Understanding decimal multiplication

We're told that Sydney knows that 427 * 23 is equal to 9,821. Use this understanding to help Sydney solve 42.7 times 2.3. So pause this video and think about what you think it's going to be.

All right, now let's do this together. You might realize that 42.7 is just 427 with a decimal between the two and the seven. Or another way to think about it is if you were to take 427 and move the decimal one to the left, you are essentially dividing by 10 to get to 42.7.

Similarly, to get from 23 to 2.3, you are also dividing by 10. So you're taking the product of two things, each of them is one-tenth of each of the things in that original product. So let's just keep that in mind for a second.

But another way to think about what this would be is it's going to be the same product; we're just going to have to divide by 10 a few times or we're going to have to move the decimal. One way to think about it is we can estimate what, say, 40 * 2 is and think about which of these is close to that. If I were to say 40 * 2, 40 * 2 is equal to 80.

So which of these are even remotely close to 80? Well, this is 9,800 something, this is 980 something, this is 98. This one is the one that is closest to 80. Now, how do we really feel good about this? Well, as I just said, when I took 427 * 23, it equals 9,821. But to go from that to 42.7 * 2.3, you have to divide by 10 twice.

So if you start with 9,821, this product over here, and if we were to divide by 10 twice, you'd move the decimal two to the left and you'd get 98.21, which is exactly what we chose.

Let's do another example here. So we are told that Dom knows that 527 * 63 is equal to 33,201. Use this understanding to help Dom find which equation gives a product of 33,201. Pause this video and think about that.

Well, there's a couple of ways that you could approach this. One way to think about it is if you just estimate these products here, which of these would get you around 3,300? So if I were to take 5 * 6, and once again I'm just rounding, I'm approximating it, so this is approximately 5.6 * 5 * 6, which is equal to 30.

Well, 30 is nowhere close to 3,300 and something. All right, if I were to say this is approximately 50 * 6, that's equal to 300. 300 is still off by more than a factor of 10. So if I were to say this is approximately equal to 50 * 60, 5 * 60 is 300; so 50 * 60 is 3,000.

Well, this is pretty close. 3,000 is not off by a factor of 10 from 33,201, so I already like this choice. Now you can also think about it this way: to go from 527 to 52.7, you are dividing by 10. To go from 63 to 63, you aren't doing anything.

So if you're taking this product and you're getting 33,201 and you want to go to this product, well, one of these needs to get divided by 10, or the product has to get divided by 10. If you took 33,201 and divide by 10, you are going to move the decimal one to the left; you're going to get 3,320.1.