1. The problem is to calculate $1 \frac{1}{4} - \frac{5}{6}$ without using a calculator and express the answer as a simplified fraction. 2. First, convert the mixed number $1 \frac{1}{4}$ to an improper fraction. Recall that $a \frac{b}{c} = \frac{ac + b}{c}$. 3. So, $1 \frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4}$. 4. Now the problem is $\frac{5}{4} - \frac{5}{6}$. 5. To subtract fractions, find a common denominator. The denominators are 4 and 6. The least common denominator (LCD) is the least common multiple (LCM) of 4 and 6. 6. The multiples of 4 are 4, 8, 12, 16, ... and the multiples of 6 are 6, 12, 18, ... The LCM is 12. 7. Convert each fraction to have denominator 12: $\frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}$ $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$ 8. Now subtract: $\frac{15}{12} - \frac{10}{12} = \frac{15 - 10}{12} = \frac{5}{12}$. 9. The fraction $\frac{5}{12}$ is already in simplest form because 5 and 12 have no common factors other than 1. Final answer: $\frac{5}{12}$