Rearrange the digits in the numbers 715 and 529 so that the difference between them is the greatest.
Before we start to solve this problem, let’s remind ourselves, what makes the difference between two numbers as large as possible? We know that the difference between two numbers is what we get when we subtract a smaller number from a larger number. Or if we look at it a different way, it’s what we add to the smaller number to get to the larger number. To make this difference as large as possible, there are two things we need to do.
To start with, we need to make the first number in the subtraction as large as possible. Then we need to make the second number in the subtraction as small as possible. And so by doing this, we end up with the greatest possible difference between the two numbers. So if we explain our problem a different way, we need to rearrange the digits in the numbers 715 and 529. So that one of our numbers is as large as possible and the other of our numbers is as small as possible.
A good way to think about this problem is notice the numbers 715 and 529. But there’s two separate groups of three digits — seven, one, and five and five, two, and nine. What’s the largest number we can make out of the digits seven, one, and five? To make the largest possible number, we need to put the largest digit into the place with a largest value and then the second largest digit into the place with a second largest value and so on.
The largest digit is seven. The largest place value is the hundreds column. So to make the largest possible number, we need to put the digit seven in the hundreds column. The next highest digit is five. So we can put that in the tens column. And we put the digit one in the ones column. So it’s worth the least. Now that we’ve made the largest possible number out of the digits seven, one, five, let’s try to make the smallest possible number out of the digits five, two, nine.
To make the smallest possible number, we need to put the digit with the smallest value into the place with the largest value. So the smallest number of hundreds is two. We then complete the other digits from left to right in order of size. So the smallest number we can make from the digits five, two, nine is 259.
Now, we may think that we’ve solved the problem. We’ve rearranged the digits in seven, one, five to make the largest possible number. And we’ve rearranged the digits in five, two, nine to make the smallest possible number. So surely the difference between the two numbers must be the greatest. But we need to think more carefully. The problem doesn’t tell us which group of digits were to make the larger and the smaller number from.
What if we make them the other way around? What if we make the smallest possible number out of the digits seven, one, five? That would be 157. And what if we made the largest possible number out of the digits five, two, nine? That would be 952. And although we’d have to subtract the numbers in the opposite order, we could still find the difference between them. So which of our two subtractions will give us the greatest difference? 751 take away 259 or 952 take away 157? Let’s work out both subtractions to find out.
There are lots of strategies we could use to find the difference between these numbers. We could count on from the larger number. We could use vertical subtraction. But let’s use number lines to count backwards. So we’ll start with 751. And we’ll partition 259 into 200, 50, and nine. 751 take away 200 equals 551. 551 take away 50 equals 501 subtract nine to leave us with 492. So when we rearrange the digits in this way, we make a difference of 492.
What happens if we rearrange them the other way? Start with 952 take away 100, which leaves us with 852 take away 50 leaving us with 802. And then subtract seven. So we’ve subtracted 100, 50, and seven. And this gives us a difference of 795. So to make the greatest difference, we need to rearrange the digits to make the second calculation. And so to answer our problem, we need to rearrange the digits in the number 715 so that it becomes 157. And we need to rearrange the digits in the number 529 so that it becomes the number 952. And then we’ll have made two numbers where the difference between them is the greatest.