Let's solve the given expression step by step:
[tex]\[ 3 \frac{1}{2} + \frac{1}{9} \times 6 \frac{3}{4} - 2 \frac{1}{3} \div 1 \frac{3}{4} \text{ of } 1 \frac{1}{2} \][/tex]
1. Convert all mixed numbers to improper fractions.
- [tex]\( 3 \frac{1}{2} = \frac{7}{2} \)[/tex]
- [tex]\( 6 \frac{3}{4} = \frac{27}{4} \)[/tex]
- [tex]\( 2 \frac{1}{3} = \frac{7}{3} \)[/tex]
- [tex]\( 1 \frac{3}{4} = \frac{7}{4} \)[/tex]
- [tex]\( 1 \frac{1}{2} = \frac{3}{2} \)[/tex]
2. Start by handling the multiplication and division first according to the order of operations (PEMDAS/BODMAS).
3. Multiply [tex]\(\frac{1}{9}\)[/tex] and [tex]\(\frac{27}{4}\)[/tex]:
[tex]\[
\frac{1}{9} \times \frac{27}{4} = \frac{27}{36} = \frac{3}{4}
\][/tex]
4. Divide [tex]\(\frac{7}{3}\)[/tex] by [tex]\(\frac{7}{4}\)[/tex]:
[tex]\[
\frac{7}{3} \div \frac{7}{4} = \frac{7}{3} \times \frac{4}{7} = \frac{4}{3}
\][/tex]
5. Now, subtract [tex]\(\frac{4}{3}\)[/tex] from the sum of [tex]\(\frac{7}{2}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\frac{7}{2} + \frac{3}{4} = \frac{14}{4} + \frac{3}{4} = \frac{17}{4} = 4.25
\][/tex]
[tex]\[
4.25 - \frac{4}{3} = \frac{17}{4} - \frac{4}{3} = \frac{51}{12} - \frac{16}{12} = \frac{35}{12} \approx 2.9167
\][/tex]
6. Finally, multiply the result [tex]\(\frac{35}{12}\)[/tex] by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
\frac{35}{12} \times \frac{3}{2} = \frac{105}{24} = \frac{35}{8} = 4.375
\][/tex]
So the detailed, step-by-step result for [tex]\(3 \frac{1}{2} + \frac{1}{9} \times 6 \frac{3}{4} - 2 \frac{1}{3} \div 1 \frac{3}{4}\)[/tex] of [tex]\(1 \frac{1}{2}\)[/tex] is [tex]\(4.375\)[/tex].
