Establish the empirical formula

The relationship between mean, median, and mode is given by: $\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}$.

$\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}$

Rearrange the formula

Subtract $\text{Median}$ from both sides to express it in terms of the given difference: $\text{Mode} - \text{Median} = 2 \times \text{Median} - 2 \times \text{Mean}$. Factor out the 2: $\text{Mode} - \text{Median} = 2(\text{Median} - \text{Mean})$.

$\text{Mode} - \text{Median} = 2(\text{Median} - \text{Mean})$

Substitute given values

The problem states the median is 24.12 less than the mode, so $\text{Mode} - \text{Median} = 24.12$. Substitute this into the equation: $24.12 = 2(\text{Median} - \text{Mean})$.

$24.12 = 2(\text{Median} - \text{Mean})$

Solve for the final value

Divide by 2: $\text{Median} - \text{Mean} = \frac{24.12}{2} = 12.06$.

$\text{Median} - \text{Mean} = 12.06$