What is the value of `[45" of "(3/7div15/14)-(6(1)/(2)div3-4)+2]`?

Options

A. `131/6`
B. `147/11`
C. `109/6`
D. `33/7`

Answer

To solve the expression \[ [45 \text{ of } ( \frac{3}{7} \div \frac{15}{14} ) - (6 \frac{1}{2} \div 3 - 4) + 2] \], we will follow the BODMAS rule (Brackets, Orders, Division and Multiplication, Addition and Subtraction).

### Step 1: Solve the division inside the brackets
First, we calculate \(\frac{3}{7} \div \frac{15}{14}\):
\[ \frac{3}{7} \div \frac{15}{14} = \frac{3}{7} \times \frac{14}{15} = \frac{3 \times 14}{7 \times 15} = \frac{42}{105} = \frac{2}{5} \]

### Step 2: Solve the mixed number division
Next, we convert \(6 \frac{1}{2}\) into an improper fraction:
\[ 6 \frac{1}{2} = \frac{13}{2} \]
Now, we calculate \(\frac{13}{2} \div 3\):
\[ \frac{13}{2} \div 3 = \frac{13}{2} \times \frac{1}{3} = \frac{13}{6} \]

### Step 3: Substitute back into the expression
Now we can substitute back into the expression:
\[ [45 \text{ of } \frac{2}{5} - \left(\frac{13}{6} - 4\right) + 2] \]

### Step 4: Simplify the expression
Calculate \(4\) as a fraction with a denominator of \(6\):
\[ 4 = \frac{24}{6} \]
So,
\[ \frac{13}{6} - 4 = \frac{13}{6} - \frac{24}{6} = -\frac{11}{6} \]

### Step 5: Substitute and simplify further
Now we have:
\[ [45 \text{ of } \frac{2}{5} - (-\frac{11}{6}) + 2] \]
This simplifies to:
\[ [45 \text{ of } \frac{2}{5} + \frac{11}{6} + 2] \]

### Step 6: Calculate \(45 \text{ of } \frac{2}{5}\)
Calculate \(45 \text{ of } \frac{2}{5}\):
\[ 45 \times \frac{2}{5} = \frac{90}{5} = 18 \]

### Step 7: Combine all parts
Now we can combine:
\[ 18 + \frac{11}{6} + 2 \]
Convert \(2\) into a fraction with a denominator of \(6\):
\[ 2 = \frac{12}{6} \]
So:
\[ 18 + \frac{11}{6} + \frac{12}{6} = 18 + \frac{23}{6} \]

### Step 8: Convert \(18\) into a fraction
Convert \(18\) into a fraction:
\[ 18 = \frac{108}{6} \]
Now combine:
\[ \frac{108}{6} + \frac{23}{6} = \frac{131}{6} \]

### Final Answer
Thus, the final value is:
\[ \frac{131}{6} \]