To solve the expression $2 \frac{1}{6} \div 3 \frac{1}{10}$, follow these steps:

1. Convert mixed numbers to improper fractions:

• $2 \frac{1}{6}$ becomes $\frac{2 \times 6 + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6}$.
• $3 \frac{1}{10}$ becomes $\frac{3 \times 10 + 1}{10} = \frac{30 + 1}{10} = \frac{31}{10}$.

2. Rewrite the division as multiplication by the reciprocal:

$\frac{13}{6} \div \frac{31}{10} = \frac{13}{6} \times \frac{10}{31}$

3. Multiply the fractions:

$\frac{13 \times 10}{6 \times 31} = \frac{130}{186}$

4. Simplify the fraction:

• The greatest common divisor (GCD) of 130 and 186 is 2.
• Divide the numerator and the denominator by 2: $\frac{130 \div 2}{186 \div 2} = \frac{65}{93}$

Since $\frac{65}{93}$ is already in its simplest form, the final answer is $\frac{65}{93}$.

Would you like any further details or explanations?

Here are some related questions:

1. How do you convert a mixed number into an improper fraction?
2. What is the process of finding the reciprocal of a fraction?
3. How do you know when a fraction is in its simplest form?
4. What steps are involved in multiplying two fractions?
5. Can you explain how to compare fractions to determine which is larger?

Tip: After multiplying fractions, always check if you can simplify the result.