To solve this expression, let's first convert each mixed number into an improper fraction and then perform the addition and subtraction. The expression is:

$3 \frac{2}{5} + 2 \frac{2}{3} + 5 \frac{7}{10}$

Step 1: Convert each mixed number to an improper fraction

1. $3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$
2. $2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}$
3. $5 \frac{7}{10} = \frac{5 \times 10 + 7}{10} = \frac{50 + 7}{10} = \frac{57}{10}$

Step 2: Find a common denominator

The denominators are 5, 3, and 10. The least common multiple of these denominators is 30.

Step 3: Convert each fraction to have a denominator of 30

1. $\frac{17}{5} = \frac{17 \times 6}{5 \times 6} = \frac{102}{30}$
2. $\frac{8}{3} = \frac{8 \times 10}{3 \times 10} = \frac{80}{30}$
3. $\frac{57}{10} = \frac{57 \times 3}{10 \times 3} = \frac{171}{30}$

Step 4: Add the fractions

Now we add the fractions:

$\frac{102}{30} + \frac{80}{30} + \frac{171}{30} = \frac{102 + 80 + 171}{30} = \frac{353}{30}$

Step 5: Simplify if possible

The fraction $\frac{353}{30}$ is already in its simplest form, as 353 and 30 have no common factors other than 1.

Final Answer:

$\frac{353}{30}$

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

Here are some related questions:

1. How do you convert mixed numbers to improper fractions?
2. What are some methods to find the least common multiple?
3. How can we simplify fractions effectively?
4. What is the process for adding or subtracting fractions with different denominators?
5. Can mixed numbers be handled directly without conversion in some cases?

Tip: Always check if fractions can be simplified after each step, as it might make calculations easier.