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16 Sep 2014, 00:58
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If a point is randomly chosen inside a circle with a radius of \(r\), what is the probability that the distance from this point to the center of the circle will be larger than \(\frac{r}{2}\)? (The area of a circle is \(\pi r^2\).)
A. \(\frac{1}{2}\)
B. \(\frac{3}{4}\)
C. \(\frac{7}{8}\)
D. \(\frac{1}{4}r^2\)
E. \(\frac{3}{4}r^2\)
A. \(\frac{1}{2}\)
B. \(\frac{3}{4}\)
C. \(\frac{7}{8}\)
D. \(\frac{1}{4}r^2\)
E. \(\frac{3}{4}r^2\)
16 Sep 2014, 00:58
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Official Solution:
If a point is randomly chosen inside a circle with a radius of \(r\), what is the probability that the distance from this point to the center of the circle will be larger than \(\frac{r}{2}\)? (The area of a circle is \(\pi r^2\).)
A. \(\frac{1}{2}\)
B. \(\frac{3}{4}\)
C. \(\frac{7}{8}\)
D. \(\frac{1}{4}r^2\)
E. \(\frac{3}{4}r^2\)
In order for a point to be farther than \(\frac{r}{2}\) from the center, it must fall within the region enclosed by the circumferences of two circles: one with a radius of \(r\) and the other with a radius of \(\frac{r}{2}\):
The probability we seek is the ratio of the area of this region to the area of the larger circle. The area of the region can be calculated as follows: Area of the region = \(\pi r^2 - \pi \left(\frac{r}{2}\right)^2 = \frac{3}{4} \pi r^2\). The area of the larger circle is \(\pi r^2\). Thus, the ratio equals \(\frac{3}{4}\).
Note that this is not a Geometry question. While it uses basic knowledge of lines and figures, it is a Probability question. There are 8 questions within GMAT Prep Focus Edition that use similar principles. Here is one example.
Answer: B
If a point is randomly chosen inside a circle with a radius of \(r\), what is the probability that the distance from this point to the center of the circle will be larger than \(\frac{r}{2}\)? (The area of a circle is \(\pi r^2\).)
A. \(\frac{1}{2}\)
B. \(\frac{3}{4}\)
C. \(\frac{7}{8}\)
D. \(\frac{1}{4}r^2\)
E. \(\frac{3}{4}r^2\)
In order for a point to be farther than \(\frac{r}{2}\) from the center, it must fall within the region enclosed by the circumferences of two circles: one with a radius of \(r\) and the other with a radius of \(\frac{r}{2}\):
The probability we seek is the ratio of the area of this region to the area of the larger circle. The area of the region can be calculated as follows: Area of the region = \(\pi r^2 - \pi \left(\frac{r}{2}\right)^2 = \frac{3}{4} \pi r^2\). The area of the larger circle is \(\pi r^2\). Thus, the ratio equals \(\frac{3}{4}\).
Note that this is not a Geometry question. While it uses basic knowledge of lines and figures, it is a Probability question. There are 8 questions within GMAT Prep Focus Edition that use similar principles. Here is one example.
Answer: B
General Discussion
23 Aug 2017, 19:48
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I think this is a high-quality question and I agree with explanation.
