The value ok `((1(1)/(9) xx 1 (1)/(20) div (21)/(38)-(1)/(3))div(2 (4)/(9)div 1(7)/(15) of ""(3)/(5)))/((1)/(5) of (1)/(5) div (1)/(125) - (1)/(125) div (1)/(5) of"" (1)/(5))` lies between......

Options

A. 0.1 and 0.15
B. 0.2 and 0.25
C. 0.15 and 0.2
D. 0.25 and 0.3

Answer

To solve the given expression step by step, we will follow the order of operations (BODMAS/BIDMAS rules).

### Step 1: Simplify the expression
The expression is:
\[ \frac{\left(\frac{1 \frac{1}{9} \times 1 \frac{1}{20}}{\frac{21}{38} - \frac{1}{3}}\right) \div \left(2 \frac{4}{9} \div 1 \frac{7}{15} \text{ of } \frac{3}{5}\right)}{\left(\frac{1}{5} \text{ of } \frac{1}{5} \div \frac{1}{125} - \frac{1}{125} \div \left(\frac{1}{5} \text{ of } \frac{1}{5}\right)\right)} \]

### Step 2: Convert mixed numbers to improper fractions
Convert the mixed numbers:
- \(1 \frac{1}{9} = \frac{10}{9}\)
- \(1 \frac{1}{20} = \frac{21}{20}\)
- \(2 \frac{4}{9} = \frac{22}{9}\)
- \(1 \frac{7}{15} = \frac{22}{15}\)

Now, the expression becomes:
\[ \frac{\left(\frac{\frac{10}{9} \times \frac{21}{20}}{\frac{21}{38} - \frac{1}{3}}\right) \div \left(\frac{22}{9} \div \frac{22}{15} \text{ of } \frac{3}{5}\right)}{\left(\frac{1}{25} \div \frac{1}{125} - \frac{1}{125} \div \frac{1}{25}\right)} \]

### Step 3: Simplify the numerator
Calculate the numerator:
1. Calculate \(\frac{21}{38} - \frac{1}{3}\):
- Find a common denominator (114):
\[ \frac{21 \times 3}{114} - \frac{1 \times 38}{114} = \frac{63 - 38}{114} = \frac{25}{114} \]

2. Now calculate \(\frac{10}{9} \times \frac{21}{20}\):
\[ \frac{10 \times 21}{9 \times 20} = \frac{210}{180} = \frac{7}{6} \]

3. Now divide by \(\frac{25}{114}\):
\[ \frac{7}{6} \div \frac{25}{114} = \frac{7}{6} \times \frac{114}{25} = \frac{798}{150} = \frac{133}{25} \]

4. Now simplify \( \frac{22}{9} \div \frac{22}{15} \text{ of } \frac{3}{5}\):
\[ \frac{22}{9} \div \frac{22}{15} = \frac{15}{9} = \frac{5}{3} \]
\[ \frac{5}{3} \text{ of } \frac{3}{5} = 1 \]

Thus, the numerator simplifies to:
\[ \frac{133}{25} \div 1 = \frac{133}{25} \]

### Step 4: Simplify the denominator
Calculate the denominator:
1. Calculate \(\frac{1}{5} \text{ of } \frac{1}{5} = \frac{1}{25}\).
2. Now calculate \(\frac{1}{25} \div \frac{1}{125} = 5\).
3. Now calculate \(\frac{1}{125} \div \frac{1}{25} = \frac{1}{5}\).
4. Now subtract:
\[ 5 - \frac{1}{5} = \frac{25}{5} - \frac{1}{5} = \frac{24}{5} \]

### Step 5: Final expression
Now, we can write the entire expression as:
\[ \frac{\frac{133}{25}}{\frac{24}{5}} = \frac{133}{25} \times \frac{5}{24} = \frac{133 \times 5}{25 \times 24} = \frac{665}{600} = \frac{133}{120} \]

### Step 6: Approximate the value
To find the approximate decimal value:
\[ \frac{133}{120} \approx 1.1083 \]

### Conclusion
The value of the expression lies between \(1.1\) and \(1.2\).