Fill in the missing digits. Six thousand what hundred seventy-two multiplied by nine equals fifty what thousand one hundred forty-eight.
This number sentence shows a four-digit number multiplied by nine. And it also shows us the answer, which is a five-digit number. Unfortunately, two of the numbers, that’s the first one and the answer, both have a missing digit. And it’s these that the question is asking us to find. Now, the way that this number sentence has been written, which is horizontally across the page, it’s tricky to see which digit multiplies by nine to give which answer. So the first thing we can do to make the problem easier is to write out the calculation as if we were doing some short multiplication. In other words, write it out vertically.
Six thousand what hundred seventy-two multiplied by nine equals fifty what thousand one hundred forty-eight. And we can see those missing digits, which are represented by boxes in the problem, we’ve written as question marks. Now, let’s quickly go through the calculation as if we were working it out for the first time, digit by digit.
First of all, we’ll multiply the ones by nine. Two times nine equals 18. So that’s where we get the eight in our ones place from. And then we must exchange 10 ones for one ten. Now, we need to multiply seven tens by nine. We know seven times 10 equals 70. So seven times nine must be seven less than 70. The answer is 63. So seven tens multiplied by nine equals 63 tens. Remember, we’ve got one ten underneath. That takes us to 64 tens. And We need to exchange our 60 tens for six hundreds.
And now, we get to the tricky part. We don’t know the digit now that we multiply by nine. But we do know that the answer to the multiplication when we’ve added six to it, because we’ve exchanged six hundreds, will give an answer with a one in the ones place. Now, this might sound complicated, but let’s go through it step by step. First of all, we’ll write out the multiples of nine that we could be dealing with. Zero times nine is zero. One times nine is nine. Two nines are 18. Now, adding nine every time is the same as adding 10 and taking away one. And there is a neat little pattern we can use here.
If we add one to the tens place and take away one from the ones place, we get the next multiple of nine. So three nines is 27. Four nines are 36. Five nines are 45. 54, 63, 72, and the last multiple of nine that we can make is 81. We don’t need to go as far as 90 because we can’t have a 10 in the hundreds place. It has to be one of the digits from zero to nine. Now, as we’ve just said, whatever the missing digit is, when we multiply by nine and then add six to it, it makes a number with a one in the ones place. So let’s add six to each of these multiples of nine. Zero add six equals six. Nine add six equals 15. 18 add six equals 24. There are lots of patterns here in these numbers if you can spot them. 27 add six equals 33, 42, 51.
We know that when a number ends in five and we add six to it, we get a number that ends in one. And we can see just by quickly finishing off the other numbers that the only multiple of nine that when we add six to it makes a number that ends in one is five times nine. Let’s replace our first missing number with the digit five. Five times nine, as we’ve just said, is 45. If we add the six that we’ve exchanged, we get 51. And we can exchange 50 of our hundreds for five thousands.
Now, all we need to do is to multiply six by nine. Five nines are 45. And we know six nines are 54. But we have five thousands underneath that we can add on to our 54 thousands. And this gives us a total of 59 thousands. And so, the number sentence should read 6572 times nine equals 59148. The missing digits are five and nine.