I have a large number of toy soldiers, which I can arrange into a rectangular array consisting of a number of rows and a number of columns. I notice that if I remove 100 toy soldiers, then I can arrange the remaining ones into a rectangular array with 5 fewer rows and 5 more columns. How many toy soldiers would I have to remove from the original configuration to be able to arrange the remaining ones into a rectangular array with 11 fewer rows and 11 more columns?
This is one of the last questions in a Mathematics Competition that I attempted. I honestly have no clue where to start when solving this question, I asked several maths teachers at my school, but they still haven't managed to find an answer. Any help with this question is appreciated.
Edit: Since I was asked to show how I attempted to solve this question, here it is.
x = number of soldiers, r = rows, c = columns
In the original configuration, the statement x = r * c is true. In the second configuration, the statement x - 100 = (r - 5) * (c + 5) is true. This statement simplified is x = cr + 5r - 5c + 75. In both of these cases, I have three unknown variables. I don't know where to go from here.
You have a sign wrong in one equation, so you should actually have the equations
$$n=rc\\ n-100=(r-5)(c+5)$$
where $r$, $c$ are the number of rows and columns, and $n$ is the total number of soldiers. Substituting $n$ gives:
$$rc-100=(r-5)(c+5)\\ rc-100=rc-5c+5r-25\\ c-r=15$$
What the question asks for is how many soldiers are removed when you reduce the number of rows by $11$ and increase the number of columns by $11$. So you want to know:
$$n-(r-11)(c+11) \\ = rc-(rc-11c+11r-121)\\ = 11c-11r+121\\ = 11(c-r)+121$$
But we already know $c-r=15$, so you need to remove $11*15+121 = 286$.