Chose the correct option
A student was anked to find the value of `(((2(1)/3+2(1)/2-1/6)div2(1)/3xx5(2)/3div1(2)/3of4(1)/4)/(3(1)/5div4(1)/2of5(1)/3+5(1)/3xx3/4div2(2)/3))`. His answer was6/7. What is the difference between the correct answer and his answer?

Options

A. `{:(9)/(14):}`
B. `{:(5)/(14):}`
C. `{:(18)/(49):}`
D. `{:(6)/(49):}`

Answer

To solve the given expression step by step, we will first convert all mixed numbers into improper fractions and then perform the necessary operations.

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(2 \frac{1}{3}\) to an improper fraction:
\[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3} \]
2. Convert \(2 \frac{1}{2}\) to an improper fraction:
\[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2} \]
3. Convert \(5 \frac{2}{3}\) to an improper fraction:
\[ 5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{17}{3} \]
4. Convert \(1 \frac{2}{3}\) to an improper fraction:
\[ 1 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3} \]
5. Convert \(4 \frac{1}{4}\) to an improper fraction:
\[ 4 \frac{1}{4} = \frac{4 \times 4 + 1}{4} = \frac{17}{4} \]
6. Convert \(3 \frac{1}{5}\) to an improper fraction:
\[ 3 \frac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{16}{5} \]
7. Convert \(4 \frac{1}{2}\) to an improper fraction:
\[ 4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2} \]
8. Convert \(5 \frac{1}{3}\) to an improper fraction:
\[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{16}{3} \]
9. Convert \(2 \frac{2}{3}\) to an improper fraction:
\[ 2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3} \]

### Step 2: Substitute the Improper Fractions into the Expression
The expression now looks like:
\[ \frac{\left(\frac{7}{3} + \frac{5}{2} - \frac{1}{6}\right) \div \left(\frac{17}{3} \times \frac{5}{3} \div \frac{5}{3} \times \frac{17}{4}\right)}{\left(\frac{16}{5} \div \frac{9}{2} \times \frac{5}{3} + \frac{16}{3} \times \frac{3}{4} \div \frac{8}{3}\right)} \]

### Step 3: Simplify the Numerator
1. Find a common denominator for \( \frac{7}{3} + \frac{5}{2} - \frac{1}{6} \):
- The LCM of 3, 2, and 6 is 6.
- Convert each fraction:
\[ \frac{7}{3} = \frac{14}{6}, \quad \frac{5}{2} = \frac{15}{6}, \quad \frac{1}{6} = \frac{1}{6} \]
- Combine:
\[ \frac{14}{6} + \frac{15}{6} - \frac{1}{6} = \frac{28}{6} = \frac{14}{3} \]

### Step 4: Simplify the Denominator
1. Simplify \( \frac{16}{5} \div \frac{9}{2} \):
\[ \frac{16}{5} \times \frac{2}{9} = \frac{32}{45} \]
2. Simplify \( \frac{16}{3} \times \frac{3}{4} \div \frac{8}{3} \):
\[ \frac{16}{3} \times \frac{3}{4} = \frac{12}{3} = 4, \quad \text{then } 4 \div \frac{8}{3} = 4 \times \frac{3}{8} = \frac{12}{8} = \frac{3}{2} \]
3. Combine:
\[ \frac{32}{45} + \frac{3}{2} = \frac{32}{45} + \frac{67.5}{45} = \frac{99.5}{45} = \frac{199}{90} \]

### Step 5: Final Calculation
Now we can substitute back into the expression:
\[ \frac{\frac{14}{3}}{\frac{199}{90}} = \frac{14}{3} \times \frac{90}{199} = \frac{1260}{597} = \frac{420}{199} \]

### Step 6: Calculate the Difference
The student's answer was \( \frac{6}{7} \). We need to find the difference:
\[ \frac{420}{199} - \frac{6}{7} = \frac{420 \times 7 - 6 \times 199}{199 \times 7} = \frac{2940 - 1194}{1393} = \frac{1746}{1393} \]

### Step 7: Final Answer
The difference between the correct answer and the student's answer is:
\[ \frac{1746}{1393} \]