I am thinking of a non-negative number in the form of a fraction with an integer numerator and a denominator of 12. When I write it as a decimal number, it will have one digit before and one digit after the decimal point, and both of these digits will be non-zero. There are multiple numbers that have both of these properties. However, if I arrange them from smallest to largest, the one I am thinking of will be the second to last.

What number am I thinking of?
Given that the denominator is 12, the numerator must be divisible by three to yield a number with a terminating decimal. After factoring out three, the denominator becomes 4. Then, the decimal places would be the following groups: $25, 50, 75$ and 00. The condition is satisfied only by the group 50, so the numerator must be an odd multiple of six. To have one non-zero digit before the decimal point, the numerator must be greater than 12 and less than 120, i.e., 18, 30, 42, .., 102, 114. The sought number is thus $\frac{102}{12}=8.5$.

Another solution. Fractions with a denominator of 12 can be expressed as mixed numbers in the form $a \frac{b}{12}$, where $1 \leqq a \leqq 9$ and $1 \leqq b \leqq 11$. The fractions $\frac{b}{12}$ are converted to decimal numbers by division and the decimal expansions are examined. We arrive at the same result.