Number puzzle - Constructing border around a rectangle

Question

If I construct a large rectangle from a certain number of smaller rectangles- x across and y down i.e. total number of rectangles is xy-, now if I was to construct a border around this large rectangle using the same number of smaller rectangles. How many smaller rectangles did I use?

I've got the relationship xy = 2x + 2y + 4, but I am struggling to find the other relationship.

Any help will be much appreciated, thank you.

Answer 1

For A is the total area inclusive the border your total area will be:

A = (x + 2) * (y + 2)

A = xy + 2x + 2y + 4

Edit:

If you would like to know the relationship between x and y for the function $xy = 2x + 2y + 4$ you need to separate one of the variables:

$xy - 2y = 2x + 4$

$y(x -2) = 2x + 4$

$y = \frac{2x + 4}{x - 2} \;\text{for} \; x>2$

Answer 2

You are quite right - there is only one equation!
But you know that $x$ and $y$ are positive whole numbers.
Also, if $x$ and $y$ get too big, then $xy$ will be too big for $2x+2y+4$.
Hunt around, try a few values of $x$ and $y$ until you find some that work.