Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

A football match between teams from North Barrican and South Barrican is played in a stadium that has a rectangular array of seats for the spectators.  There are 11 North Barrican supporters in each row, and 14 South Barrican supporters in each column. This leaves 17 empty seats.  What is the smallest possible number of seats in the stadium?  Since our seats are in a rectangular array, we know that there are some number of rows and some number of columns. So let's say that there are rows and c columns.  Now we can count the number of seats in two different ways. One way is just r times c because it's a rectangular array.  The other way we have 11 times r plus 14 times c plus 17 extra seats. So those have to be equal because they're both counting the number of seats.  Now let's subtract over these first two to get this right here. Now this looks like it almost factors. It's like really, really close to factoring.  It just is missing a constant term at the end. So I'm just going to add in 11 times 14 on both sides so that we can factor this.  If you want to look it up, this trick is called Simon's favorite factoring trick. So now this side is going to factor as r minus 14 times c minus 11.  And we can check that we get rc minus 11r minus 14c and then plus 11 times 14. On this side, the 171 stays there.  So since r and c are both integers, we know that this quantity is an integer and this quantity is an integer. And they both divide 171.  So we have a few ways to create 171, and they're like this. We have six different ways. Now we just need to try out all of these and see which one minimizes the number of seats.  Well, if we choose this one, then c is 171 plus 11, and that's just going to be huge. Like that times 1 plus 14 is going to be gigantic. Similar story for this one over here.  Also for this one and for this one. So really the only two we need to try are these two. So let's try r minus 14 equals 9 and c minus 11 equals 19.  And then try r minus 14 equals 19 and c minus 11 equals 9. Well we try out both of those options. In the first case we get 690 seats, and in the second case we get 660 seats.  So our final answer is 660.

Video Summary

The video transcript describes a math problem involving seat arrangement in a stadium. North Barrican fans occupy 11 seats per row, and South Barrican fans occupy 14 seats per column, leaving 17 seats vacant. The challenge is to determine the minimum number of seats using given conditions and equations. By equating two ways of counting seats (\(r \cdot c\) and \(11r + 14c + 17\)), the problem is transformed using Simon's favorite factoring trick. By testing integer solutions for \(r\) and \(c\) that minimize \(r \cdot c\), the smallest possible seat count is found to be 660.

Keywords

math problem

seat arrangement

Simon's favorite factoring trick

integer solutions

smallest seat count