What is the simplified value of `[{(3-2/(1 + 1/(4 + 2/3))) xx 1 13/21 -: 3 2/7}-1/3] xx 9/10 =?`

Options

A. 1
B. 2
C. `1/5`
D. `3/10`

Answer

To solve the expression \[ \left[ \left( 3 - \frac{2}{1 + \frac{1}{4 + \frac{2}{3}}} \right) \times \frac{1 \frac{13}{21}}{3 \frac{2}{7}} - \frac{1}{3} \right] \times \frac{9}{10} \], we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Simplify the innermost fraction**:
\[ 4 + \frac{2}{3} = \frac{12}{3} + \frac{2}{3} = \frac{14}{3} \]
**Hint**: When adding fractions, find a common denominator.

2. **Substitute back into the expression**:
\[ 1 + \frac{1}{4 + \frac{2}{3}} = 1 + \frac{1}{\frac{14}{3}} = 1 + \frac{3}{14} = \frac{14}{14} + \frac{3}{14} = \frac{17}{14} \]
**Hint**: Remember to convert whole numbers to fractions for easy addition.

3. **Now simplify the fraction**:
\[ \frac{2}{1 + \frac{1}{4 + \frac{2}{3}}} = \frac{2}{\frac{17}{14}} = 2 \times \frac{14}{17} = \frac{28}{17} \]
**Hint**: Dividing by a fraction is the same as multiplying by its reciprocal.

4. **Substituting back into the expression**:
\[ 3 - \frac{28}{17} = \frac{51}{17} - \frac{28}{17} = \frac{23}{17} \]
**Hint**: Convert whole numbers to fractions with a common denominator for subtraction.

5. **Now, simplify \(1 \frac{13}{21}\) and \(3 \frac{2}{7}\)**:
\[ 1 \frac{13}{21} = \frac{34}{21}, \quad 3 \frac{2}{7} = \frac{23}{7} \]
**Hint**: Convert mixed numbers to improper fractions.

6. **Now perform the multiplication**:
\[ \left( \frac{23}{17} \times \frac{34}{21} \right) \div \frac{23}{7} = \frac{23}{17} \times \frac{34}{21} \times \frac{7}{23} \]
The \(23\) cancels out:
\[ = \frac{34 \times 7}{17 \times 21} = \frac{238}{357} \]
**Hint**: When dividing fractions, multiply by the reciprocal.

7. **Now, simplify \(\frac{238}{357}\)**:
\[ \frac{238 \div 119}{357 \div 119} = \frac{2}{3} \]
**Hint**: Always check for common factors to simplify fractions.

8. **Subtract \(\frac{1}{3}\)**:
\[ \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \]
**Hint**: Keep the denominators the same when subtracting fractions.

9. **Finally, multiply by \(\frac{9}{10}\)**:
\[ \frac{1}{3} \times \frac{9}{10} = \frac{9}{30} = \frac{3}{10} \]
**Hint**: Multiply the numerators and denominators separately.

### Final Answer:
\[ \frac{3}{10} \]