Complete 0.0407 multiplied by 1000 what 4.07 multiplied by 10 using the symbol for is less than, is equal to, or is greater than.
This problem is asking us to compare the value of two different multiplications, 0.0407 multiplied by 1000 and 4.07 multiplied by 10. And we need to choose the correct symbol to use in the blank in between them. Is the first expression less than the second? Does it have the same value as the second? Or is it greater than the second calculation? We need to find the answer to both multiplications to be able to compare them properly.
Perhaps the easier of the two calculations to begin with is the one on the right. This is because the decimal is a little shorter. It only has two decimal places, whereas the other number has four. And also because we’re multiplying by 10. We can use what we know about place value here to find the answer. We know that each column to the left of another when we’re talking about place value is 10 times greater. So, for example, two ones become worth 10 times as much if we move that two one place to the left.
So if we want the number 4.07 to become 10 times larger, we need to shift the digits one place to the left. That way, each of our three digits will be worth 10 times as much. Four ones become four tens. The placeholder of zero in the tens place becomes a placeholder of zero in the ones place. And seven hundredths become worth seven-tenths. 4.07 multiplied by 10 is 40.7.
We can use a similar method of shifting the digits to find the answer to 0.0407 multiplied by 1000. If we know that shifting the digits one place to the left is the same as multiplying by 10, then shifting the digits two places to the left is the same as multiplying by 10 lots of 10, or 10 and then 10 again. That’s the same as multiplying by 100. And a shift of the digits a third time to the left is the same as multiplying by 10 then by 10 then by 10. And that’s the same as multiplying by 1000.
So we need to move our digits three places to the left. And as we do so, we may end up losing some of our zeros. So we’ll start by shifting the digits one place. Just like before, this is the same as multiplying by 10. Now we can see that this placeholder zero here is pointless. We don’t need it. So we don’t need to write this.
Let’s shift the digits a second time. This is the same as if we’d multiply it by 100 now. And again, this placeholder zero in the tens place is not necessary. Finally, so that we’ve multiplied by 1000, we need to shift the digits one more place again. The answer is exactly the same as the second calculation.
It may have looked like we were going to get a different answer. But because we shifted our longer decimal by more places — this is because we had to multiply by 1000 — the answer ended up being the same. Both calculations have a value of 40.7. And so the symbol that we can use in between both calculations is the equal sign. 0.0407 multiplied by 1000 is equal to 4.07 multiplied by 10.