If the perimeter of square region S and the perimeter of circular region C are equal, then the ratio of the area of S to the area of C is closest to:


A. \(\frac {3}{2}\)

B. \(\frac {4}{3}\)

C. \(\frac {3}{4}\)

D. \(\frac {2}{3}\)

E. \(\frac {1}{2}\)

If Perimeter of square = Perimeter of Circle, then:
4a = 2\(\pi\)r, where a = side of square and r is radius of the circle
a/r = \(\pi\)/2

Area of S/Area of C = \(a^2\)/ \(\pi\) \(r^2\)
= \((\pi/2)^2\) * 1/\(\pi\)
= \(\pi\)/4 = 3.14/4
= \(\approx\) 3/4 = C

Hi All,

This question is perfect for TESTing VALUES.

We're told that the PERIMETER of a square is equal to the PERIMETER (meaning the 'circumference') of a circle. We're asked to figure out the approximate ratio of the area of the square to the area of the circle.

Let's TEST VALUES. Since we're dealing with a circle, let's work "pi" into our math right from the beginning....

Perimeter = 4pi

For the Square:
Perimeter = 4pi
Side length = pi
Area = (pi)^2

For the Circle:
Perimeter = 4pi
Radius = 2
Area = 4pi

Area of Square/Area of Circle = (pi)^2/4pi = pi/4 = about 3.14/4 = about 3/4

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Equating the perimeter of square (4x, where x is the length of the side) and the perimeter of the circle (circumference = \(2\pi{r}\)), we get \(4x=2\pi{r}\), which gives \(x=\frac{\pi{r}}{2}\).

Hope it's clear.
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\(x=\frac{\pi{r}}{2}\).

\(\frac{A_{square}}{A_{circle}}=\frac{x^2}{\pi{r^2}}=\frac{(\frac{\pi{r}}{2})^2}{\pi{r^2}}=\frac{\frac{\pi^2 r^2}{4}}{\pi{r^2}}=\frac{\pi^2{r^2}}{4\pi{r^2}}=\frac{\pi}{4}\approx(\frac{3}{4})\).

Hope it helps.
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Hi ameyaprabhu,

The questions that you'll face on the Official GMAT are almost all based on patterns of some type (even if you don't immediately realize that a pattern is there). In the Quant section, TESTing VALUES can often be used to define a pattern, so you might try using that approach in these types of situations. My explanation (2 posts above your post) shows how to approach this prompt in that way.

GMAT assassins aren't born, they're made,
Rich
_________________

Solution

Given:

• The perimeter of square region S and the perimeter of circular region C are equal.

To find:

• The ratio of the area of S to the area of C is closest to which option among the given ones.

Approach and Working:

• Let side of square region S is a and radius of circular region C is r.

o Perimeter of square = Perimeter of circle
o 4a= 2*pi*r
o 2a= pi*r
o \(a= \frac{{pi * r}}{2}\)

• Area of S : Area of C = \(a^2 :pi * r^2\)

o = \(\frac{{pi^2 * r^2}}{4} : pi * r^2\)
o = \(\frac{pi}{4}\)= ¾

Hence, the correct answer is option C.

Answer: C
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