Let's do a couple more of problems involving place value and number order. So let's say I we're to give you the digits. The digits I'm giving you are zero, four and five. And what I want you to do is first of all figure out all of the three digit numbers that can be predicted with zero, four and five and then I want you to tell me which of those are the smallest and which of those are the largest. And one thing you keep in mind is, you can't have zero as the first digit, right because if it's zero is the first digit you really don’t have a three digit number for example the number is zero, four, five this really isn’t a three digit number, this is just 45 which is a two digit number. So that one is not allowed, so with that said maybe you want to pause it and try to figure out all of the ways you can construct a three digit number that doesn’t start with zero with the digits zero, four and five and you should do that right, now.
Well, I'm assuming that you have unpause, so let's see all of the numbers that you can do. And actually we could do both parts of this problem simultaneously. How can we get a really large number? Well the way we get the largest number is by putting our largest digit in the largest place value. So if this is a three digit number we’ll put it in the hundreds place, so the way we construct the largest possible number we put the largest number in the hundreds place. So we say 500, let's put the second large number in the tenths place 40 and then we have to use the zero, so we’ll put that there.
So this is going to be a larger, so you'll see when we list other ones, I hope it make sense why because this, you bring the largest number in the hundreds and the second largest number in the tenths. Most figure out all of the other combinations that we can get of this three numbers. So it could be 504, all right just switching the four and zero, the zero and the four. We could get or we could have switched the four and the five here, so we get 450 right? Or we could switch the five and the zero here and you get 405 and that’s all of them. I mean there we're this, this I guess we could call them permutation and you haven’t learned that yet, so I'm going too into it but the other two would have been this but this don’t count because they start with the zero so we can't count those.
So we already figured out that this is the largest and once again we know 540 is larger than 504 because 40 is larger than four. And as, you know, the logic we used when we originally solved this problem. So that’s the largest, so what's the smallest? Well I just happened to put them in, in order from largest to smallest, this is the smallest and why is that? We put the smallest possible digit in the hundreds place. And we couldn’t put zero by rule and then after we put that there we put the smallest possible digit that we could put in the tenths place and that was a zero because that was allowed and so we got 405.
Let's do another similar problem. Let's say I have the digits zero, seven, two, eight and I'm asking you what is the largest four digit number that I can construct out of this four digits? And ones again the rule is that zero can't be the first digit although that wouldn’t help you much anyway. So what's the largest number I can build here? So it's going to be a four digit number, so this is going to be and you're going to have thousands place, hundreds place, tens place and ones place, right. So just like we talked about in that that problem we did a few seconds ago. How do we construct the largest number? Well, what place value matters the most? The thousands place matters the most, right. And if we're going back to counting marbles you’d care about how many of the buckets of thousands you have that those—that’s where most of the marbles takes place, so its thousands place matters a lot.
The thousands place because whatever digit here you're essentially saying that times of thousands, you got to put the largest number there. So the largest number out of this, that’s eight, so we cross that out we use that. And then the hundreds place, we already use the eight you want to put the highest number that you could fit there in the hundreds because whatever numbers is here is going to be multiplied by a hundred. So out of this will send that collection number seven. And then we use the same logic for the tenths place. What's the next largest number there? That’s going to be the two and then we're left with the zero and if that was the incompletely obviously to you I encourage you to try all the different ways to rearrange this numbers and you'll see that when you put the largest number in the thousands place because you're saying that’s 8, 000 right.
Let see if we would switch this and then you would only had 7,800 and that would, would that would have been much smaller than 8,700. Let's do a similar exercise but now let's try to figure out the smallest digit we could form. So let's say the numbers are three, seven, and nine. So what's the smallest four digit number that I could form out of these three numbers? So let's do the places again there’s thousands, hundreds, tens and ones. So if we want to create the smallest, there is two ways of thinking about it, we could put the largest number in the ones place and then the second larger is in the ten and so fort. Or we could say let's put the smallest number in the thousand, we want the smallest number of thousands. So we could either do 3,000, 7,000, 4,000 or 9,000, we want a smaller number so let's do 3,000, right that’s clearly the smallest of our choices.
So and we use three and now we have to pick between 700, 400, or 900. So what's the smallest, 700, 400, or 900? Well sure 400 is, use it there and then we want, we have to just picked between 70 or 90, well we want use 70 because we're looking for a small number. And then of course you have the nine there and ones again I encourage you to try, try to get a smaller number that than what we just did. And hope that makes sense that we want the smallest possible number you want the smallest number in front of the thousand. So ones you use that up and you want the smallest number on the hundreds.
Let's see if we can see some patterns between numbers. So let's say, I had the number—unless you was just going to write down number I'm not going to read it to you because I would like to give you. And so let's say I have that number 6,342 and tats the digits and 6,442, so I have two questions for you. Which number is larger? And how much larger is that number? Well first we could read them out this is 6,342, this is 6,442. So we immediately see where these two numbers different are 6,300 and 6,400 the 42’s are the same so they're different in the hundreds place.
Right, if we we're to do, if we we're to expand this out. This is 6,000 plus 300 plus 40 plus two, right. This number on the right is 6,000 plus and we make sure I do the colors right, 400 plus 40 plus two. And so we see the only difference between the numbers it takes place in the hundreds place. Here we have 6,000 is equal to 6,000, 40 is equal to 40, two is equal to two and here we have 300 versus 400, right.
So this number is clearly the larger of the two number and how much larger is it? well we just have to compare the hundreds because everything else is equal, 400 versus 300 how much larger is 400 than 300? Well it's 100 larger, all right? You have to 100 so you could write 6,342 plus 100 is equal to 6,442. Another way you could have looked at it, without expanding it. You could have said “Well, I have 300 here I have 400 here just looking at place value. So I need to get one more hundred to get to this value”. So you can kind of do that mental arithmetic, you add 100 here, you would add up one to this digit so you get six, four, four two or 6,442.
Let's do one more like that. Let's say, I had the numbers, I could read them out 6,442 just like we have in the previous one and let's say we have the number 6,542. And I want to expand, well let's sure let's expand then out. I want to do it as I do in all different colors but we could say this is 6,000 plus 400 plus 40 plus two, this is 6,000 plus 500 plus 40 plus two and what's the difference here? Well I didn’t even have to expand it out, to show you the difference we have a four in the hundreds place here. We have a five in the hundreds place here. So this digit in the hundreds place is one more than this digit in the hundreds place.
So essentially we would do go from this entire number to that entire number we have to increment the hundreds digit by one or we need one extra hundred. So ones again the number on the right is larger and it's larger by a hundred. We could also see that here when we expand it, to go from 400 to 500 you add a hundred. So let's see if we can use some of these insights to see if we can do some quick mental math. So if I we're to ask you what number, what number is a 100 more than 6,442? Well, you could do this in your head, how do you think about it? Well, if I'm doing 100 more and that means in the hundreds place I'm going to add one, right. Because incremental digit in the hundreds place means a hundred, so essentially to add a hundred to this, I go to the hundreds place and add one, so 6,442 we could do that in our head plus 100 is equal to 6,542.
Let's see if we could do that same mental math to see what number is a 100 more than 6,542? So what number is—I'm running out of space, is a 100 more—and that’s little hard to do, what number is a 100 more than 6,542? So once again we have to, if we want to add a 100 to something we increment to the 100 digit by one. So this is the hundreds digit so it's 6,642, right. Similarly if we want to know what number is a thousand more than 6,542 we would just increment the thousands digit right? If we wanted to add a thousand, we could increment—one of the thousands place and add one, we get 7,542.
If someone asked you what number is 10 less than 6,542? Well, do you go to the tenths place and you would subtract one from the digit. So if you wanted to subtract 10 minus 10 and it would be 6,500 and we've decreased this digit by one, 32 anyway don’t want to confuse you and I think that’s about it for this video. I’ll see you in the next one see you soon.