Grades 11-12 Video Solutions 2021-video 2021 11-12/17

Video Transcription

Problem number 17 states, in the 5x5 square shown, the sum of the numbers in each row and in each column is the same.  There is a number in every cell, but some of the numbers are not shown. What is the number in the cell marked with a question mark?  So here we have our original grid. And let's look at the rows and columns that are populated.  We know that the sum of the values in each row and column must be the same, so we'll call these sums a, b, c, and d. And we can set them equal to one another.  Let's label all the unknown values in each row and column as w, x, y, and z. So column A has a sum of 16, w, 12, 25, y, and 4.  Column B has a sum of 22, x, 1, z, and the mystery number. Row C has a sum of 20, w, 21, x, and 2. And row D has a sum of 24, y, 5, z, and 6.  So now with all of our equations, let's try and set up a system of equations such that all of the unknowns will cancel out.  Since A is equal to B, which is equal to C, which is equal to D, we can say that A plus B equals C plus D.  And since there exists one of each unknown variable, w, x, y, z, in each of those sets of equations, setting them equal to one another will make them cancel out.  So when we substitute in the variables, this is what we get. And all the variables are going to cancel out, and we get 68 plus the mystery number is equal to 78.  So the mystery number is 10.  So the question asked us, what is the number in the cell marked with a question mark? The answer is 10, letter B.  .

Video Summary

The problem involves a 5x5 grid where the sums of numbers in each row and column are equal, but some numbers are missing. To find the number represented by a question mark, the speaker labels the unknown numbers as variables and sets up equations based on the given sums. By organizing and equating these sums, a system of equations is created, eventually showing that the mystery number is 10. The answer to the problem is that the number in the cell marked with a question mark is 10.