GMAT Diagnostic Test Question 33

Question

GMAT Diagnostic Test Question 33 
Field: geometry
Difficulty: 650

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A circle is inscribed in a half circle with a diameter of π. What is the ratio of the area of the half circle to the area not covered by the inscribed circle?

A. 1: 1
B. 1: 2
C. 1: 4
D. 3: 4
E. 2: 1

Bunuel · Answer

BELOW IS REVISED VERSION OF THIS QUESTION. A circle is inscribed right in the middle of a semicircle with a diameter of \pi as shown below. What is the ratio of the area of the semicircle to the area not covered by the inscribed circle? Semicircle 2.png A. 4:1 B. 3:2 C. 3:1 D. 4:3 E. 2:1 Semicircle.png Since the radius of the big circle ( \frac{\pi}{2} ) is twice the radius of the inscribed circle ( \frac{\pi}{4} ) then its area is 4 times greater then the area of the inscribed circle (because in the area formula the radius is squared. For example the area of a circle with radius of 2 is 4\pi , which is 4 times greater than the radius of a circle with the radius of 1, which is \pi ). Thus the are of the semicircle is 4/2=2 times greater than the area of the inscribed circle: so, the area of the semicircle is 2 units, the area of the inscribed circle is 1 unit, and the area of the semicircle not covered by the inscribed circle is also 1 unit. Ratio: 2/1. Answer: E. Hope it's clear.

KevinBrink · Answer

Dear Bunuel,

I understand the explanation, but you are adding a circle is inscribed in the MIDDLE of a semi circle. If you add MIDDLE the question clears, however this is not in the original question.

Regards

Kevin

Bunuel · Answer

The original question was revised.

TirthankarP · Answer

I downloaded the v6 of the diagnostic test (which I assume is the latest). The options for this question are different and 2:1 is not at all there as an option.
But answer still says that it is E.

Anshulmodi · Answer

Yes it still has the error; there is no such option.

Anshulmodi · Answer

I might have used a long method to answer the question, but here's how I solved it:-

Area of Semi-circle = Area of Entire Circle / 2

Area of Entire Circle = πr^2 --> π*(π/2)^2 --> π*(π^2/4) --> π^3/4

Area of Semi-circle = π^3/4 / 2 --> π^3/4*2 --> π^3/8

Area of Inscribed Circle = πr^2 --> π*(π/4)^2 --> π*(π^2/16) --> π^3/16

Area not covered by the Inscribed Circle
= Area of Semi-circle - Area of Inscribed Circle
= π^3/8 - π^3/16
= π^3/16

Finally :

Area of Semi-circle / Area not covered by the Inscribed Circle

= π^3/8 / π^3/16
= π^3/8 * 16/π^3
= 2:1

I have laid down each step in detail as I am new with typing math in text so tried my best not to good up

I hope this is also a right way to solve the problem.

Anshulmodi · Answer

Dear Bunuel,

Is the statement "A circle is inscribed right in the middle of a semicircle with a diameter of π as shown below" a little ambiguous ?

As in the first read I mis-understood that π is the  diameter of the inscribed circle  and not the semicircle.

Saabs · Answer

Please correct me if I'm wrong, but I did not do any calculations for this and answered it correctly in 45 seconds. As the area of the half circle is larger than the area not covered by the circle, the answer has to be a ratio which has a larger number on the left side. Answer E is the only option where the left number of the ratio is larger than the right.

bumpbot · Answer

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GMAT Diagnostic Test Question 33

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