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Find two whole numbers with as great a difference as possible where both numbers result in 850 when rounded to the nearest ten.

This question asks us to find two whole numbers. But it gives us two conditions. The first is that the numbers that we come up with have to have as great a difference as possible. In other words, when we subtract one of our numbers from the other, it makes the largest possible answer. The second condition, and this is the one what we need to start with, is that both numbers need to result in 850 when we round them to the nearest ten. This second condition is the one that limits our answer. If we just had to find two whole numbers with as great a difference as possible, we’d have to come up with a really really large number. But we have this second condition. Our numbers result in 850 when we round them to the nearest ten.

Let’s draw a number line. And we’ll put the number we’re rounding to 850 right in the centre. Which numbers will round up to 850? 849, 848, 847, 846. 845 is halfway between 840 and 850. But we still round this number up. But when we get to 844, that’s nearer to 840. So we’d round that down. We can’t include this. These are all the numbers that round up to 850 when we round them to the nearest ten. But which numbers round down? 851, 852, 853, and 854. When we get to 855, that’s halfway between 850 and 860. Remember, we said when the digit is a five, we round up. So we can’t include the number 855.

Let’s go back to the question. We have to find two whole numbers that result in 850 when rounded to the nearest ten. This means our two whole numbers must come from this list here. There’s no other numbers we could choose. And we have to find two whole numbers with as great a difference as possible. We don’t need to think too hard about this. And we certainly don’t need to work any subtractions out. The two numbers with the greatest difference are the two numbers at either end of our number line. They have the biggest space between them, the biggest difference. There’s no other pair of arrows that we could draw on this number line that would have a greater difference than this. The difference is actually nine. And there are no other pairs that have a difference of nine. And so we’ve managed to find two whole numbers that have as great a difference as possible where both numbers result in 850 when we round them to the nearest ten.

And those two numbers, and remember we can write these in any order, are 845 and 854.