The area of a rectangle has increased by 5%. The length of one side was decreased by 10%. By what percentage was the other side increased? Give your answer correct to the nearest integer.
Let the original length of one side of the rectangle be $x$. Since the area has increased by 5%, the new area of the rectangle is $1.05$ times the original area, or $x(x) = 1.05x^2$. Dividing both sides by $1.05$, we find $x^2 = \frac{1}{1.05}$, so $x = \sqrt{\frac{1}{1.05}}$. Therefore, the new length of that side is $0.9767x$, which is a decrease of $100\%-97.67\% = 2.33\%$. Since the length of the other side has increased proportionally, the answer is $\boxed{2}$.
Let's assume the original length of the rectangle is represented by L, and the original width is represented by W.
The area of a rectangle is given by the formula A = L * W.
When the area is increased by 5%, the new area becomes 1.05 times the original area. Mathematically, this can be written as:
1.05 * A = L * W
Now, if one side (let's say the length) is decreased by 10%, the new length becomes 0.9 times the original length. Mathematically, we can write this as:
New Length = 0.9 * Original Length
Let's substitute this value in the equation for the new area:
1.05 * A = (0.9 * L) * W
Next, let's divide both sides of the equation by the original area (A) to simplify it:
1.05 = 0.9 * (L / A) * W
Now, let's find the value of (L / A) by rearranging the equation:
(L / A) = (1.05 / 0.9) * (1 / W)
(L / A) = 1.16667 * (1 / W)
(L / A) ≈ 1.16667 * (1 / W)
Finally, we can find the percentage increase of the other side (width) by subtracting 1 from (L / A) and multiplying by 100 to get the percentage:
Percentage Increase ≈ ((L / A) - 1) * 100
Percentage Increase ≈ (1.16667 * (1 / W) - 1) * 100
Therefore, the other side (width) is increased by approximately 17% (rounded to the nearest integer).