To solve the addition of two mixed numbers $2 \frac{3}{10}$ and $4 \frac{2}{5}$ in two different ways, let's break it down step by step.

Mixed Number Form Method

1. Align the whole numbers and fractions: $2 \frac{3}{10} + 4 \frac{2}{5}$ This can be rewritten as: $(2 + \frac{3}{10}) + (4 + \frac{2}{5})$

2. Add the whole numbers separately: $2 + 4 = 6$

3. Add the fractions: The fractions are $\frac{3}{10}$ and $\frac{2}{5}$. We need to express $\frac{2}{5}$ with a denominator of 10: $\frac{2}{5} = \frac{4}{10}$

4. Now, add the fractions: $\frac{3}{10} + \frac{4}{10} = \frac{7}{10}$

5. Final result: Combine the whole number and fractional parts: $6 + \frac{7}{10} = 6 \frac{7}{10}$

---

Improper Fraction Method

1. Convert the mixed numbers to improper fractions: $2 \frac{3}{10} = \frac{23}{10}, \quad 4 \frac{2}{5} = \frac{22}{5}$ Now express $\frac{22}{5}$ with a denominator of 10: $\frac{22}{5} = \frac{44}{10}$

2. Add the improper fractions: $\frac{23}{10} + \frac{44}{10} = \frac{67}{10}$

3. Convert back to a mixed number: $\frac{67}{10} = 6 \frac{7}{10}$

---

Thus, in both methods, the final answer is: $6 \frac{7}{10}$

Would you like more details on either method or have any questions?

---

Here are 5 related questions for you:

1. How do you subtract mixed numbers?
2. How do you convert a mixed number to an improper fraction?
3. What is the process to simplify fractions?
4. How do you add mixed numbers with different denominators?
5. How do you handle improper fractions in multiplication or division?

Tip: Always ensure to find a common denominator when adding or subtracting fractions!