Introduction to division with partial quotients

In this video, we want to compute what 833 divided by seven is. So, I encourage you to pause this video and see if you can figure that out on your own. All right, now let's work through it together.

You might have appreciated this is a little bit more difficult than things that we have done in the past. In this video, I'm going to show you a method that your parents have probably not seen. But you'll see that it's kind of fun, and it's called division with partial quotients, which is a very fancy word. But, as I said, it'll be fun.

The first thing I will do is I will rewrite this as 833 divided by seven. You can view these as the same expression. The reason why we do it this way is it formats it so it's a little bit easier to do our division with partial quotients.

So, the way that division with partial quotients works—and once again, it's not the way that your parents probably learned how to do it—is you just say, "Hey, how many times can 7 go into 833?" I don't have to get it exactly; I just want to go under 833.

My brain immediately thinks, well, 700 is less than 833, so we're going to go into 833 at least 100 times. What we would do is we would write that 100 up here. We want to be very careful about our place value. You could view this column as the hundreds column, this is the tens column, this is the ones column.

Then, we want to see how much do we have left over. How close did seven times a hundred get us? So, what we do is we multiply 100 times seven to get 700, and then we can subtract that 700 from 833 to figure out how much more we have left.

So, 833 minus 700 is 133. We could then say, "All right, we still have another 133 to go." So, how many more times can seven go into this? Well, seven goes into 133. You don't, once again, have to have it exactly. If you know 7 times 10 is equal to 70—actually, let's go with that—we know we can go at least 10 times.

So, let's write that up here. We're going at least 10 times. To figure out how much more we have left, let's multiply 10 times 7 to get 70. Then we can subtract and see that we have—let's see—3 minus 0 is 3, and 13 tens minus 7 tens is 6 tens, so we have 63 left.

So, 7 definitely can go into 63. We're going to keep doing this until we have a number less than 7 over here. So, let's see. How many times does 7 go into 63? You might know from your multiplication tables that 7 times 9 is 63, so you could get it exactly.

You could just write that up here; we have 9 more times to go into the number. You would say, "9 times 7 is 63," and you can say, "Hey, we got exactly there!" We have nothing left over. As long as this number is less than seven, you know that you can't divide seven anymore into our original number. So you're done.

How many times does seven go into eight hundred and thirty-three? Well, we said it went a hundred times. Then we were able to go another 10 times, and then we were able to go another nine times. So, what we want to do is add these numbers.

You want to add 100 plus 10 plus 9. When you add up all of them, what do you get? You get 9 ones, 110, 100—you get 119. So, this is equal to 119. All I did is I added these up.

Now I want to be very clear that you could do division with partial quotients and not do it exactly like this. That's kind of why it's fun. So, let's do it another way.

Let's say we want to figure out again how many times does 7 go into 833. We could have said maybe it goes 150 times. What you could have said is, "All right, my current guess or estimate is 150 times." Then I could multiply 150 times 7.

How would I do that? Let's see: 0 times 7 is 0. 5 times 7 is 35. You can carry the 3, so to speak. 1 times 7 is 7 plus that 3 is going to be 10, and so that gets us to 10, 50.

Well, over here we just finished overshooting it. It doesn't go 150 times; there's nothing left over, so 150 is too high. We would want to backtrack that, and then you could go, "Well, maybe I could go to 110." So, let's try that out.

110—and now let's multiply: 0 times 7 is 0. 1 times 7 is 7. 1 times 7 is 7. So, 110 times 7 is 770. So that works. It’s less than 833, but let's see what we have left.

So we subtract and get a 3 here, and then let's see—83 tens minus 77 tens is 6 tens—and actually, that got us there a lot faster. So then you could just know that, hey, 7 goes into 63 nine times.

But let's say we didn't know that. We could say, "All right, let's say I'm going to estimate it goes 8 times." So you would put an 8 up here, and then you'd say how much do we have left over? 8 times 7 is 56. You subtract, and then 63 minus 56 is exactly 7.

Then they say, "Okay, look I can go one more time," so I'd write that up there. 1 times 7 is 7, and then you see we have nothing left over, so we are done.

So how many times did it go in? 1 plus 8 is 9 plus 110 is 119. So hopefully, you find that interesting. I really want you to think about why this is working.

We're just trying to see how many times can we go in without overshooting it, and then what's left over? So, how many more times can we go in? And then what's left over? Then how many more times can we go in until what we have left over is less than seven, so that we can't go into it any more times?