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.
If any linear dimension (such as radius, side length, height, etc.) of a two-dimensional figure is multiplied by $k$ while the shape of the figure remains the same, the area of the figure is multiplied by $k^2$.  Since the new diameter is 2 times the original diameter, the new area is $2^2=4$ times the old area.  Therefore, the ratio of original area to new area is $\boxed{\frac{1}{4}}$.
\frac{1}{4}
Level 4
Prealgebra
Problem: 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.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the ratio of the original area to the enlarged area, we start by recalling the formula for the area of a circle, which is A = πr², where r is the radius. 

Let the original diameter be d. Therefore, the original radius r is d/2. The area of the original garden is A_original = π(d/2)² = π(d²/4).

The new diameter is twice the old diameter, so the new diameter is 2d. The new radius r_new is 2d/2 = d. The area of the enlarged garden is A_enlarged = π(d)² = π(d²).

Now, we need to find the ratio of the original area to the enlarged area. This is given by:

Ratio = A_original / A_enlarged = (π(d²/4)) / (π(d²)).

The π cancels out, simplifying the ratio to:

Ratio = (d²/4) / (d²) = 1/4.

Thus, the ratio of the original area to the enlarged area is 1/4. The final answer is therefore expressed as a common fraction: $\frac{1}{4}$.<|END_OF_LATENT|> The final answer is $\frac{1}{4}$.