A rectangle has integer side lengths. One pair of opposite sides is increased by $30\%$ and the other pair of sides is decreased by $20\%$. The new side lengths are also integers. What is the smallest possible area, in square units, of the new rectangle?
I want to find the smallest possible area of the new rectangle, so I need to minimize the product of the new side lengths. Let me call the original side lengths $x$ and $y$, where $x$ and $y$ are positive integers. Then the new side lengths are $1.3x$ and $0.8y$, which are also integers. This means that $x$ must be a multiple of $10$ and $y$ must be a multiple of $5$. The smallest values of $x$ and $y$ that satisfy these conditions are $x = 10$ and $y = 5$. The area of the new rectangle is then $1.3x \cdot 0.8y = 1.3 \cdot 10 \cdot 0.8 \cdot 5 = 52$ square units. 