To solve this problem, we’ll add the mixed numbers $2 \frac{12}{15}$ and $3 \frac{1}{3}$ by first converting them into improper fractions and then finding a common denominator for addition.

1. Convert Mixed Numbers to Improper Fractions:

   • For $2 \frac{12}{15}$: $2 \frac{12}{15} = \frac{(2 \times 15) + 12}{15} = \frac{30 + 12}{15} = \frac{42}{15}$

   • For $3 \frac{1}{3}$: $3 \frac{1}{3} = \frac{(3 \times 3) + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3}$

2. Find a Common Denominator:

   • The denominators are 15 and 3. The least common multiple of 15 and 3 is 15.

   • Convert $\frac{10}{3}$ to have a denominator of 15: $\frac{10}{3} = \frac{10 \times 5}{3 \times 5} = \frac{50}{15}$

3. Add the Fractions: $\frac{42}{15} + \frac{50}{15} = \frac{42 + 50}{15} = \frac{92}{15}$

4. Convert Back to a Mixed Number:

   • Divide 92 by 15 to get 6 with a remainder of 2. $\frac{92}{15} = 6 \frac{2}{15}$

Final Answer:

$2 \frac{12}{15} + 3 \frac{1}{3} = 6 \frac{2}{15}$

Would you like more details on any part of this solution?

Additional Related Questions:

1. How can you add mixed numbers with different denominators more efficiently?
2. How do you simplify fractions within a mixed number?
3. What other methods can be used to convert mixed numbers to improper fractions?
4. Why is finding the least common multiple helpful in adding fractions?
5. How can you subtract mixed numbers with different denominators?

Quick Tip:

Always check if your final answer (especially in fractions) can be simplified to its lowest terms!