A third ant walks around the perimeter of another rectangle sheet of paper. The dimensions of this sheet of paper are given in decimal numbers expressed to the hundredth place, with the tenths and hundredths digits being non-zero. If the ant travel between 18 and 19 centimeters, what are the dimensions of the sheet of paper? Give one possibility.

Answer 1

Let the length and width of the rectangle be $x$ and $y$ centimeters, respectively. The perimeter of the rectangle is $2x+2y$ centimeters, so the ant walks a distance of $2x+2y$ centimeters around the perimeter. We know that $18 < 2x + 2y < 19$.

Since $x$ and $y$ are given as decimal numbers expressed to the hundredth place, we know that they can be written in the form $a.b$, where $b$ is not equal to zero. Expressing $2x+2y$ in this form, we have $$2x+2y=2(a+b)+2(b+c)=2a+4b+2c.$$However, we want to express $2x+2y$ in the form $x.y$, where $y$ is not equal to zero. Taking the product of both sides by $0.1$ (which will shift the digits one place to the right), we obtain $$0.2x+0.2y=0.2a+0.4b+0.2c=a.b,$$from which it follows that $a+b$ is equal to $2a$, or $a=b$. Since neither $a$ nor $b$ can be zero, the only possibility is $a=b=1$. We plug this value back into the equation $0.2x+0.2y=0.2a+0.4b+0.2c$ and obtain $0.2x+0.2y=0.6$. Thus, $x+y=\boxed{3}.$