Division with partial quotients example

Let’s say we want to figure out what 473 divided by 5 is, and like always, why don’t you pause this video and try to work through it? If you’re familiar with the idea of division with partial quotients, I encourage you to try it out that way.

All right, now let’s work on this together. Let me just rewrite this in a format that makes it a little bit easier for me to do my division with partial quotients. I’m just going to write 473 divided by 5. What’s neat about division with partial quotients is that there’s a lot of flexibility in how you do it. There’s no exactly one right way.

So, you could say, “All right, how many times can 5 go into 473 without going over 473?” You might be tempted. Someone might say, “Oh, maybe you can go 100 times.” But you could quickly see that if you said 100 times, and you say, “How much do I have left over?” You do 100 times 5, you’d get 500, and you’d say, “Oh wait, wait, I overshot 473.” So 100 is too high.

Then you would back up a little bit. You could say, “All right, I’m gonna go for low numbers and just slowly chip away at it.” You might say that 5 goes into 473, you know, it goes more than 20 times, so let’s just start with 20. Well, what is 20 times 5? 20 times 5 is 100. Let me be very clear what that is: that is 20 times our 5 right over there.

So clearly, we did not overshoot it. But how many more times can 5 go into 473 than 20? To figure that out, we figure out how much more we have left over to go. So we subtract the 100 from 473, and we have 373.

How many times can 5 go into 373? Well, if 20 times 5 was 100, well gee, this is 300 here, so maybe we can go another 60 times. So let me put a 60 right over there. So what is 60 times 5? Well, 6 tens times 5 would be 30 tens, which is 300.

So how much further do we have to go? We still have 73 left. 373 minus 300 is 73. How many times does 5 go into 73? Well, let’s see. I know 5 times 12 is 60. So let me try 12. I will put a 12 over here, and so 12 times 5 is 60.

How much further do we have to go? Let’s see. 73 minus 60 is 13, and 5 can go into 13, so we can get a few more fives in there. Just to be clear what we’re doing: We’ve already said, “Hey, we can go 20 times and then another 60 and then another 12 times.” If we go into 473 all of those times, we still have 13 left over.

So we could fit a few more fives in there. Now, how many times does 5 go into 13? Well, 5 times 2 is 10. 5 times 3 would be too much, so we could say 2 times. So that gets us 2 times 5, which is 10, and so what we have left over is a 3.

Then we can’t fit any more fives in; we can’t divide 5 into 3, so that’s going to give us our remainder. How many times does 5 go into 473? Well, we were able to get it in there 20 times, and then another 60 times, and then another 12 times, and then another 2 times.

So what we can do is we can add all of these together. And what do we get? 2 plus 12 is 14, plus 60 is 74, plus 20 is 94, and then we have a remainder of 3. So this is equal to 94, and then I’ll write this in red with a remainder of 3. Did I do that right?

2 plus 14, let’s see. 2 plus 12 is 14, 74, yep, 94. Now let me just show you that there’s more than one way to do this, and I’ll draw a little line here so we don’t get too confused.

We could have done something like this: we could have said what is 473 divided by 5? We might have immediately recognized that, all right, 5 goes into 48 times, so 5 would go into 480 times—at least probably more times than that—but let’s just start with 80.

How close does that get us to 473? Well, to figure that out, we multiply 80 times 5 to get 400. Can we fit any more fives into the difference? So let’s figure out the difference. We will subtract, and just to be very clear, this was 80 times five. The reason why we did that is, see how close we got to 473?

Now we want to figure out how much we have left over, how many more fives can we fit in? So you subtract 400 from 473, and we are going to get 73. Well, we can for sure fit more fives into 73. We might say, “Hey, maybe let’s try to fit another 10 fives in there,” so let’s stick another 10 fives in there.

Now, how close are we? Well, 10 times 5 is 50. To be clear, what we just did here is we multiplied 10 times 5. Now can we fit any more fives in? Well, to figure that out, we would need to subtract and figure out what we have left.

So 73 minus 50, that’s going to be 23, and we can for sure fit more fives into 23. We know that 5 times 4 is equal to 20, so let’s put four more fives in there. Four times five is going to be 20.

Let’s see how close we are now. To be very clear, that is four times our 5. So what do we get to? What do we have left? Can we fit any more fives in? Let’s see. 23 minus 20 is just 3, and that’s less than 5, so we can’t fit any more fives in.

So there we have it: We were able to get 85s into 473. We saw that we still had some left over. We were able to fit another 10 into that leftover, and then we still saw that we had some left over and we were able to fit another 4 into that leftover, and then we had a leftover of 3. We weren’t able to fit any more 5s in.

In total, we were able to fit 94 fives into 473, and then we had three left over, which is exactly what we had there.