Rounding whole numbers: missing digit

- What digits could replace the question mark in the hundreds place to make this statement true? Four thousand question mark hundreds 29 rounds to 5000 if we round to the nearest thousand. So we want a number whose nearest thousand is 5000. It's closer to 5000 than any other thousand. And what we know about our number so far is it starts with 4000 then we'll fill in the digit for the hundreds and ends with 29. So, we want to know what digits from zero to nine can we fill in here so that this number is closer to 5000 than any other thousand. Well the only other thousand this could be close to is 4000 because we're starting with a four here, all of our numbers will be between 4000 and 5000, so let's draw that, let's show that on a number line. Here's a number line that goes from 4000 to 5000 and then the hundreds here are marked so this would be 4100, 4200, 4300, 4400, so on all the way to 4900 and then finally, 5000. So if we want to fill in digits here, then we can graph them on our number line and see what are they nearest to. Are they nearest to 5000 or are they nearest to 4000? Just, for example, if we plot a number right here, whatever number this point represents, what is its nearest thousand? Well, it is literally nearest to 5000, so it would round to 5000, that would be its nearest thousand. But, if for example we had a point that was somewhere like this, this points nearest thousand would be 4000. This point would round down to 4000 because we can look and see, it is closer to 4000 than it is to 5000. It is nearer. So, let's look back now at our number and try to fill in some digits and see where they land on this number line. Let's draw in the first possible digit we could fill in here, is a zero. So this would be 4000 zero hundreds 29, well that would be between 4000 and 4100, it's greater than 4000 but not quite all the way to 4100, so we could put that probably somewhere, estimating a little bit, about right here, a little closer to 4000 than 4100. And now looking at this number, we can see it is much closer to 4000, its nearest thousand is definitely 4000. It takes us far longer to get to 5000, so zero will not work. So, we can take away, it's not going to be a zero. And I don't know that we need to try all 10 numbers, what we can do is look down here, let's look at this graph and see which section of it, which section of this number line is closer to 5000. And the answer will be if we find right in the middle, here, 4500, anything greater than this will round up to 5000. From the halfway point up will round up to 5000. So, what points knowing that this right here is 4500, what numbers or what digits can we fill in here to get a number that's greater than 4500? Because anything less than that's going to round down, be closer to 4000. And so, we could try, let's just put a five in. 4529 would be right after 4500, somewhere about like this, that is closer, just barely but it's closer to 5000, so five works. Five is a digit that we could use. What about four, if we fill in a four here, we have 4400, well that's going to be just to the left of 4500, this line is 4400, here's 4500, so that point will come somewhere around here, which just barely rounds down to 4000, it's close to the middle but a little bit closer, a little bit nearer to 4000, its nearest thousand is 4000. So, four will not work, five does work and from here I think we can figure out our solution almost without using the number line. If 4400 was too small, well then 4300 is definitely too small and 200 and 100 and we already saw zero hundreds was way here, way down here which is way too small, and now we know 4529 works, that's big enough, so anything bigger will only get closer to 5000. Anything bigger than 5 int he hundreds place will only move us closer, farther to the right, so any digit five or greater would work here, would round up to 5000. So the digits that are five or greater are five, and then six, seven, eight and nine. Those are the possible digits that we could fill into our hundreds place so that our number rounds to 5000 as its nearest thousand. The solutions are five, six, seven, eight and nine.