Jennifer estimates the product of two three-digit numbers by rounding each of them to the nearest 100 and then multiplying. What would Jennifer’s estimate for the product 431 multiplied by 891 be?
In this problem, we’re not being asked to find the answer to 431 times 891. Instead, we need to find an estimate for it. We need to make the calculation a little bit easier to help us find an answer that’s quite close to the actual answer to give us an indication of what it might be. And the way that we can estimate the answer is by using the strategy that Jennifer uses. We’re told what she does to find it.
Now, there are two steps to Jennifer’s method. First, we’re told that she rounds each number to the nearest 100. And then, secondly, we’re told that she then multiplies them. So, let’s use these two steps of Jennifer’s to help us estimate 431 multiplied by 891. Step one, we need to round each number to the nearest 100. Now, our first number in the calculation is 431. And the two multiples of 100 that this number’s in-between are 400 and 500. But which is it nearer to? Which should we round it to?
To help us think about this, labeling the midway point is always useful. Halfway between 400 and 500 is 450. 431 is less than 450, might be about here on our number line. And so, we can see that the nearest multiple of 100 is 400. We can round this number down to 400. So, in our estimate, the first number in our multiplication is going to be 400.
The second number that we need to round is 891. Perhaps you can see straightaway which multiple of 100 this is close to; this is only nine away. Let’s draw it on our number line. Anyway, 891 is in-between 800 and 900. Even though perhaps we don’t need to label it this time, the halfway point is 850. And as we’ve said already, 891 is only nine away from the next multiple of 100. And so, this time we’re going to round the number up to the nearest multiple of 100. So, in our estimation, the second number of the multiplication is going to be 900.
Our calculation was 431 multiplied by 891. And it’s turned into 400 multiplied by 900, which is a lot easier to work out. We can use facts we already know to help us find the answer. Firstly, what’s four multiplied by nine? Well, we know that 10 fours are 40, so nine fours must be four less than 40. Nine fours or, four times nine, is 36.
Here’s where we can use our skills at multiplying by 100 to help. If we make the first number in our multiplication 100 times bigger — So, instead of four, we now write 400 and we multiply that by nine — then our answer is going to be 100 times bigger. It’s going to be 3600. Look at the two zeros on the end. And we can use the same method then to find a final answer. Instead of 400 times nine, we’re going to make the second number in our multiplication 100 times bigger as well.
So, instead of 3600, the answer is going to be 360000. We’ve shifted the digits another two places to the left. And so, there are another two zeros in the answer. So, here, we’ve used two steps to estimate the answer to 431 multiplied by 891. First, we rounded each number to the nearest 100. Our calculation then became 400 multiplied by 900. And then, we multiplied these two numbers together to find the estimated product. Our estimate for the product 431 multiplied by 891 is 360000.