Let's break down and solve the given mathematical expression from the image.

The expression provided is:

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

Step 1: Simplify the numerator

The numerator of the fraction is:

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

To add these two fractions, we need a common denominator. The least common denominator (LCD) of 5 and 3 is 15. Convert each fraction:

$\frac{2}{5} = \frac{6}{15}, \quad \frac{2}{3} = \frac{10}{15}$

Now, add the fractions:

$\frac{6}{15} + \frac{10}{15} = \frac{16}{15}$

Step 2: Simplify the denominator

The denominator of the fraction is:

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

We know that $\frac{3}{3} = 1$. So the expression becomes:

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

Now, find the LCD of 2 and 5, which is 10. Convert the fractions:

$\frac{1}{2} = \frac{5}{10}, \quad \frac{2}{5} = \frac{4}{10}$

Now, add them:

$1 + \frac{5}{10} + \frac{4}{10} = 1 + \frac{9}{10} = \frac{19}{10}$

Step 3: Combine the simplified numerator and denominator

Now we have:

$\frac{\frac{16}{15}}{\frac{19}{10}}$

To divide fractions, multiply by the reciprocal of the denominator:

$\frac{16}{15} \times \frac{10}{19} = \frac{160}{285}$

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 5:

$\frac{160 \div 5}{285 \div 5} = \frac{32}{57}$

Step 4: Add $\frac{1}{3}$

Now we need to add $\frac{32}{57} + \frac{1}{3}$. To do this, find the LCD of 57 and 3, which is 57. Convert the fractions:

$\frac{1}{3} = \frac{19}{57}$

Now, add the fractions:

$\frac{32}{57} + \frac{19}{57} = \frac{51}{57}$

Simplify the fraction:

$\frac{51}{57} = \frac{51 \div 3}{57 \div 3} = \frac{17}{19}$

Final Answer:

The value of the expression is $\frac{17}{19}$.

However, none of the provided options match $\frac{17}{19}$. It seems the options might not have the correct answer listed.

Would you like further details on this, or do you have any questions?

Related Questions:

1. How do you simplify complex fractions with multiple terms in the numerator and denominator?
2. How can you find the least common denominator for adding fractions with different denominators?
3. What is the best way to approach fractions with mixed operations (addition and division)?
4. How do you determine if a fraction can be simplified further?
5. Can you solve other similar problems with more complex expressions?

Tip:

Always simplify both the numerator and denominator individually before combining them when dealing with complex fractions.