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The standard algorithm for multiplying whole numbers involves breaking the numbers down into their place values and multiplying each place value separately. This process is illustrated through three examples: one complete walkthrough, one where the viewer is asked to identify mistakes in incorrect solutions, and one where the viewer is asked to fill in a missing number. Created by Sal Khan.
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- We're not going to do a few examples questions from the Khan academy exercise on the standard algorithms. We're asked, which of the following correctly multiplies 74 times 8 using the standard algorithm. So pause this video and see if you can work on that before we do it together. All right, now let's just remind ourselves what the standard algorithm is. In fact, let's just remind ourselves what an algorithm is. An algorithm is a series of steps that you can do to do something, so you'll often hear about a computer algorithm. But you can also have algorithms in math, just a method for doing something. And the standard algorithm, that's the typical, or the standard, way that a lot of people will tackle a multiplication question or computation like this. But just as a reminder, in the standard algorithm if we're multiplying 74 times 8 we would write the eight in the ones place right below the four in the ones place, and then you multiply each of these places times the eight. So you would start with the four times the eight, you would get 32. 32 you can express as two ones and three tens. So we'll put that three up there. And then you would multiple the seven times the eight. Seven times eight is 56, and that's going to be 56 tens because it's seven 10s times eight is 56 tens. Plus the three 10s you had before get you to 59 tens, and so you would write over here. That's 59 tens, and so this would be 592. Now when you look at the choices, that's exactly what happened here in choice C. Just for kicks we can see what went wrong in these other ones. Let's see, in this first one, when we multiplied the, when we multiplied the four ones times eight ones, according to this, this person somehow got three ones and two tens, and 23. Four times eight is not 23, you can rule that one out. And here when they multiplied the four ones times the eight ones, that would be the 32. So it's two ones and then another three tens, so there should've been a three up here. And so that way when you multiplied the seven tens times eight, you get 56 but then you add this other three tens so you would really need to get to 59 tens. So that's why that one didn't work. Let's do another example here. And this is going to be with a different type of question. So here we are told that, Don starts to use the standard algorithm to solve 418 times five. His work is shown below. What number should Don replace Y with? Pause this video and see if you can figure it out. Okay, so the way to think about this, this might at first confuse you a little bit, because Y, and why is there a Y there, in the first place? But what they're really trying to get at is making sure that you, or we, understand what Don is trying to do when he's trying to do the standard algorithms. So as we just highlighted in last example, the way that we would tackle this with the standard algorithm, and actually let me write it down, 418 times five. We would say eight ones times five ones is going to be 40 ones. 40 ones we can write as zero ones and four tens. And so that looks like the place where they, where Don stopped computing. So he's on his way to solving the whole thing he just partially computed it so far. But just by doing that, we know what the Y should be. The Y should be four. So what number should Don replace the Y with? He should replace it with a four, and it's representing Y tens, or four tens. And of course you could keep going with this computation. If Don were to say, okay, I have one ten times five, that'd be five tens, plus another four tens, that's nine tens. And then last but not least, if Don wanted to figure out, well, he's got four hundreds here times five is twenty hundreds, which we can express as zero hundreds and two thousands. Or you just do this as 20 hundreds. So they're not asking us to do the entire computation. We're just trying to figure out what Don did essentially in this first step. What number should he have written here instead of a Y? So Y could be replaced with a four. It's representing the four, which is really in the tens place, so four tens.