Sure, let's go through the problem step-by-step.
We need to simplify the expression:
[tex]\[
\left[2 \div\left(1+1 \div \frac{1}{2}\right)\right] \times\left[3 \div\left(\frac{5}{6} \text { of } \frac{3}{2} \div 1 \frac{1}{4}\right)\right]
\][/tex]
### Step 1: Simplify the Inner Expression in the First Bracket
First, simplify the expression inside the first set of brackets:
[tex]\[
2 \div \left(1 + 1 \div \frac{1}{2}\right)
\][/tex]
Consider the term:
[tex]\[
1 \div \frac{1}{2} = 2
\][/tex]
So the expression now becomes:
[tex]\[
2 \div \left(1 + 2\right) = 2 \div 3 = \frac{2}{3}
\][/tex]
### Step 2: Simplify the Inner Expression in the Second Bracket
Next, simplify the expression inside the second set of brackets:
[tex]\[
3 \div \left(\frac{5}{6} \text{ of } \frac{3}{2} \div 1\frac{1}{4}\right)
\][/tex]
First, we change [tex]\( 1\frac{1}{4} \)[/tex] to an improper fraction:
[tex]\[
1\frac{1}{4} = 1 + \frac{1}{4} = \frac{5}{4}
\][/tex]
Now we handle the expression:
[tex]\[
\frac{5}{6} \text{ of } \frac{3}{2} \div \frac{5}{4}
\][/tex]
First, simplify the "of" operation which implies multiplication:
[tex]\[
\frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4}
\][/tex]
Next, we divide by [tex]\(\frac{5}{4}\)[/tex]:
[tex]\[
\frac{5}{4} \div \frac{5}{4} = 1
\][/tex]
Therefore:
[tex]\[
3 \div 1 = 3
\][/tex]
### Step 3: Multiply the Two Simplified Parts
Now we have the simplified results from the two brackets:
[tex]\[
\frac{2}{3} \times 3 = \frac{2}{3} \times \frac{3}{1} = 2
\][/tex]
### Final Result
[tex]\[
\left[2 \div\left(1+1 \div \frac{1}{2}\right)\right] \times\left[3 \div\left(\frac{5}{6} \text { of } \frac{3}{2} \div 1 \frac{1}{4}\right)\right] = 2
\][/tex]
In conclusion, the simplified result of the given mathematical expression is:
[tex]\[
2
\][/tex]
