Let's go through each of the problems step by step. These are mixed fraction addition problems, so we'll convert mixed fractions to improper fractions, find the common denominator, and simplify the answers where necessary.

Problem 1:

$43 \frac{1}{2} + \frac{2}{5} + \frac{8}{5} + 1 \frac{3}{5}$

1. Convert mixed fractions to improper fractions:

   • $43 \frac{1}{2} = \frac{87}{2}$
   • $1 \frac{3}{5} = \frac{8}{5}$

2. Add all the fractions together. First, find the least common denominator (LCD) of 2 and 5, which is 10. Convert fractions to have the same denominator:

   • $\frac{87}{2} = \frac{435}{10}$
   • $\frac{2}{5} = \frac{4}{10}$
   • $\frac{8}{5} = \frac{16}{10}$
   • $\frac{8}{5} = \frac{16}{10}$ (already converted)

3. Now add them all together: $\frac{435}{10} + \frac{4}{10} + \frac{16}{10} + \frac{16}{10} = \frac{435 + 4 + 16 + 16}{10} = \frac{471}{10}$

4. Convert back to a mixed fraction: $\frac{471}{10} = 47 \frac{1}{10}$

Problem 2:

$13 \frac{3}{10} + \frac{1}{2}$

1. Convert mixed fraction to improper fraction:

   • $13 \frac{3}{10} = \frac{133}{10}$

2. Find the least common denominator (LCD) of 10 and 2, which is 10:

   • $\frac{1}{2} = \frac{5}{10}$

3. Add the fractions: $\frac{133}{10} + \frac{5}{10} = \frac{138}{10}$

4. Simplify: $\frac{138}{10} = 13 \frac{8}{10} = 13 \frac{4}{5}$

Problem 3:

$3 \frac{1}{2} + \frac{1}{5}$

1. Convert mixed fraction to improper fraction:

   • $3 \frac{1}{2} = \frac{7}{2}$

2. Find the least common denominator (LCD) of 2 and 5, which is 10:

   • $\frac{7}{2} = \frac{35}{10}$
   • $\frac{1}{5} = \frac{2}{10}$

3. Add the fractions: $\frac{35}{10} + \frac{2}{10} = \frac{37}{10}$

4. Convert back to a mixed fraction: $\frac{37}{10} = 3 \frac{7}{10}$

Problem 4:

$4 \frac{1}{3} + 3 \frac{2}{5}$

1. Convert mixed fractions to improper fractions:

   • $4 \frac{1}{3} = \frac{13}{3}$
   • $3 \frac{2}{5} = \frac{17}{5}$

2. Find the least common denominator (LCD) of 3 and 5, which is 15:

   • $\frac{13}{3} = \frac{65}{15}$
   • $\frac{17}{5} = \frac{51}{15}$

3. Add the fractions: $\frac{65}{15} + \frac{51}{15} = \frac{116}{15}$

4. Convert back to a mixed fraction: $\frac{116}{15} = 7 \frac{11}{15}$

Problem 5:

$3 \frac{2}{5} + \frac{2}{5}$

1. Convert mixed fraction to improper fraction:

   • $3 \frac{2}{5} = \frac{17}{5}$

2. Add the fractions (same denominator): $\frac{17}{5} + \frac{2}{5} = \frac{19}{5}$

3. Convert back to a mixed fraction: $\frac{19}{5} = 3 \frac{4}{5}$

Would you like further details on any of these steps or explanations?

Here are five related questions for practice:

1. How do you add fractions with different denominators?
2. What is a mixed fraction, and how do you convert it to an improper fraction?
3. What is the least common denominator (LCD), and how is it used in fraction addition?
4. How do you simplify fractions after adding them?
5. How do you convert improper fractions back to mixed fractions?

Tip: When adding mixed fractions, always convert them to improper fractions first for easier calculation.