplification, and then we'll think about how to get from this product to this product. So we can start with-- so we're going to say 3 times 1 is 3. 3 times 2 is 6. 3 times 1 is 3. 3 times 121 is 363. And now we're going to go to the tens place, so this is a 40 right over here. So since we're in the tens place, let's put a 0 there. 40 times 1 is 40. 40 times 20 is 800. 40 times 100 is 4,000. And we've already known how to do this in the past, and now we can just add all of this together. And we get-- let me do a new color here-- 3 plus 0 is 3. 6 plus 4 is 10. 1 plus 3 plus 8 is 12. 1 plus 4 is 5. So 121 times 43 is 5,203. Now, how is this useful for figuring out this product? Well, to go from 1.21 to 121, we're essentially multiplying by 100. Right? We're moving the decimal two places over to the right. And to go from 0.043 to 43, what are we doing? We're removing the decimal, so we're multiplying by ten, hundred, thousand. We're multiplying by 1,000. So to go from this product to this product or to this product, we essentially multiplied by 100, and we multiplied by 1,000. So then to go back to this product, we have to divide. We should divide by 100 and then divide by 1,000, which is equivalent to dividing by 100,000. But let's do that. So let's rewrite this number here, so 5,203. Actually let me write it like this just so it's a little bit more aligned, 5,203. And we could imagine a decimal point right over here. If we divide by 100-- so you divide by 10, divide by 100-- and then we want to divide by another 1,000. So divide by 10, divide by 100, divide by 1,000. So our decimal point is going to go right over there, and we're done. 1.21 times 0.043 is 0.05203. So one way you could think about it is just multiply these two numbers as if there were no decimals there. Then you could count how many digits are to the right of the decimal, and you see that there are one, two, three, four, five digits to the right of the decimal, and so in your product, you should have one, two, three, four, five digits to the right of the decimal. Why is that the case? Well, when you ignored the decimals, when you just pretended that this was 121 times 43, you essentially multiplied this times 100,000-- by 100 and 1,000-- and so to get from the product you get without the decimals to the one that you need with the decimals, you have to then divide by 100,000 again. Multiplied by 100,000 is essentially equivalent to moving the decimal place five places to the right, and then dividing by 100,000 is equivalent to the moving the decimal five digits to the left. So divide by 10, divide by 100, divide by 1,000, divide by 10,000, divide by 100,000. And either way, we are done. This is what we get.