The value of `(3 1/4-4/5" of "5/6)/(4 1/3div1/5-(3/10+21 1/5))-(2 1/3" of "1 1/2)` is

Options

A. 9
B. 13
C. 12
D. 11

Answer

To solve the expression

\[ \frac{(3 \frac{1}{4} - \frac{4}{5} \text{ of } \frac{5}{6})}{(4 \frac{1}{3} \div \frac{1}{5} - ( \frac{3}{10} + 21 \frac{1}{5}))} - (2 \frac{1}{3} \text{ of } 1 \frac{1}{2}), \]

we will follow the order of operations (BODMAS/BIDMAS).

### Step 1: Convert Mixed Numbers to Improper Fractions

1. Convert \(3 \frac{1}{4}\) to an improper fraction:
\[ 3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4} \]

2. Convert \(4 \frac{1}{3}\) to an improper fraction:
\[ 4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3} \]

3. Convert \(21 \frac{1}{5}\) to an improper fraction:
\[ 21 \frac{1}{5} = \frac{21 \times 5 + 1}{5} = \frac{105 + 1}{5} = \frac{106}{5} \]

4. Convert \(2 \frac{1}{3}\) to an improper fraction:
\[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]

5. Convert \(1 \frac{1}{2}\) to an improper fraction:
\[ 1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2} \]

### Step 2: Calculate the Numerator

Now we calculate the numerator:
\[ \frac{13}{4} - \frac{4}{5} \text{ of } \frac{5}{6} \]

Calculating \(\frac{4}{5} \text{ of } \frac{5}{6}\):
\[ \frac{4}{5} \times \frac{5}{6} = \frac{4 \times 5}{5 \times 6} = \frac{4}{6} = \frac{2}{3} \]

Now substitute this back into the numerator:
\[ \frac{13}{4} - \frac{2}{3} \]

Finding a common denominator (12):
\[ \frac{13 \times 3}{12} - \frac{2 \times 4}{12} = \frac{39}{12} - \frac{8}{12} = \frac{31}{12} \]

### Step 3: Calculate the Denominator

Now we calculate the denominator:
\[ 4 \frac{1}{3} \div \frac{1}{5} - \left( \frac{3}{10} + \frac{106}{5} \right) \]

Calculating \(4 \frac{1}{3} \div \frac{1}{5}\):
\[ \frac{13}{3} \div \frac{1}{5} = \frac{13}{3} \times 5 = \frac{65}{3} \]

Now calculate \(\frac{3}{10} + \frac{106}{5}\):
Finding a common denominator (10):
\[ \frac{3}{10} + \frac{106 \times 2}{10} = \frac{3}{10} + \frac{212}{10} = \frac{215}{10} \]

Now substitute this back into the denominator:
\[ \frac{65}{3} - \frac{215}{10} \]

Finding a common denominator (30):
\[ \frac{65 \times 10}{30} - \frac{215 \times 3}{30} = \frac{650}{30} - \frac{645}{30} = \frac{5}{30} = \frac{1}{6} \]

### Step 4: Combine Numerator and Denominator

Now we have:
\[ \frac{\frac{31}{12}}{\frac{1}{6}} = \frac{31}{12} \times 6 = \frac{31 \times 6}{12} = \frac{186}{12} = \frac{31}{2} \]

### Step 5: Calculate the Final Expression

Now we calculate:
\[ \frac{31}{2} - \frac{7}{3} \]

Finding a common denominator (6):
\[ \frac{31 \times 3}{6} - \frac{7 \times 2}{6} = \frac{93}{6} - \frac{14}{6} = \frac{79}{6} \]

### Final Answer

The value of the expression is:
\[ \frac{79}{6} \text{ or } 13 \frac{1}{6} \]