In The Figure Shown, If The Area of The Shaded Region is 3 Times The Area of The Smaller Circular Region GMAT Problem Solving

Question: In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

1. 4
2. 3
3. 2
4. √3
5. \(\sqrt{2}\)

‘In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide”. 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.

Solution and Explanation:

Approach Solution 1:
The problem statement provides the diagram of a shaded portion of a circular ring.
We will have to label the diagram first. As seen below, we can also use a specific formula for area of the shaded region in a ring.

To determine the area of the shaded ring we can use the formula, where a = radius of the smaller circle and b = radius of the larger circle:
Area of shaded ring = π(b^2 – a^2)
This particular problem statement mentions that the area of the shaded ring is 3 times the area of the smaller circular region. We know that the area of the smaller region is πa^2, so we can create the following equation:
π(b^2 – a^2) = 3πa^2
b^2 – a^2 = 3a^2
b^2 = 4a^2
b = 2a
Since the radius of the larger circle is twice the radius of the smaller circle, the circumference of the larger circle is also twice the circumference of the smaller circle.

Correct Answer: C

Approach Solution 2:
Large circle's area: Small circle's area?
Let the Small circle's area = x
Hence, the Shaded region's area = 3x
Large circle's area = 3x + x = 4x
Large circle circumference/ small circle's circumference?
Let Small circle's radius
r=1
Area of Small: π\(r^2\)=π
Area of Large = (4x) = (4∗π)=4π
Radius, R, of Large circle: 4π=π\(R^2\)
=>R=2
Circumference, Small: 2πr=2π
Circumference, Large: 2πR=4π
Large/Small=4π/2π=2
Hence, the large circle's circumference is two times the small circle's circumference. The correct answer is C.

Correct Answer: C

Approach Solution 3:

Let the radius of the larger circle be R and the radius of the smaller circle be r.
So, the area of the shaded region = π(R^2−r^2)

But π(R^2−r^2)=3πr^2

that implies R^2=4r^2

that implies R=2r

 Now, the ratio of the circumference of a larger circle to that of a smaller circle = 2πr/2πR​=R/r​=2r/r​=2:1

Correct Answer: C

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