`4(1)/(10)-[2(1)/(2)-{(5)/(6)-((2)/(5)+(3)/(10)-(4)/(15))}]`

Answer

To solve the expression \( 4\left(\frac{1}{10}\right) - \left[2\left(\frac{1}{2}\right) - \left\{\frac{5}{6} - \left(\frac{2}{5} + \frac{3}{10} - \frac{4}{15}\right)\right\}\right] \), we will follow the order of operations, which is to solve the innermost brackets first and work our way outwards.

### Step-by-Step Solution:

1. **Solve the innermost bracket**:
\[ \frac{2}{5} + \frac{3}{10} - \frac{4}{15} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 5, 10, and 15 is 30.

- Convert each fraction:
\[ \frac{2}{5} = \frac{12}{30}, \quad \frac{3}{10} = \frac{9}{30}, \quad \frac{4}{15} = \frac{8}{30} \]

- Now substitute back into the expression:
\[ \frac{12}{30} + \frac{9}{30} - \frac{8}{30} = \frac{12 + 9 - 8}{30} = \frac{13}{30} \]

2. **Substitute back into the curly bracket**:
\[ \frac{5}{6} - \frac{13}{30} \]
Again, we need a common denominator. The LCM of 6 and 30 is 30.

- Convert \(\frac{5}{6}\):
\[ \frac{5}{6} = \frac{25}{30} \]

- Now substitute back:
\[ \frac{25}{30} - \frac{13}{30} = \frac{25 - 13}{30} = \frac{12}{30} = \frac{2}{5} \]

3. **Substitute back into the square bracket**:
\[ 2\left(\frac{1}{2}\right) - \frac{2}{5} \]
Calculate \(2\left(\frac{1}{2}\right) = 1\).

- Now substitute:
\[ 1 - \frac{2}{5} \]
Convert \(1\) to a fraction with a denominator of 5:
\[ 1 = \frac{5}{5} \]
Now perform the subtraction:
\[ \frac{5}{5} - \frac{2}{5} = \frac{3}{5} \]

4. **Substitute back into the main expression**:
\[ 4\left(\frac{1}{10}\right) - \frac{3}{5} \]
Calculate \(4\left(\frac{1}{10}\right) = \frac{4}{10} = \frac{2}{5}\).

- Now substitute:
\[ \frac{2}{5} - \frac{3}{5} = \frac{2 - 3}{5} = \frac{-1}{5} \]

### Final Answer:
\[ \frac{-1}{5} \]