Simplifying the Fraction Expression

The problem requires simplifying the expression: $ \left( \frac{15}{24} \div \frac{15}{2} \right) \div \left( \frac{1}{8} \times \frac{16}{3} + \frac{1}{8} \right) + \frac{2}{11} \div \frac{19}{23} \text{ of } \frac{23}{11} $ We will solve this step-by-step using the order of operations (PEMDAS/BODMAS).

Step 1: Simplify the First Parenthesis

Calculate the expression inside the first parenthesis:

• $ \frac{15}{24} \div \frac{15}{2} $
• Convert division to multiplication by the reciprocal: $ \frac{15}{24} \times \frac{2}{15} $
• Cancel common factors: $ \frac{1}{24} \times \frac{2}{1} = \frac{2}{24} $
• Simplify the fraction: $ \frac{1}{12} $

Step 2: Simplify the Second Parenthesis

Calculate the expression inside the second parenthesis:

• First, perform the multiplication: $ \frac{1}{8} \times \frac{16}{3} = \frac{16}{24} $
• Simplify the multiplication result: $ \frac{16 \div 8}{24 \div 8} = \frac{2}{3} $
• Now, perform the addition: $ \frac{2}{3} + \frac{1}{8} $
• Find a common denominator (24): $ \frac{2 \times 8}{3 \times 8} + \frac{1 \times 3}{8 \times 3} = \frac{16}{24} + \frac{3}{24} $
• Add the fractions: $ \frac{16 + 3}{24} = \frac{19}{24} $

Step 3: Perform Division Between Parentheses

Now substitute the simplified parenthesis results back into the expression:

• $ \frac{1}{12} \div \frac{19}{24} $
• Convert division to multiplication: $ \frac{1}{12} \times \frac{24}{19} $
• Cancel common factors: $ \frac{1}{1} \times \frac{2}{19} = \frac{2}{19} $

Step 4: Simplify the 'of' Term

Calculate the term involving 'of' and division:

• $ \frac{2}{11} \div \left( \frac{19}{23} \text{ of } \frac{23}{11} \right) $
• Calculate 'of' (multiplication) first: $ \frac{19}{23} \times \frac{23}{11} $
• Cancel common factors: $ \frac{19}{1} \times \frac{1}{11} = \frac{19}{11} $
• Now perform the division: $ \frac{2}{11} \div \frac{19}{11} $
• Convert division to multiplication: $ \frac{2}{11} \times \frac{11}{19} $
• Cancel common factors: $ \frac{2}{1} \times \frac{1}{19} = \frac{2}{19} $

Step 5: Combine Results

Add the results from Step 3 and Step 4:

• $ \frac{2}{19} + \frac{2}{19} $
• Add the fractions: $ \frac{2 + 2}{19} = \frac{4}{19} $

The simplified value of the expression is $ \frac{4}{19} $.