Question

A poster advertising a student election candidate is too large according to the election rules. The candidate is told she must reduce the length and width of the poster by \(20.0 \% .\) By what percentage will the area of the poster be reduced?

Short answer

Answer: The area of the poster is reduced by 36%.

Step-by-step solution

Assume initial length and width

Let's assume the initial dimensions of the poster are \(l\) and \(w\).

Calculate the reduced length and width

To reduce the dimensions by \(20\%\), we multiply the initial length and width by \(0.8\). The new reduced dimensions would be: \(l_{new} = 0.8l\) \(w_{new} = 0.8w\)

Calculate the initial and new area

The initial area of the poster is given by \(A_{old} = lw\). The new area after reducing the dimensions is given by \(A_{new} = l_{new}w_{new}\).

Calculate the percentage of area reduced

To calculate the percentage reduction in the area, we will find the difference in the areas, and then divide this difference by the initial area, and finally multiply by \(100\%\). The percentage reduction in area can be calculated as follows: Percentage Reduction \(= \frac{A_{old} - A_{new}}{A_{old}} \times 100\%\)

Substituting values and finding the percentage of area reduced

Now, we substitute the values obtained in the previous steps into the formula for the percentage reduction in the area: Percentage Reduction \(= \frac{lw - (0.8l)(0.8w)}{lw} \times 100\%\) Percentage Reduction \(= \frac{lw - 0.64lw}{lw} \times 100\%\) Percentage Reduction \(= (1 - 0.64) \times 100\%\) Percentage Reduction \(= 0.36 \times 100\%\) Thus, the poster's area will be reduced by \(36\%\).