The value of `((1)/(5)-:(1)/(5)xx(1)/(5))/((1)/(5)-:(1)/(5)"of"(1)/(5))-4(1)/(5)-:105` is

Options

A. A)0
B. B)2
C. C)10
D. D)5

Answer

To solve the expression \(\frac{\frac{1}{5} \div \frac{1}{5} \times \frac{1}{5}}{\frac{1}{5} \div \frac{1}{5} \text{ of } \frac{1}{5} - 4 \cdot \frac{1}{5} \div 105}\), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

**Step 1: Simplify the numerator**
- The numerator is \(\frac{1}{5} \div \frac{1}{5} \times \frac{1}{5}\).
- First, calculate \(\frac{1}{5} \div \frac{1}{5} = 1\).
- Now, multiply by \(\frac{1}{5}\):
\[ 1 \times \frac{1}{5} = \frac{1}{5}. \]

**Step 2: Simplify the denominator**
- The denominator is \(\frac{1}{5} \div \frac{1}{5} \text{ of } \frac{1}{5} - 4 \cdot \frac{1}{5} \div 105\).
- First, calculate \(\frac{1}{5} \div \frac{1}{5} = 1\).
- Next, calculate \(1 \text{ of } \frac{1}{5} = \frac{1}{5}\).
- So, the first part of the denominator simplifies to \(\frac{1}{5}\).

- Now, calculate \(4 \cdot \frac{1}{5} \div 105\):
- \(4 \cdot \frac{1}{5} = \frac{4}{5}\).
- Now, divide by 105:
\[ \frac{4}{5} \div 105 = \frac{4}{5} \times \frac{1}{105} = \frac{4}{525}. \]

- Now, combine the two parts of the denominator:
\[ \frac{1}{5} - \frac{4}{525}. \]
- To subtract these fractions, find a common denominator. The common denominator is 525:
\[ \frac{1}{5} = \frac{105}{525}. \]
- Now perform the subtraction:
\[ \frac{105}{525} - \frac{4}{525} = \frac{101}{525}. \]

**Step 3: Combine the results**
- Now we have:
\[ \frac{\frac{1}{5}}{\frac{101}{525}}. \]
- Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{1}{5} \times \frac{525}{101} = \frac{525}{505}. \]

**Step 4: Simplify the final fraction**
- Simplifying \(\frac{525}{505}\):
\[ \frac{525 \div 5}{505 \div 5} = \frac{105}{101}. \]

### Final Answer:
The value of the expression is \(\frac{105}{101}\).