0,(12) : 0,1(6) =x:0, (3)

To solve this equation, we first need to understand the notation being used. The notation seems to represent the numbers in a base, with parentheses indicating a repeating decimal.

The equation appears to be:

\[ 0.121212... : 0.166666... = x : 0.33333... \]

Let's convert these repeating decimals into fractions:

1. Convert \(0.121212...\) to a fraction: Let \(y = 0.121212...\). Then, \(100y = 12.121212...\). Subtracting the original \(y\): \[ 100y - y = 12.121212... - 0.121212... \] \[ 99y = 12 \implies y = \frac{12}{99} = \frac{4}{33} \]

2. Convert \(0.166666...\) to a fraction: Let \(z = 0.166666...\). Then, \(10z = 1.66666...\). Subtracting gives: \[ 10z - z = 1.66666... - 0.16666... \] \[ 9z = 1 \implies z = \frac{1}{9} \]

3. Convert \(0.33333...\) to a fraction: \(0.33333... = \frac{1}{3}\).

Now substituting these fractions back into our equation:

\[ \frac{\frac{4}{33}}{\frac{1}{9}} = \frac{x}{\frac{1}{3}} \]

To simplify the left side:

\[ \frac{4}{33} \div \frac{1}{9} = \frac{4}{33} \times 9 = \frac{36}{33} = \frac{12}{11} \]

So we have:

\[ \frac{12}{11} = \frac{x}{\frac{1}{3}} \]

Now cross-multiplying to solve for \(x\):

\[ 12 \cdot \frac{1}{3} = 11x \implies 4 = 11x \implies x = \frac{4}{11} \]

Thus, the solution to the equation is:

\[ x = \frac{4}{11} \]