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

Video Transcription

Problem number 12. Justin can make a 4 by 4 square where the sum of the numbers in all four rows and all four columns is the same from the three pieces shown and  one additional piece. Which of the following pieces is needed to complete his square? So the first thing to notice is that there is one of the pieces with  a single row of four highlighted here and if we imagine fitting this piece into a 4 by 4 square, we notice that this piece, this row of four, must always  be either a row or a column in the final grid. And so that means that the common sum or the common column sum in the final 4 by 4 grid is equal to the sum of  these four numbers. The sum of these four numbers is 2 plus 1 plus 3 plus 1 equals 7. So that means that all four rows and all four columns in Justin's  grid must sum to 7. And so that means that since the grid has four rows, then the sum of all the numbers in the grid, which is equal to the sum of the four  rows in the grid, is equal to 4 times 7, which is 28. Because remember, all of the four rows have the same sum. And using this information, we can calculate the  sum of the numbers on the missing piece because we know the sum of the numbers in the whole grid and we know the sum of the numbers in each of the three pieces.  We can see that the missing piece sum is 28 minus 7 minus 8 minus 8, which is equal to 5. And those 7, 8, and 8 come from the totals of each of those three  pieces shown. We can compute the sum of the numbers in each of the options and see that they are 5, 3, 4, 6, and 7. We need the missing piece to sum to 5, and  the only one that does that is A.

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