The value of `( (1)/(6) + [ 4 (3)/(4) -(3 (1)/(6) - 2 (1)/(3) ) ] )/( ( (1)/(5)" of "(1)/(5) div (1)/(5) ) div ( (1)/(5) div (1)/(5) xx (1)/(5) ) )` lies between:

Options

A. `4.2 and 4.3`
B. `4.0 and 4.1`
C. `4.3 and 4.4`
D. `4.6 and 4.8`

Answer

To solve the given expression step by step, let's break it down systematically.

### Step 1: Simplify the Numerator
The numerator is given as:
\[ \frac{1}{6} + \left[ 4 \frac{3}{4} - \left( 3 \frac{1}{6} - 2 \frac{1}{3} \right) \right] \]

First, convert the mixed fractions to improper fractions:
- \(4 \frac{3}{4} = \frac{16 + 3}{4} = \frac{19}{4}\)
- \(3 \frac{1}{6} = \frac{18 + 1}{6} = \frac{19}{6}\)
- \(2 \frac{1}{3} = \frac{6 + 1}{3} = \frac{7}{3}\)

Now substitute these values back into the expression:
\[ \frac{1}{6} + \left[ \frac{19}{4} - \left( \frac{19}{6} - \frac{7}{3} \right) \right] \]

### Step 2: Simplify the Inner Bracket
Now, simplify the inner bracket:
\[ \frac{19}{6} - \frac{7}{3} \]
To subtract these fractions, find a common denominator (which is 6):
\[ \frac{19}{6} - \frac{14}{6} = \frac{5}{6} \]

Now substitute this back into the expression:
\[ \frac{1}{6} + \left[ \frac{19}{4} - \frac{5}{6} \right] \]

### Step 3: Simplify the Remaining Bracket
Now simplify:
\[ \frac{19}{4} - \frac{5}{6} \]
Again, find a common denominator (which is 12):
\[ \frac{19 \times 3}{12} - \frac{5 \times 2}{12} = \frac{57}{12} - \frac{10}{12} = \frac{47}{12} \]

Now substitute this back into the expression:
\[ \frac{1}{6} + \frac{47}{12} \]

### Step 4: Add the Two Fractions
To add these fractions, find a common denominator (which is 12):
\[ \frac{1 \times 2}{12} + \frac{47}{12} = \frac{2}{12} + \frac{47}{12} = \frac{49}{12} \]

### Step 5: Simplify the Denominator
Now, simplify the denominator:
\[ \left( \frac{1}{5} \text{ of } \frac{1}{5} \div \frac{1}{5} \right) \div \left( \frac{1}{5} \div \frac{1}{5} \times \frac{1}{5} \right) \]

First, simplify the left part:
\[ \frac{1}{5} \text{ of } \frac{1}{5} = \frac{1}{5} \times \frac{1}{5} = \frac{1}{25} \]
Then divide by \(\frac{1}{5}\):
\[ \frac{1}{25} \div \frac{1}{5} = \frac{1}{25} \times 5 = \frac{5}{25} = \frac{1}{5} \]

Now simplify the right part:
\[ \frac{1}{5} \div \frac{1}{5} = 1 \quad \text{and} \quad 1 \times \frac{1}{5} = \frac{1}{5} \]

Now substitute back into the denominator:
\[ \frac{1}{5} \div \frac{1}{5} = 1 \]

### Step 6: Final Calculation
Now we have:
\[ \frac{\frac{49}{12}}{1} = \frac{49}{12} \]

### Step 7: Approximate the Value
To find the approximate value:
\[ \frac{49}{12} \approx 4.0833 \]

### Conclusion
The value of the expression lies between 4 and 5.