To solve the given problem $1 \frac{1}{5} \times 1 \frac{3}{5} + 7 \frac{1}{5} \div 4 \frac{4}{5}$, we will follow the order of operations: multiplication and division first, then addition.

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Step 1: Convert all mixed numbers to improper fractions

1. $1 \frac{1}{5} = \frac{6}{5}$
2. $1 \frac{3}{5} = \frac{8}{5}$
3. $7 \frac{1}{5} = \frac{36}{5}$
4. $4 \frac{4}{5} = \frac{24}{5}$

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Step 2: Perform the operations

Multiplication: $1 \frac{1}{5} \times 1 \frac{3}{5}$

$\frac{6}{5} \times \frac{8}{5} = \frac{48}{25}$

Division: $7 \frac{1}{5} \div 4 \frac{4}{5}$

$\frac{36}{5} \div \frac{24}{5} = \frac{36}{5} \times \frac{5}{24} = \frac{36 \cdot 5}{5 \cdot 24} = \frac{36}{24} = \frac{3}{2}$

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Step 3: Add the results

Convert $\frac{3}{2}$ to a fraction with denominator 25 for addition: $\frac{3}{2} = \frac{37.5}{25} = \frac{75}{50} \approx \frac{21}{50}$

Step 1: Simplify individual parts

Mixed fraction conversions:

1