Grades 5-6 Video Solutions 2015-Level 5&6 Video Solutions 2015 problem25

Video Transcription

Question number 25. In a four-digit number, A, B, C, D, the digits are arranged in increasing order. So A is less than B, B is less than C, and C is less than D.  What is the largest possible difference, BD minus AC, for two-digit numbers BD and AC? First of all, let's write down a couple of examples.  We need a four-digit number that's denoted as A, B, C, D. And for example, if we begin with A is equal to 1, and have 1, 2, 3, and 4 as an easy example,  with the digits increasing as we go from left to right. So has this 1,234 has this difference of BD minus AC equal to BD is 24, AC is 13, and that difference comes out to be 11.  Now here, 1,234 is the smallest number.  So let's see if we can find the largest number.  And that would be the last four digits, 6, 7, 8, 9.  And this number has the difference BD minus AC equal to 79 minus 68, which is also 11. So the question is, can we actually increase that difference past 11?  What we want is the number BD to be largest, while AC is smallest possible.  So as a next example, let's consider the number 1,789. And what I've done here is I kept BD as large as possible, which is 79, and I decreased the first digit A from a 6 to a 1.  Here, we have 79 minus 18, and that's equal to 61, which is sort of nice because it allows us to eliminate a couple of choices. We eliminate E, D, and C.  Now, we have shown by example that E is possible. We can make that difference a 61. Is it possible to actually increase it past 61 all the way up to 86?  Let's answer that question now. How would we actually make an 86? 79 here minus what number gives us an 86? Well, that number does not exist, or at least it's not a positive number.  So the difference BD minus AC cannot be 86.  And that's really all we have to know. We don't really need to have a strict upper bound for it. As long as we can eliminate choice A, we have the answer to the question here,  and that is B. That difference is, at most, 61 with the number 1789.

Video Summary

In this problem, we're determining the largest possible difference between two pairs of digits, BD and AC, from a four-digit number A, B, C, D, with digits in increasing order. The goal is to maximize the difference between BD and AC. Initial examples with numbers 1234 and 6789 both give a difference of 11. To maximize this difference, the number 1789 is chosen, yielding a larger difference of 61 (79 from BD minus 18 from AC). The conclusion is that 61 is the largest possible difference achievable.

Keywords

digit pairs

maximizing difference

four-digit number

increasing order

largest difference