Track / Stage 5 / 89 of 400 #689 of 2000

Problem 689

AIME late

Algebra Difficulty 5.2 Find the answer

3. (16 points) In a garden plot, it was decided to create a rectangular flower bed. Due to a lack of space, the length of the flower bed was reduced by $10 \%$ , and the width was reduced by $20 \%$ . As a result, the perimeter of the flower bed decreased by $12 \%$ . However, this was not enough, so it was decided to reduce the length by $20 \%$ and the width by $10 \%$ . By what percentage did the perimeter of the rectangular flower bed decrease from the original version?

Your answer

Official solution

Answer: 18.

Solution. Let $x$ be the length of the flower bed, $y$ be the width of the flower bed. After the reduction: $0.9 x$ - length of the flower bed, $0.8 y$ - width of the flower bed, $2(0.9 x + 0.8 y)$ - perimeter. We get the equation: $2(0.9 x + 0.8 y) = 0.88 \cdot 2(x + y)$ or $x = 4 y$ . The original perimeter: $10 y$ . After the second reduction: $0.8 x = 3.2 y$ - length of the flower bed, $0.9 y$ - width of the flower bed, $2(3.2 y + 0.9 y) = 8.2 y$ - perimeter. The perimeter decreased by $1.8 y$ or by $18\%$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.