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This problem trains for: SAT-I, CAHSEE, GMAT, AMC-8, Math Kangaroo 7-8, Math Kangaroo 9-10.
By scaling a circle its area has decreased by 30%. By approximately how much percent has the circumference decreased?
The shortest solution is to note that the ratio between the areas of two similar figures is equal to the square of the ratio of similarity.
Conversely, the similarity ratio is equal to the square root of the ratio of the areas.
Since the circumference is a linear measurement, it is proportional to the radius. Hence, the circumferences are in the same ratio as the similarity ratio.
and the decrease is approximately 16%
If the solution above is not easy to comprehend, here is a step-by-step longer one:
The area of the circle with radius R is:
The circumference of the same circle is:
If the final area is related to the initial area as in:
Taking the square root of both sides:
and now multiplying both sides to get the relation between circumferences:
Therefore, the new circumference is 16% smaller than the initial one.