The value of `5 (5)/(6) div 3(1)/(2) xx 2 (1)/(10) + (3)/(5) ` of ` 7(1)/(2) div (2)/(3) - (2)/(3) div (8)/(15) xx 1 (1)/(5)`

Options

A. `8 (3)/(4)`
B. `2 (3)/(5)`
C. `8(3)/(5 )`
D. `1(3)/(4)`

Answer

To solve the expression given in the question, we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
The expression is:
\[ 5 \frac{5}{6} \div 3 \frac{1}{2} \times 2 \frac{1}{10} + \frac{3}{5} \]
and
\[ 7 \frac{1}{2} \div \frac{2}{3} - \frac{2}{3} \div \frac{8}{15} \times 1 \frac{1}{5} \]

First, we convert the mixed numbers to improper fractions:
- \( 5 \frac{5}{6} = \frac{35}{6} \)
- \( 3 \frac{1}{2} = \frac{7}{2} \)
- \( 2 \frac{1}{10} = \frac{21}{10} \)
- \( 7 \frac{1}{2} = \frac{15}{2} \)
- \( 1 \frac{1}{5} = \frac{6}{5} \)

Now, the expression becomes:
\[ \frac{35}{6} \div \frac{7}{2} \times \frac{21}{10} + \frac{3}{5} \]
and
\[ \frac{15}{2} \div \frac{2}{3} - \frac{2}{3} \div \frac{8}{15} \times \frac{6}{5} \]

### Step 2: Solve the First Part
Now, we will solve the first part:
\[ \frac{35}{6} \div \frac{7}{2} \times \frac{21}{10} + \frac{3}{5} \]

1. **Division of Fractions**:
\[ \frac{35}{6} \div \frac{7}{2} = \frac{35}{6} \times \frac{2}{7} = \frac{35 \times 2}{6 \times 7} = \frac{70}{42} = \frac{5}{3} \]

2. **Multiplication**:
\[ \frac{5}{3} \times \frac{21}{10} = \frac{5 \times 21}{3 \times 10} = \frac{105}{30} = \frac{7}{2} \]

3. **Adding the Second Fraction**:
\[ \frac{7}{2} + \frac{3}{5} \]
To add these fractions, we need a common denominator. The LCM of 2 and 5 is 10:
\[ \frac{7}{2} = \frac{35}{10}, \quad \frac{3}{5} = \frac{6}{10} \]
Thus,
\[ \frac{35}{10} + \frac{6}{10} = \frac{41}{10} \]

### Step 3: Solve the Second Part
Now, we will solve the second part:
\[ \frac{15}{2} \div \frac{2}{3} - \frac{2}{3} \div \frac{8}{15} \times \frac{6}{5} \]

1. **Division of Fractions**:
\[ \frac{15}{2} \div \frac{2}{3} = \frac{15}{2} \times \frac{3}{2} = \frac{45}{4} \]

2. **Second Division**:
\[ \frac{2}{3} \div \frac{8}{15} = \frac{2}{3} \times \frac{15}{8} = \frac{30}{24} = \frac{5}{4} \]

3. **Multiplication**:
\[ \frac{5}{4} \times \frac{6}{5} = \frac{6}{4} = \frac{3}{2} \]

4. **Subtraction**:
\[ \frac{45}{4} - \frac{3}{2} \]
Convert \( \frac{3}{2} \) to have a common denominator of 4:
\[ \frac{3}{2} = \frac{6}{4} \]
Thus,
\[ \frac{45}{4} - \frac{6}{4} = \frac{39}{4} \]

### Step 4: Combine Both Parts
Now we combine the results from both parts:
\[ \frac{41}{10} \text{ and } \frac{39}{4} \]

To add these, we need a common denominator. The LCM of 10 and 4 is 20:
\[ \frac{41}{10} = \frac{82}{20}, \quad \frac{39}{4} = \frac{195}{20} \]
Thus,
\[ \frac{82}{20} + \frac{195}{20} = \frac{277}{20} \]

### Final Answer
The final answer is:
\[ \frac{277}{20} = 13 \frac{17}{20} \]