The expression provided involves adding two mixed numbers: $3 \frac{2}{5} + 1 \frac{3}{7}$. Let’s solve this step by step.

Step 1: Convert mixed numbers to improper fractions.

1. Convert $3 \frac{2}{5}$: $3 \frac{2}{5} = \frac{3 \cdot 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$

2. Convert $1 \frac{3}{7}$: $1 \frac{3}{7} = \frac{1 \cdot 7 + 3}{7} = \frac{7 + 3}{7} = \frac{10}{7}$

Now the expression becomes: $\frac{17}{5} + \frac{10}{7}$

Step 2: Find the least common denominator (LCD).

The denominators are $5$ and $7$. The LCD of $5$ and $7$ is $35$.

Step 3: Rewrite each fraction with the LCD as the denominator.

1. Rewrite $\frac{17}{5}$: $\frac{17}{5} = \frac{17 \cdot 7}{5 \cdot 7} = \frac{119}{35}$

2. Rewrite $\frac{10}{7}$: $\frac{10}{7} = \frac{10 \cdot 5}{7 \cdot 5} = \frac{50}{35}$

Now the expression becomes: $\frac{119}{35} + \frac{50}{35}$

Step 4: Add the fractions.

Since the denominators are the same, we add the numerators: $\frac{119}{35} + \frac{50}{35} = \frac{119 + 50}{35} = \frac{169}{35}$

Step 5: Convert the improper fraction back to a mixed number.

Divide $169$ by $35$: $169 \div 35 = 4 \, \text{remainder } 29$ So: $\frac{169}{35} = 4 \frac{29}{35}$

Final Answer:

$3 \frac{2}{5} + 1 \frac{3}{7} = 4 \frac{29}{35}$

Would you like a detailed breakdown of any step? Let me know if you have questions!

Here are 5 related questions to expand your understanding:

1. How do you convert mixed numbers to improper fractions?
2. What is the least common denominator, and how is it calculated?
3. How do you simplify improper fractions to mixed numbers?
4. Can you subtract $3 \frac{2}{5} - 1 \frac{3}{7}$?
5. How do you add fractions with unrelated denominators?

Tip: Always double-check your work by simplifying your answer to its lowest terms!