A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]

draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));

fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));

fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,gray(.8));

fill((7,1)--(8,1)--(8,2)--(7,2)--cycle,gray(.8));

fill((1,4)--(2,4)--(2,5)--(1,5)--cycle,gray(.8));

fill((3,3)--(6,3)--(6,6)--(3,6)--cycle,gray(.8));

fill((7,4)--(8,4)--(8,5)--(7,5)--cycle,gray(.8));

fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,gray(.8));

fill((4,7)--(5,7)--(5,8)--(4,8)--cycle,gray(.8));

fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,gray(.8));

label("T",(1.5,7),S);

label("S",(6,4.5),W);

[/asy] If the ratios $9:\text{S}$ and $\text{S}:\text{T}$ are both equal to 3 and $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares, what is the total shaded area?
We are given that $\frac{9}{\text{S}}=\frac{\text{S}}{\text{T}}=3.$ \[\frac{9}{\text{S}}=3\] gives us $S=3,$ so \[\frac{\text{S}}{\text{T}}=3\] gives us $T=1$. There are 8 shaded squares with side length $\text{T}$ and there is 1 shaded square with side length $\text{S},$ so the total shaded area is $8\cdot(1\cdot1)+1\cdot(3\cdot3)=8+9=\boxed{17}.$
17
Level 5
Prealgebra
Problem: A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]

draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));

fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));

fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,gray(.8));

fill((7,1)--(8,1)--(8,2)--(7,2)--cycle,gray(.8));

fill((1,4)--(2,4)--(2,5)--(1,5)--cycle,gray(.8));

fill((3,3)--(6,3)--(6,6)--(3,6)--cycle,gray(.8));

fill((7,4)--(8,4)--(8,5)--(7,5)--cycle,gray(.8));

fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,gray(.8));

fill((4,7)--(5,7)--(5,8)--(4,8)--cycle,gray(.8));

fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,gray(.8));

label("T",(1.5,7),S);

label("S",(6,4.5),W);

[/asy] If the ratios $9:\text{S}$ and $\text{S}:\text{T}$ are both equal to 3 and $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares, what is the total shaded area?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the side lengths S and T, we start with the given ratios. The ratio 9:S = 3 implies that S = 9/3 = 3 feet. Next, we use the ratio S:T = 3. Substituting S, we have 3:T = 3, which gives T = 3/3 = 1 foot. 

Now, we calculate the areas of the shaded squares. The area of the large shaded square with side length S is S^2 = 3^2 = 9 square feet. Each of the eight smaller shaded squares has a side length T, so the area of one small square is T^2 = 1^2 = 1 square foot. 

Since there are eight smaller squares, their total area is 8 * 1 = 8 square feet. 

Finally, we sum the areas of the large shaded square and the eight smaller squares to find the total shaded area: 9 + 8 = 17 square feet. 

Thus, the total shaded area is 17.<|END_OF_LATENT|> The final answer is $17$.