Question

The length of the edging that surrounds circular garden K is $\frac{1}{2}$ the length of the edging that surrounds circular garden G. What is the area of garden K? (Assume that the edging has negligible width.)
(1) The area of G is $25\pi$ square meters.
(2) The edging around G is $10\pi$ meters long.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is D Each statement alone is sufficient.

• It is to be noted that the length of the edging around a circular garden is equal to the circumference of the circle. The formula for the circumference of a circle, where C is the circumference and d is the diameter, is . The formula for the area of a circle, where A is the area and r is the radius, is . In any circle, r is equal to . If the length of the edging around K is equal to the length of the edging around G, then the circumference of K is equal to the circumference of G.

1. Since the area of G is square meters, ; so $r=5$. So, if the radius of G is $5$, the diameter is $10$, and the circumference of G is equal to . Since the circumference of K is half that of G, then the circumference of K is , making the diameter of K equal to $5$. If the diameter of K is $5$, the radius of K is $2.5$, and the area of K is or SUFFICIENT.
2. If the edging around G is meters long, then the circumference of G is . The area of K can then by found by proceeding as in (1); SUFFICIENT.

• The correct answer is D; each statement alone is sufficient.