To solve the problem \(3 \frac{1}{2} \div 1 \frac{4}{6}\), we need to convert the mixed numbers to improper fractions and then perform the division. Let's go step-by-step:

### Step 1: Convert Mixed Numbers to Improper Fractions

1. Convert \(3 \frac{1}{2}\) to an improper fraction:
   \[
   3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}
   \]

2. Convert \(1 \frac{4}{6}\) to an improper fraction:
   \[
   1 \frac{4}{6} = \frac{1 \times 6 + 4}{6} = \frac{10}{6}
   \]

   Simplify \(\frac{10}{6}\) by dividing both numerator and denominator by 2:
   \[
   \frac{10}{6} = \frac{5}{3}
   \]

### Step 2: Perform the Division of Fractions

To divide by a fraction, multiply by its reciprocal:
\[
\frac{7}{2} \div \frac{5}{3} = \frac{7}{2} \times \frac{3}{5}
\]

### Step 3: Multiply the Fractions

Multiply the numerators and the denominators:
\[
\frac{7 \times 3}{2 \times 5} = \frac{21}{10}
\]

### Step 4: Convert Back to a Mixed Number

Convert \(\frac{21}{10}\) to a mixed number:
\[
\frac{21}{10} = 2 \frac{1}{10}
\]

### Final Answer

\[
3 \frac{1}{2} \div 1 \frac{4}{6} = 2 \frac{1}{10}
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you convert a mixed number to an improper fraction?
2. What is the reciprocal of a fraction, and how is it used in division?
3. How can you simplify fractions before performing operations with them?
4. What are some strategies for converting improper fractions back to mixed numbers?
5. How can you perform operations with fractions that have different denominators?

**Tip:** Always simplify fractions whenever possible to make calculations easier.

Learn how to solve the division problem involving mixed numbers: 3 ½ divided by 1 4/6. This problem covers key concepts in fractions, including converting mixed numbers to improper fractions, dividing fractions, and converting results back to mixed numbers. Suitable for students in Grades 6-8.

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