The value of `3(1)/(5) - [2(1)/(2) - ((5)/(6) - ((2)/(5) + (3)/(10) - (4)/(15))]` is :

Options

A. `6/5`
B. `(11)/(10)`
C. `9/(10)`
D. `(13)/(5)`

Answer

To solve the expression \(3\left(\frac{1}{5}\right) - \left[2\left(\frac{1}{2}\right) - \left(\frac{5}{6} - \left(\frac{2}{5} + \frac{3}{10} - \frac{4}{15}\right)\right)\right]\), we will follow these steps:

### Step 1: Convert mixed fractions to improper fractions
First, we convert the mixed fractions into improper fractions:
- \(3\left(\frac{1}{5}\right) = \frac{15 + 1}{5} = \frac{16}{5}\)
- \(2\left(\frac{1}{2}\right) = \frac{2 \cdot 1 + 1}{2} = \frac{3}{2}\)

### Step 2: Simplify the innermost expression
Now, we simplify the innermost expression:
\[ \frac{2}{5} + \frac{3}{10} - \frac{4}{15} \]
To do this, we need a common denominator. The least common multiple (LCM) of \(5\), \(10\), and \(15\) is \(30\).

- Convert each fraction:
- \(\frac{2}{5} = \frac{2 \cdot 6}{5 \cdot 6} = \frac{12}{30}\)
- \(\frac{3}{10} = \frac{3 \cdot 3}{10 \cdot 3} = \frac{9}{30}\)
- \(\frac{4}{15} = \frac{4 \cdot 2}{15 \cdot 2} = \frac{8}{30}\)

Now, substituting back:
\[ \frac{12}{30} + \frac{9}{30} - \frac{8}{30} = \frac{12 + 9 - 8}{30} = \frac{13}{30} \]

### Step 3: Substitute back into the expression
Now we substitute this back into the larger expression:
\[ \frac{5}{6} - \frac{13}{30} \]
Again, we need a common denominator. The LCM of \(6\) and \(30\) is \(30\).

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

Now, we can simplify:
\[ \frac{25}{30} - \frac{13}{30} = \frac{25 - 13}{30} = \frac{12}{30} = \frac{2}{5} \]

### Step 4: Substitute back into the expression
Now we substitute this back into the expression:
\[ \frac{3}{2} - \frac{2}{5} \]
We need a common denominator again. The LCM of \(2\) and \(5\) is \(10\).

Convert each fraction:
- \(\frac{3}{2} = \frac{3 \cdot 5}{2 \cdot 5} = \frac{15}{10}\)
- \(\frac{2}{5} = \frac{2 \cdot 2}{5 \cdot 2} = \frac{4}{10}\)

Now we can simplify:
\[ \frac{15}{10} - \frac{4}{10} = \frac{15 - 4}{10} = \frac{11}{10} \]

### Step 5: Final Result
Thus, the value of the expression is:
\[ \frac{11}{10} \]