Grades 7-8 Video Solutions 2024-2024_7-8

Video Transcription

Problem number 21. The digits 0 through 9 can be drawn with horizontal and vertical segments as shown.  Greg chooses 3 different digits. In total, his digits have 5 horizontal segments and 10 vertical segments. What is the sum of his 3 digits?  Using this diagram, we can make a table of each of the digits and how many horizontal and vertical segments it has.  This will be useful to us later on when we need to figure out how these sum to 5 and 10.  In particular, we need 3 of these horizontal numbers, which are all between 0 and 3, to sum to 5.  The only way you can have 3 numbers that sum to 5 with 0, 1, 2, or 3 is these 3 possibilities, 2, 2, 1, 3, 1, 1, or 3, 2, 0. So let's check all of them and see which ones work.  First notice that 2, 2, 1 cannot work because there is only a single number, the digit 0, that has 2 horizontal segments. And Greg has to choose 3 different digits.  So we cannot have 2 0s and there is only 1 number with 2 horizontal segments. So the horizontal segments cannot be 2, 2, and 1.  3, 1, 1 also does not work because there is only 2 numbers with 1 horizontal segment, 4 and 7.  In that case, we would need 5 more vertical segments because 4 has 3 vertical segments and 7 has 2 vertical segments.  So that means that we need 10-3-2 equals 5 vertical segments in the last number. But none of our digits have 5 vertical segments, so that means this is impossible.  This means that we must have 1 number with 3 horizontal segments, 1 number with 2 horizontal segments, and 1 number with 0 horizontal segments.  And there is only 1 number with 0 horizontal segments, that is the digit 1. And that 1 has 2 vertical segments.  So that means that from the remaining 2 digits, we need 10-2 equals 8 vertical segments from the remaining 2 digits.  So now we need to look and see which 2 of these numbers add up to 8. But notice that all of the vertical values are between 2 and 4.  And so the only way you can get 8 with 2 numbers from 2 to 4 is 4 plus 4.  And that means that the other digits must be 0 and 8 because they're the only ones with 4 vertical segments.  And that means that if we go back to our original diagram, we can remove all the numbers that are not the ones that we decided on.  0, 1, and 8, we can check that they have 5 horizontal segments from the 0 and the 8, and 10 vertical segments.  And so that means that these are the 3 digits that have exactly 5 horizontal and 10 vertical segments. The sum of these digits is 0 plus 1 plus 8, which is equal to 9.  So the answer is A, 9.

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