Determine the value of the equation: $1 \frac{1}{8}-\frac{3}{4}$

Step 1 ：Convert the mixed number $1 \frac{1}{8}$ into an improper fraction. The denominator remains the same, which is 8. The numerator is calculated by multiplying the whole number by the denominator and then adding the numerator of the fractional part. So, $1 \frac{1}{8}$ becomes $\frac{8*1+1}{8} = \frac{9}{8}$.

Step 2 ：We have the equation $\frac{9}{8}-\frac{3}{4}$. Before we can subtract these fractions, we need to make sure they have the same denominator. The least common denominator (LCD) of 8 and 4 is 8.

Step 3 ：Rewrite $\frac{3}{4}$ as $\frac{6}{8}$ (by multiplying both the numerator and the denominator by 2). Now, we have the equation $\frac{9}{8}-\frac{6}{8}$.

Step 4 ：Subtracting the numerators, we get $\frac{9-6}{8} = \frac{3}{8}$. So, the value of the equation $1 \frac{1}{8}-\frac{3}{4}$ is $\frac{3}{8}$.

Step 5 ：Check our result: $1 \frac{1}{8}-\frac{3}{4} = \frac{9}{8}-\frac{6}{8} = \frac{3}{8}$. So, our result is correct.

Step 6 ：\(\boxed{\frac{3}{8}}\) is the final answer.