GMAT Sample Questions & Answers
GMAT sample questions and answers written by our admissions experts
Problem solving: Geometry, Circles
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?
Before we examine the different approaches to this question, we should have a clear idea of what is happening in this case:
There are two circles; a small one and a big one and the small one is completely contained within the big circle.
The shaded area represents the difference in the area between them:
Area of big circle = Area of small circle + shaded area
The shaded area, according to the question, is 3 times the area of the small circle:
Shaded area = 3 x area of small circle.
Substituting in the above equation:
Area of big circle = Area of small circle + 3 x area of small circle
Thus: area of big circle = 4 x area of small circle
So how many times is the circumference of the big circle greater than circumference of the small circle?
The academic approach
We need to recall the basic formulas that relate to circles:
Area of circle: A=πR^2
Circumference of circle: C=2πR
Where r is the radius of the circle.
In terms of the areas, we saw above that: area of big circle = 4 x area of small circle
We can now say that: πR_BIG^2=4× πR_SMALL^2
Cancelling p in both sides, we will have R_BIG^2=4× R_SMALL^2
Taking the square root on both sides: RBIG = 2 x RSMALL
Now, the circumference of big circle is 2πR_BIG
But if we replace RBIG with 2 x RSMALL, we will have:
Circumference of big circle is 2π2×R_SMALL.
However, π2×R_SMALL= 2πR_SMALL, which is the circumference of the small circle.
Therefore, the circumference of the big circle is 2 x the circumference of the small circle and the correct answer is C.
A simpler approach: similarity
The small and big circles have identical shape (quite obviously!). We could see the big circle as a ‘magnified version’ of the small circle. When this happens, we have ‘similarity’: all linear measures of both circles are proportional: radii, diameters, circumferences.
That means if the radius of the big circle is k times bigger than the radius of the small circle, then:
The diameter of the big circle will be k times bigger that the diameter of the small circle
The circumference of the big Circle will be k times bigger that the circumference of the small circle
What about the two areas? They will also be proportional, but to k2. Therefore:
The area of the big circle will be k2 times bigger that the area of the small circle.
Thus, in our case, k2 = 4 and k = 2. So, the circumference of the big circle is 2x the circumference of the small one. The correct answer is C.
Read More: How Can You Score 800 On The GMAT?
Problem solving: Algebra, Plug-in numbers, Quant
If 1 < x < y < z, which of the following has the greatest value?
A. z(x + 1)
B. z(y + 1)
C. x(y + z)