### Instruction: A circular garden is enlarged so that the new diameter is twice the old diameter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction.### Response: To solve this problem, I need to recall the formula for the area of a circle, which is A = pi * r^2, where r is the radius.
Since the diameter is twice the original, that means the radius is also twice the original.
Let me label the old radius as r and the new radius as 2r.
Then the old area is A_old = pi * r^2 and the new area is A_new = pi * (2r)^2.
I want to find the ratio of the original area to the enlarged area, which is A_old / A_new.
I can simplify this expression by canceling out the pi factors and expanding the square.
A_old / A_new = (pi * r^2) / (pi * (2r)^2) = (pi * r^2) / (pi * 4r^2) = r^2 / 4r^2 = 1/4.
So the ratio of the original area to the enlarged area is 1/4.