A designer enlarged both the length and the width of a rectangular carpet by 50 percent. The new carpet was too large so the designer was asked to reduce its length and its width by 20 percent. By what percent was the area of the final carpet design greater than the area of the original design?
To find the percent by which the area of the final carpet design is greater than the area of the original design, we need to calculate the areas of both the original and final designs.
Let's assume the original length of the rectangular carpet is L and the original width is W.
When the length and width are increased by 50 percent, the new dimensions become (L + 0.5L) and (W + 0.5W), which simplify to 1.5L and 1.5W.
The area of the original design is A1 = L * W.
The area of the enlarged design is A2 = 1.5L * 1.5W = 2.25LW.
Next, the designer is asked to reduce the length and width of the enlarged carpet by 20 percent. This means the new length and width become (1.5L - 0.2 * 1.5L) and (1.5W - 0.2 * 1.5W), which simplify to 1.3L and 1.3W.
The area of the final design is A3 = 1.3L * 1.3W = 1.69LW.
Now, to find the percent increase in area, we divide the difference between the final area and the original area by the original area, and then multiply by 100.
Percent Increase = ((A3 - A1) / A1) * 100
= ((1.69LW - LW) / LW) * 100
= (0.69LW / LW) * 100
= 69%
Therefore, the area of the final carpet design is 69% greater than the area of the original design.