The problem asks for the area of a rectangle with a height of $1 \frac{1}{4}$ inches and a base of $\frac{5}{6}$ inches.

First, let's convert the mixed number to an improper fraction:

$1 \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}$

Now, the area $A$ of the rectangle is calculated by multiplying the height by the base:

$A = \text{Height} \times \text{Base} = \frac{5}{4} \times \frac{5}{6}$

To multiply these fractions:

$A = \frac{5 \times 5}{4 \times 6} = \frac{25}{24} \text{ square inches}$

So, the area of the rectangle is $\frac{25}{24}$ square inches.

Would you like more details on this solution, or do you have any questions?

Here are five related questions to expand your understanding:

1. How do you convert a mixed number to an improper fraction?
2. How would the area change if the base were doubled while keeping the height the same?
3. What is the decimal equivalent of the area $\frac{25}{24}$ square inches?
4. If another rectangle has the same area but a different base, how would you find its height?
5. How would you find the perimeter of this rectangle given the height and base?

Tip: Always remember to simplify fractions where possible before performing operations; it can make the calculations easier!