The student was asked to find the value of `9 4/9-:11 1/3 of 1/6+(1 1/3xx1 4/5-:3/5)xx2 1/6 of 2/3 / 4/3 of 2/3 .His answer was 19 1/4. What is the difference between his answer and the correct answer?

Options

A. `7 3/4`
B. `6 1/3`
C. `7 1/2`
D. ` 6 2/3`

Answer

To solve the given problem step by step, we will first rewrite the mixed numbers and perform the necessary calculations.

### Step 1: Convert Mixed Numbers to Improper Fractions
- Convert \(9 \frac{4}{9}\) to an improper fraction:
\[ 9 \frac{4}{9} = \frac{9 \times 9 + 4}{9} = \frac{81 + 4}{9} = \frac{85}{9} \]
- Convert \(11 \frac{1}{3}\) to an improper fraction:
\[ 11 \frac{1}{3} = \frac{11 \times 3 + 1}{3} = \frac{33 + 1}{3} = \frac{34}{3} \]
- Convert \(1 \frac{1}{3}\) to an improper fraction:
\[ 1 \frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3} \]
- Convert \(1 \frac{4}{5}\) to an improper fraction:
\[ 1 \frac{4}{5} = \frac{1 \times 5 + 4}{5} = \frac{5 + 4}{5} = \frac{9}{5} \]
- Convert \(2 \frac{1}{6}\) to an improper fraction:
\[ 2 \frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6} \]

### Step 2: Substitute the Improper Fractions into the Expression
The expression now looks like this:
\[ \frac{85}{9} \div \frac{34}{3} \cdot \frac{1}{6} + \left(\frac{4}{3} \cdot \frac{9}{5} \div \frac{3}{5}\right) \cdot \frac{13}{6} \cdot \frac{2}{3} \div \frac{4}{3} \cdot \frac{2}{3} \]

### Step 3: Simplify Each Part
1. **First Part:**
\[ \frac{85}{9} \div \frac{34}{3} \cdot \frac{1}{6} = \frac{85}{9} \cdot \frac{3}{34} \cdot \frac{1}{6} \]
- Calculate:
\[ = \frac{85 \cdot 3}{9 \cdot 34 \cdot 6} = \frac{255}{1836} = \frac{85}{612} \]

2. **Second Part:**
\[ \frac{4}{3} \cdot \frac{9}{5} \div \frac{3}{5} = \frac{4}{3} \cdot \frac{9}{5} \cdot \frac{5}{3} = \frac{4 \cdot 9}{3 \cdot 3} = \frac{36}{9} = 4 \]
- Now multiply by \(\frac{13}{6} \cdot \frac{2}{3} \div \frac{4}{3} \cdot \frac{2}{3}\):
\[ 4 \cdot \frac{13}{6} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{3}{2} = \frac{13}{6} \cdot 2 = \frac{26}{6} = \frac{13}{3} \]

### Step 4: Combine Both Parts
Now we combine both parts:
\[ \frac{85}{612} + \frac{13}{3} \]
Convert \(\frac{13}{3}\) to have a common denominator of 612:
\[ \frac{13}{3} = \frac{13 \cdot 204}{3 \cdot 204} = \frac{2652}{612} \]
Now add:
\[ \frac{85}{612} + \frac{2652}{612} = \frac{2737}{612} \]

### Step 5: Convert to Mixed Number
To convert \(\frac{2737}{612}\) to a mixed number:
\[ 2737 \div 612 \approx 4.47 \quad \text{(4 whole parts)} \]
Calculating the remainder:
\[ 2737 - (612 \cdot 4) = 2737 - 2448 = 289 \]
Thus, the mixed number is:
\[ 4 \frac{289}{612} \]

### Step 6: Find the Difference from the Student's Answer
The student's answer was \(19 \frac{1}{4} = \frac{77}{4}\).
Convert \(\frac{77}{4}\) to have a common denominator of 612:
\[ \frac{77}{4} = \frac{77 \cdot 153}{4 \cdot 153} = \frac{11841}{612} \]
Now calculate the difference:
\[ \frac{11841}{612} - \frac{2737}{612} = \frac{11841 - 2737}{612} = \frac{9114}{612} = 14 \frac{1}{2} \]

### Final Answer
The difference between the student's answer and the correct answer is:
\[ \boxed{14 \frac{1}{2}} \]