An easy algebric question

Given that $x$ and $y$ are the length and width of a rectangle, and that we are finding the relationship between $x$ and $y$ on the area of a rectangle, we can assume that:

$A=xy$

So, since this is an initial condition, let's change this to:

$A_1=x_1 y_1$

In the second case, $x_1$ is increased by $10\%$ , and $y_1$ is decreased by $10\%$ . So, for $x_2$ and $y_2$ , we have:

$x_2 = x_1(1.00+0.10) = (1.10)x_1$

$y_2 = y_1(1.00-0.10) = (0.90)y_1$

So, if we assume that $A_2=x_2 y_2$ , then plugging these values into that equation gives:

$A_2=(1.10)x_1 (0.90)y_1 = (0.99)x_1y_1$

So, since $x_1 y_1=A_1$ , we can find the relationship between the two areas, which is:

$A_2 = (0.99)A_1$

So, the new area is $99\%$ of the old one, meaning that the area decreased by $1\%$ .