The product of two numbers is 18612. Given that one of the numbers is 396, find the other number.
If the product of two numbers is 18612, then we know that this five-digit number is the result when we multiply those two numbers together. In the second sentence of the problem, we’re told that one of the numbers is 396. And the problem asks us to find the other number. How can we solve a problem where one of the numbers in a multiplication is missing? Or we can turn the problem the other way around and use the inverse operation to help.
The inverse or the opposite to multiplying is dividing. So we can take 18612. And if we divide it by 396, then we can identify our missing number. And to divide 18612 by 396, we’re going to need to use long division. Because we’re dividing by 396, which is such a large number, there’s no point looking at just one digit, two digits. And even when we look at the first three digits, it’s too small. There are no lots of 396 in 186. Instead, we need to look at the first four digits. And because we don’t know all the different multiples of 396, we could go through each one in turn until we find the number that gets close to 1861.
But ideally, we don’t want to have to do this. Can we use estimation to help us? 396 is very close to 400, and we can use this fact to help us. We can count quickly in multiples of 400 to see how close we can get to 1861. 400, 800, 12 100s or 1200, 16 100s or 1600. This is as close as we can get to 1861. So it looks like we might be able to fit four lots of 396 into this number. Let’s multiply 396 by four to see what the answer is. Six fours are 24. We know 10 fours are 40. So nine fours are 36. We have two that we’ve exchanged, so that takes us to 38. And finally, three fours are 12 plus the three that we’ve exchanged equals 15.
And we can see that if we add another lot of 396, it’s going to be too large. So we can only fit four lots of 396 into this number. Let’s calculate the remainder. Four lots of 396, as we’ve just found out, is 1584. So if we subtract this from 1861, we can find out what we’ve got left. We can’t subtract four ones from one one. So we’re going to need to exchange. We’ll take one 10 and exchange it for 10 ones. Now, we have 11 ones. We can take away our four ones. This leaves us with seven ones. In the tens column, we can’t subtract eight tens from five tens. So again, we’re going to have to exchange.
This time, we’ll take 100. And we’ll exchange it for 10 tens. Now that we have 15 tens, we can subtract the eight tens. This leaves us with seven tens. In the hundreds column, seven hundreds subtract five hundreds leaves us with two hundreds. And there’s nothing left in the thousands column. Our remainder is 277. There are no lots of 396 in 277. So we’re going to have to include another digit. And we have one more digit at the top. So if we bring this down, we can make our three-digit number into a four-digit number. The digit from the top is a two. And our calculation now becomes, how many lots of 396 are there in 2772?
Rather than going through every multiple of 396, let’s use our estimation skills again. How many lots of 400 could we fit into this number? Well, when we were counting in 400, we got as far as 16 100s or 1600. So we could carry on from there. Five lots of 400 is 2000. If we add another lot of 400, this would be six lots of 400. That’s 2400. And seven lots of 400 is 2800. Now, if we were dividing by 400, we’d have to stop at 2400 because 2800 is too large. But remember, we’re not dividing by 400. We’re dividing by a number less than 400. So it could possibly be that there are seven lots of 396 in this number. Let’s see what both six lots and seven lots of 396 are. And we’ll decide which one is right.
Firstly, six sixes are 36. We know 10 sixes are 60. So nine sixes must be six less than this. The answer is 54. We’ve also exchanged three. This takes our total to 57 tens. And in the hundreds column, three sixes are 18. Plus the five hundreds that we’ve exchanged, and our total is 23 hundreds. So we can say six lots of 396 is 2376. And this definitely fits into 2772.
But what if we add one more lot of 396? Perhaps there are seven lots of 396 in this number. Six ones plus six ones equals 12 ones. Seven tens plus nine tens equals 16 tens. Plus the one we’ve exchanged, 17 tens. Three hundreds plus three hundreds equals six hundreds. Plus the one we’ve exchanged, seven hundreds. And two thousands plus zero thousands equals two thousands. This is the number we need to divide.
So we’ve found that there are seven lots of 396 in this number. And there is no remainder. So we can write seven at the top. And just to prove that there’s no remainder, we can subtract seven lots of 396, which is 2772, from the number that we’re dividing, which, of course, is exactly the same. And this leaves us with nothing. We’ve divided our numbers exactly. And we can use this answer to complete the problem. If the product of two numbers is 18612 and one of the numbers is 396, we’ve used long division to find that the other number and the answer to our problem is 47.