Multiplying decimals using estimation

So let's see if we can come up ways to compute what 2.8 times four point seven three is. So pause this video and try to work it out. Actually, I'll give you a hint: try to figure out just using the digits, not even paying attention to the decimals, the digits that the product would have, and then use estimation to think about where to place the decimal in your product so you get a reasonable answer.

All right, now let's do this together. So let's just imagine that we were multiplying these numbers without decimals. So that would be a situation where we would have 473 times 28. We could try to compute that. So we could think about, let's multiply everything times the 8. So 3 times 8 is 24. 7 times 8 is 56 plus 2 is 58. Then 4 times 8 is 32 plus 5 is 37.

Next, we could multiply everything times the 2. I'll cross those out so I don't get confused. 3 times 2 is going to be 6, and we have to be very careful. We are now in the tens place, so we want a zero here. 3 times 2 tens is going to be six tens. 7 times 2 is 14. Four times two is eight plus one is nine.

We add everything together, and we get 4 plus 0 is 4. 8 plus 6 is 14. Then 1 plus 7 plus 4 is 12. After that, we get 1 plus 3 is 4 plus 9 is equal to 13. So we know that the final answer has the digits 1, 3, 2, 4, 4 in that order: 1, 3, 2, 4, 4.

Now we have to think about where we would put a decimal for this to be a reasonable answer. Here's where estimation is useful. We know that two point eight times four point seven three is going to be roughly equal to what? Well, two point eight is pretty close to three, so I'll estimate two point eight as being three. Four point seven three is, if I had to estimate it, I’d say hey, it'd be pretty close to 5. So this should be pretty close to 3 times 5, which should be close to 15.

If I were to put the decimal there, that's way more than 15, so that doesn't seem reasonable. Even if I were to put the decimal there, one thousand three hundred twenty-four point four is still way more than fifteen. Far to put the decimal there, still way more than fifteen. If I were to put the decimal there, hey, that actually feels about right: thirteen and 244 thousandths is approaching fifteen. It's in the ballpark and actually the closest, 'cause if we were to put the decimal there, then we go to one point three two four four, which is a lot less than fifteen.

So if we want this to be roughly equal to fifteen, we definitely would want to put the decimal right over there. This is the most reasonable computation we can do because we know the digits are going to be 1, 3, 2, 4, 4. This helps us put the decimal in the future.

We're going to come up with ways of doing it where you don't necessarily have to estimate, but I encourage you that estimation is always key. If you ever in your life forget some type of method or process for multiplying decimals, it's the estimation that allows you to understand whether you're coming up with a reasonable answer.

This is really important because of the decimal. There’s—there’s a remember reading a news story a couple of years ago where someone put in a stock trade where they got the decimal wrong, and because of that, they essentially sold 10 times as many shares as they were supposed to. So they lost hundreds of millions of dollars. So anyway, decimals are important.