(5 2/3 - 2 1/4) ÷ (5/12 + 1 5/6) ÷ (2 3/4 × 3 4/5)= (use cancellation method and KFC)

To solve the expression \((5 \frac{2}{3} - 2 \frac{1}{4}) \div ( \frac{5}{12} + 1 \frac{5}{6} ) \div (2 \frac{3}{4} \times 3 \frac{4}{5})\) using the cancellation method and KFC (Keep, Flip, Change), we will first convert all mixed numbers to improper fractions, then simplify step by step.

### Step-by-Step Solution

1. **Convert Mixed Numbers to Improper Fractions:**

   - \(5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{17}{3}\)
   - \(2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}\)
   - \(1 \frac{5}{6} = \frac{1 \times 6 + 5}{6} = \frac{11}{6}\)
   - \(2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}\)
   - \(3 \frac{4}{5} = \frac{3 \times 5 + 4}{5} = \frac{19}{5}\)

2. **Rewrite the Expression:**

   \[
   \left( \frac{17}{3} - \frac{9}{4} \right) \div \left( \frac{5}{12} + \frac{11}{6} \right) \div \left( \frac{11}{4} \times \frac{19}{5} \right)
   \]

3. **Perform the Subtraction and Addition:**

   - **Subtraction:**
   
     \[
     \frac{17}{3} - \frac{9}{4}
     \]

     Find a common denominator for the fractions, which is 12:

     \[
     \frac{17}{3} = \frac{17 \times 4}{3 \times 4} = \frac{68}{12}, \quad \frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}
     \]

     So,

     \[
     \frac{68}{12} - \frac{27}{12} = \frac{41}{12}
     \]

   - **Addition:**
   
     \[
     \frac{5}{12} + \frac{11}{6}
     \]

     Convert \(\frac{11}{6}\) to have a common denominator of 12:

     \[
     \frac{11}{6} = \frac{11 \times 2}{6 \times 2} = \frac{22}{12}
     \]

     So,

     \[
     \frac{5}{12} + \frac{22}{12} = \frac{27}{12} = \frac{9}{4} \quad (\text{simplified})
     \]

4. **Simplify the Expression with KFC (Keep, Flip, Change):**

   The expression is now:

   \[
   \frac{41}{12} \div \frac{9}{4} \div \left( \frac{11}{4} \times \frac{19}{5} \right)
   \]

   - **First Division (Keep, Flip, Change):**

     \[
     \frac{41}{12} \div \frac{9}{4} = \frac{41}{12} \times \frac{4}{9} = \frac{41 \times 4}{12 \times 9} = \frac{164}{108}
     \]

     Simplify \(\frac{164}{108}\) by dividing by their greatest common divisor (which is 4):

     \[
     \frac{164 \div 4}{108 \div 4} = \frac{41}{27}
     \]

   - **Second Division with Multiplication:**

     The next part of the expression:

     \[
     \frac{11}{4} \times \frac{19}{5} = \frac{11 \times 19}{4 \times 5} = \frac{209}{20}
     \]

     Now apply KFC again:

     \[
     \frac{41}{27} \div \frac{209}{20} = \frac{41}{27} \times \frac{20}{209} = \frac{41 \times 20}{27 \times 209}
     \]

     This simplifies to:

     \[
     \frac{820}{5631}
     \]

5. **Final Answer:**

   The simplified result of the expression \((5 \frac{2}{3} - 2 \frac{1}{4}) \div ( \frac{5}{12} + 1 \frac{5}{6} ) \div (2 \frac{3}{4} \times 3 \frac{4}{5})\) is:

   \[
   \frac{820}{5631}
   \]

   Which cannot be simplified further.

Would you like more details or have any questions?

### Related Questions:

1. How do you find the least common multiple of two denominators?
2. What is the process for converting a mixed number into an improper fraction?
3. How can you simplify fractions by finding the greatest common divisor?
4. What are the steps involved in dividing fractions?
5. How does the Keep, Flip, Change method work in fraction division?

**Tip:** When working with mixed numbers, always convert them to improper fractions before performing arithmetic operations.

Learn how to solve the complex fraction expression (5 2/3 - 2 1/4) ÷ (5/12 + 1 5/6) ÷ (2 3/4 × 3 4/5) using the cancellation method and conversion of mixed numbers to improper fractions. This problem involves fraction operations and is suitable for Grades 6-8 students.

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