A Circle with a Radius R is Inscribed into a Square with a Side K.

Question: A circle with a radius R is inscribed into a square with a side K. if the ratio of the area of square to the area of the circle is P and the ratio of the perimeter of the square to that of the circle is Q, which of the following must be true?

1. \(\frac{P}{Q}>1\)
2. \(\frac{P}{Q}=1\)
3. \(1>\frac{P}{Q}>\frac{1}{2}\)
4. \(\frac{P}{Q}=\frac{1}{2}\)
5. \(\frac{P}{Q}<\frac{1}{2}\)

“A circle with a radius R is inscribed into a square with a side K.” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer

Approach 1

Let us assume any number, K = 6

For the inscribed circle, we will get:

\(R=\frac{K}{2}=\frac{6}{2}=3\)

So, now we will find the area of the square.

Area of square = \((side)^2\)

Area of square = \((6)^2\) = 36

Area of circle = \(\pi*R^2\)= \(\pi*(3)^2\) = 9\(\pi\)

So, according to the question:

P = \(\frac{AreaofSquare}{Areaofcircle}=\frac{36}{9\pi}=\frac{4}{\pi}\)

Perimeter of the Square = 24

Perimeter of the circle = \(2\pi*R=6\pi\)

So, the value of Q = \(\frac{Perimeterofsquare}{Perimeterofcircle}=\frac{24}{6\pi}=\frac{4}{\pi}\)

Therefore,

\(\frac{P}{Q}=1\)

Correct option: B

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