The perimeter of a square is equal to the perimeter of a rectangle. GMAT Problem-Solving

Question: The perimeter of a square is equal to the perimeter of a rectangle. The length of the rectangle is three times longer than is width having total area of 1200 sq.meter. What will be the total cost if the total area of the square is covered with stones having a dimension of 50 cm.sq. each and if Rs. 50 is charged for placing a stone in the square?

A. 300,000
B. 320,000
C. 400,000
D. 520,000
E. 350,000

Answer: B

Solution and Explanation:

Approach Solution 1:
After determining the area and perimeter of the square, find the perimeter of the rectangle. square-area conversion to square centimeters (metres); and calculate the overall cost for setting a certain number of stones on the bigger square.
1) The rectangle's perimeter
Area of a rectangle where width = x and length is three times more than width:
3x * x = 1200
3(x²) = 1200
x² = 400
X = 20
length of a rectangle: 3x = 60m
Rectangular width: 20m x
Rectangle perimeter equals 2L plus 2W, or 160m.
2) Calculate the square's area and perimeter in square centimeters
A rectangle's perimeter equals a square's perimeter, where s is the square's side.
4s = 160m
S = 40m
We require a large square's sq cm space.
Side = 40m * 100cm/ 1m = 4000cm
Area = 4000² = 16 * 10⁶ sqcm
3) Total price
To determine how many squares would fit, we must divide a large square area by the area of smaller squares. We will then multiply by a unit cost that is exactly equivalent in value to the size, in square centimeters, of the small squares.
To put it another way, we will divide the area by 50 to determine the number of tiles, and then multiply that number by 50 to determine the cost of those tiles.
As a result, the price is the whole area in rupees since (* 50) and (/ 50) return us to the starting point.
COST = 320,000 Rupees
Correct option: B

Approach Solution 2:
To answer this GMAT question, apply the data provided in the question. These issues pertain to many different branches of mathematics. This query relates to basic mathematics. It is challenging to choose the best option due to the way the options are presented. Applicants must be able to understand the proper strategy for getting the desired response. There is only one correct answer out of the five options offered.
Let's denote the side length of the square as "s" and the width of the rectangle as "w". Then, the length of the rectangle would be 3w since it is three times longer than its width.
The perimeter of the square is given by 4s, while the perimeter of the rectangle is given by 2(l + w) = 2(3w + w) = 8w, since the rectangle has two sides of length w and two sides of length 3w.
We know that these perimeters are equal, so we can set up the equation:
4s = 8w
Simplifying, we get:
s = 2w
Next, we can use the formula for the area of a rectangle to find the value of w:
Area of rectangle = length x width = (3w)(w) = 3w²
We know that the total area of the rectangle is 1200 sq meters, so:
3w² = 1200
w² = 400
w = 20
Therefore, the width of the rectangle is 20 meters, and its length is 3 times longer, or 60 meters.
Since the side length of the square is twice the width of the rectangle, it must be:
s = 2w = 2(20) = 40
So, the area of the square is:
Area of square = s² = 40² = 1600 sq meters
Now, we can calculate the number of stones needed to cover the square:
Number of stones = Area of square / Area of each stone
= 1600 sq meters / (0.5 x 0.5 sq meters)
= 6400 stones
Finally, the cost of placing each stone is Tk. 50, so the total cost would be:
Total cost = Number of stones x Cost per stone
= 6400 stones x Tk. 50/stone
= Tk. 320,000
Correct option: B

Approach Solution 3:
Let's start by finding the dimensions of the rectangle. We know that its area is 1200 sq meters, so we can set up an equation:
length x width = 1200
Since the length is three times longer than the width, we can substitute 3w for the length:
3w x w = 1200
Simplifying, we get:
3w² = 1200
w² = 400
w = 20
Therefore, the width of the rectangle is 20 meters, and its length is 3 times longer, or 60 meters.
Now, we can use the fact that the perimeter of the square is equal to the perimeter of the rectangle. The perimeter of the square is 4s, and the perimeter of the rectangle is 2(l + w) = 2(3w + w) = 8w. So:
4s = 8w
Substituting w = 20, we get:
4s = 8(20)
s = 40
Therefore, the side length of the square is 40 meters.
Next, we can calculate the number of stones needed to cover the square. The area of the square is:
Area of square = s² = 40² = 1600 sq meters
Each stone has an area of 50 cm x 50 cm = 0.25 sq meters. So, the number of stones needed is:
Number of stones = Area of square / Area of each stone
= 1600 sq meters / 0.25 sq meters
= 6400 stones
Finally, the cost of placing each stone is Tk. 50, so the total cost would be:
Total cost = Number of stones x Cost per stone
= 6400 stones x Tk. 50/stone
= Tk. 320,000
Therefore, the total cost of covering the square with stones is Tk. 320,000.
Correct option: B

“The perimeter of a square is equal to the perimeter of a rectangle." - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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