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