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

Options

A. `(21)/(10)`
B. `(7)/(5)`
C. `(7)/(3)`
D. `(8)/(3)`

Answer

To solve the expression \( 5 \left( \frac{1}{5} \right) - \left[ 3 \left( \frac{1}{2} \right) - \left\{ \frac{5}{6} - \left( \frac{3}{5} + \frac{1}{10} - \frac{4}{15} \right) \right\} \right] \), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Simplify the first term**:
\[ 5 \left( \frac{1}{5} \right) = 1 \]

**Hint**: Multiply the whole number by the fraction.

2. **Simplify the innermost bracket**:
\[ \frac{3}{5} + \frac{1}{10} - \frac{4}{15} \]
To simplify this, we need to find a common denominator. The LCM of 5, 10, and 15 is 30.

Converting each fraction:
\[ \frac{3}{5} = \frac{18}{30}, \quad \frac{1}{10} = \frac{3}{30}, \quad \frac{4}{15} = \frac{8}{30} \]
Now substituting back:
\[ \frac{18}{30} + \frac{3}{30} - \frac{8}{30} = \frac{18 + 3 - 8}{30} = \frac{13}{30} \]

**Hint**: Always convert fractions to a common denominator before adding or subtracting.

3. **Substitute back into the expression**:
\[ 1 - \left[ 3 \left( \frac{1}{2} \right) - \left( \frac{5}{6} - \frac{13}{30} \right) \right] \]

4. **Calculate \( 3 \left( \frac{1}{2} \right) \)**:
\[ 3 \left( \frac{1}{2} \right) = \frac{3}{2} \]

**Hint**: Again, multiply the whole number by the fraction.

5. **Now simplify the bracket**:
\[ \frac{5}{6} - \frac{13}{30} \]
The LCM of 6 and 30 is 30.
Converting:
\[ \frac{5}{6} = \frac{25}{30} \]
Now substituting back:
\[ \frac{25}{30} - \frac{13}{30} = \frac{25 - 13}{30} = \frac{12}{30} = \frac{2}{5} \]

**Hint**: Convert fractions to a common denominator for subtraction.

6. **Substituting back**:
\[ 1 - \left( \frac{3}{2} - \frac{2}{5} \right) \]

7. **Calculate \( \frac{3}{2} - \frac{2}{5} \)**:
The LCM of 2 and 5 is 10.
Converting:
\[ \frac{3}{2} = \frac{15}{10}, \quad \frac{2}{5} = \frac{4}{10} \]
Now substituting back:
\[ \frac{15}{10} - \frac{4}{10} = \frac{15 - 4}{10} = \frac{11}{10} \]

**Hint**: Always convert fractions to a common denominator before performing operations.

8. **Final calculation**:
\[ 1 - \frac{11}{10} = \frac{10}{10} - \frac{11}{10} = -\frac{1}{10} \]

**Hint**: When subtracting fractions, ensure you keep track of the signs.

### Final Answer:
The expression simplifies to:
\[ -\frac{1}{10} \]