The average of Amy's, Ben's, and Chris's ages is 9. Four years ago,  Chris was the same age as Amy is now. In 3 years, Ben's age will be $\frac{2}{3}$ of Amy's age at that time. How many years old is Chris now?
Let Amy's, Ben's, and Chris's ages be $a$, $b$, and $c$, respectively. We have the equations \begin{align*} \tag{1}
\frac{a+b+c}{3}=9 \Rightarrow a+b+c&=27 \\ \tag{2}
c-4&=a\\ \tag{3}
b+3&=\frac{2}{3}(a+3)
\end{align*} From Equation (3), we have $b=\frac{2}{3}(a+3)-3$. We substitute Equation (2) into Equation (3) to eliminate $a$, to get $b=\frac{2}{3}(c-1)-3$. Substituting this last equation and Equation (2) into Equation (1) to eliminate $a$ and $b$, we have  \[[c-4]+[\frac{2}{3}(c-1)-3]+c=27\] Solving for $c$, we find that $c=13$. Thus, Chris's age is $\boxed{13}$.
13
Level 4
Algebra
Problem: The average of Amy's, Ben's, and Chris's ages is 9. Four years ago,  Chris was the same age as Amy is now. In 3 years, Ben's age will be $\frac{2}{3}$ of Amy's age at that time. How many years old is Chris now?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>Let A be Amy's current age, B be Ben's current age, and C be Chris's current age. The average age of Amy, Ben, and Chris is given as 9, which leads to the equation:

(A + B + C) / 3 = 9. 

Multiplying both sides by 3 gives:

A + B + C = 27. 

Next, we consider the information about Chris's age. Four years ago, Chris's age was C - 4, and at that time, he was the same age as Amy is now, which gives us:

C - 4 = A. 

From this, we can express A in terms of C:

A = C - 4. 

Now, we analyze the relationship between Ben's and Amy's ages in three years. In three years, Amy's age will be A + 3, and Ben's age will be B + 3. The problem states that Ben's age will be $\frac{2}{3}$ of Amy's age at that time:

B + 3 = (2/3)(A + 3). 

We can rearrange this equation to express B in terms of A:

B + 3 = (2/3)A + 2. 

Subtracting 3 from both sides gives:

B = (2/3)A - 1. 

Now we have expressions for A and B in terms of C. We substitute A = C - 4 into the equation for B:

B = (2/3)(C - 4) - 1. 

Expanding this gives:

B = (2/3)C - (8/3) - 1 = (2/3)C - (8/3) - (3/3) = (2/3)C - (11/3). 

Now we have A, B, and C expressed in terms of C:

A = C - 4, 
B = (2/3)C - (11/3). 

Substituting these expressions into the equation A + B + C = 27:

(C - 4) + ((2/3)C - (11/3)) + C = 27. 

Combining like terms gives:

C - 4 + (2/3)C + C - (11/3) = 27. 

This simplifies to:

(1 + 1 + 2/3)C - 4 - 11/3 = 27. 

Converting 1 to a fraction gives:

(3/3 + 3/3 + 2/3)C - 4 - 11/3 = 27, 

which simplifies to:

(8/3)C - 4 - 11/3 = 27. 

Now, we convert -4 to a fraction:

(8/3)C - (12/3) - (11/3) = 27. 

Combining the constants gives:

(8/3)C - (23/3) = 27. 

Adding (23/3) to both sides results in:

(8/3)C = 27 + (23/3). 

Converting 27 to a fraction gives:

(8/3)C = (81/3) + (23/3) = (104/3). 

Multiplying both sides by (3/8) gives:

C = (104/3) * (3/8) = 104/8 = 13. 

Thus, Chris's current age is 13.<|END_OF_LATENT|> The final answer is $13$.