To solve the addition of the mixed numbers $3 \frac{1}{4} + 2 \frac{1}{4} + 1 \frac{1}{12}$, we follow these steps:

1. Convert the mixed numbers to improper fractions:

   • $3 \frac{1}{4} = \frac{3 \cdot 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}$
   • $2 \frac{1}{4} = \frac{2 \cdot 4 + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$
   • $1 \frac{1}{12} = \frac{1 \cdot 12 + 1}{12} = \frac{12 + 1}{12} = \frac{13}{12}$

2. Find a common denominator for the fractions:

   The least common denominator (LCD) for 4 and 12 is 12.

   • Convert $\frac{13}{4}$ to a fraction with denominator 12: $\frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12}$

   • Convert $\frac{9}{4}$ to a fraction with denominator 12: $\frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}$

3. Add the fractions:

   • Add $\frac{39}{12} + \frac{27}{12} + \frac{13}{12}$: $\frac{39 + 27 + 13}{12} = \frac{79}{12}$

4. Convert the improper fraction back to a mixed number:

   • $\frac{79}{12}$ can be converted as follows: $79 \div 12 = 6 \text{ R } 7 \implies 6 \frac{7}{12}$

Thus, the sum of $3 \frac{1}{4} + 2 \frac{1}{4} + 1 \frac{1}{12}$ is $6 \frac{7}{12}$.

Would you like more details or have any questions about this solution?

Here are some related questions to further your understanding:

1. How do you find the least common denominator (LCD) of two or more fractions?
2. How can you convert a mixed number to an improper fraction?
3. What is the process for converting an improper fraction back to a mixed number?
4. How do you simplify the addition of fractions with different denominators?
5. What are some strategies for adding mixed numbers efficiently?
6. Can you explain how to subtract mixed numbers with different denominators?
7. What is the importance of converting mixed numbers to improper fractions for addition?
8. How do you handle the addition of mixed numbers with large whole parts and fractions?

Tip: When adding mixed numbers, converting them to improper fractions first can simplify the calculation process.