Let's solve the expression step-by-step:
[tex]\[
\frac{1}{3}\left(2 \frac{1}{2}+3 \frac{1}{3}\right) \div \frac{2}{9}\left(3 \frac{1}{8}-1 \frac{1}{12}\right)
\][/tex]
### Step 1: Convert Mixed Numbers to Improper Fractions
- Convert [tex]\(2 \frac{1}{2}\)[/tex]:
[tex]\[
2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}
\][/tex]
- Convert [tex]\(3 \frac{1}{3}\)[/tex]:
[tex]\[
3 \frac{1}{3} = 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3}
\][/tex]
### Step 2: Sum the Improper Fractions
[tex]\[
2 \frac{1}{2} + 3 \frac{1}{3} = \frac{5}{2} + \frac{10}{3}
\][/tex]
To add these fractions, find a common denominator:
[tex]\[
\frac{5}{2} = \frac{15}{6} \quad \text{and} \quad \frac{10}{3} = \frac{20}{6}
\][/tex]
Thus,
[tex]\[
\frac{5}{2} + \frac{10}{3} = \frac{15}{6} + \frac{20}{6} = \frac{35}{6}
\][/tex]
### Step 3: Convert the Denominator Mixed Numbers to Improper Fractions
- Convert [tex]\(3 \frac{1}{8}\)[/tex]:
[tex]\[
3 \frac{1}{8} = 3 + \frac{1}{8} = \frac{24}{8} + \frac{1}{8} = \frac{25}{8}
\][/tex]
- Convert [tex]\(1 \frac{1}{12}\)[/tex]:
[tex]\[
1 \frac{1}{12} = 1 + \frac{1}{12} = \frac{12}{12} + \frac{1}{12} = \frac{13}{12}
\][/tex]
### Step 4: Subtract the Improper Fractions
[tex]\[
3 \frac{1}{8} - 1 \frac{1}{12} = \frac{25}{8} - \frac{13}{12}
\][/tex]
To subtract these fractions, find a common denominator:
[tex]\[
\frac{25}{8} = \frac{75}{24} \quad \text{and} \quad \frac{13}{12} = \frac{26}{24}
\][/tex]
Thus,
[tex]\[
\frac{25}{8} - \frac{13}{12} = \frac{75}{24} - \frac{26}{24} = \frac{49}{24}
\][/tex]
### Step 5: Calculate Numerator and Denominator of the Given Expression
The numerator of the expression:
[tex]\[
\frac{1}{3} \left(2 \frac{1}{2} + 3 \frac{1}{3}\right) = \frac{1}{3} \times \frac{35}{6} = \frac{35}{18}
\][/tex]
The denominator of the expression:
[tex]\[
\frac{2}{9} \left(3 \frac{1}{8} - 1 \frac{1}{12}\right) = \frac{2}{9} \times \frac{49}{24} = \frac{98}{216} = \frac{49}{108}
\][/tex]
### Step 6: Divide Numerator by Denominator
[tex]\[
\frac{\frac{35}{18}}{\frac{49}{108}} = \frac{35}{18} \times \frac{108}{49} = \frac{35 \times 108}{18 \times 49}
\][/tex]
Simplify numerator and denominator:
[tex]\[
\frac{35 \times 108}{18 \times 49} = \frac{3780}{882} = \frac{4.285714285714286 \quad \text{(approximately)}}
\][/tex]
Therefore, the final result can be calculated as:
[tex]\[
4.285714285714286
\][/tex]
