The Area of a Rectangle and the Square of its Perimeter are in the GMAT Problem Solving

Question: The area of a rectangle and the square of its perimeter are in the ratio 1 : 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio

(A) 1:4
(B) 2:9
(C) 1:3
(D) 3:8
(E) 4:9

Correct Answer: A
Solution and Explanation:
Approach Solution 1:
Let the length and breadth be x and y..
so Area = xy and perimeter = 2(x+y)..
Area: square of perimeter = 1:25 =>
\(xy:(2(x+y)^2= 1:25..........4(x+y)^2= 25xy.......x^2+y^2+2xy/xy= 25/4\)
\(x^2/xy+y^2/xy+2xy/xy= 25/4 \)
\(x/y+y/x+2= 25/4 \)

Let the ratio we are looking for x/y be a
\(a+1/a= 25/4−17/4..........a^2/1/a= 17/4..........4a^2+4= 17a..........4a^2−17a+4= 0 \)
\(4a^2−16-a+4= 0...........4a(a−4)−1(a−4)=0......(4a−1)(a−4)= 0 \)
So either a= 4 or a= 1/4

Approach Solution 2:
Perimeter of rectangle= 2(a+b)
Area of rectangle= ab
\([2(a+b)]^2/ab= 25/1\)
\(4(a^2+b^2+2ab)/ab= 25/1\)
\(a/b+b/a+2= 25/4\)
\(a/b+b/a= 17/4\)
\(a/b+b/a= 4/1+¼\)
\(b/a= ¼\)

Approach Solution 3:
Let 'a' and 'b' be the length of sides of the rectangle, (a > b)
Area of the rectangle = a*b
Perimeter of the rectangle = 2*(a+b)
=>a*b/(2*(a + b))2 = 1/25
=> 25ab = 4(a + b)2
=> 4a2 - 17ab + 4b2 = 0
=> (4a - b)(a - 4b) = 0
=> a = 4b or b/4

We initially assumed that a > b, therefore a ≠ 4.
Hence, a = 4b
=> b: a = 1: 4

“The area of a rectangle and the square of its perimeter are in the”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Review".To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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