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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.

Simply:

equation

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:

equation

The circumference of the same circle is:

equation

If the final area is related to the initial area as in:

equation

equation

Taking the square root of both sides:

equation

and now multiplying both sides to get the relation between circumferences:

equation

equation

equation

Therefore, the new circumference is 16% smaller than the initial one.