The value of `(1/3+6/7div6/3+8/9-2/3xx4/3)`

Options

A. `16/21`
B. `{:17/21:}`
C. `{:15/21:}`
D. `{:9/21:}`

Answer

To solve the expression \( \left( \frac{1}{3} + \frac{6}{7} \div \frac{6}{3} + \frac{8}{9} - \frac{2}{3} \times \frac{4}{3} \right) \), we will follow the order of operations (BODMAS/BIDMAS rules). Here’s the step-by-step solution:

### Step 1: Identify the operations
The expression contains addition, division, and multiplication. According to the BODMAS rule, we will perform division and multiplication before addition and subtraction.

### Step 2: Simplify the division
The division \( \frac{6}{7} \div \frac{6}{3} \) can be rewritten as multiplication by the reciprocal:
\[ \frac{6}{7} \div \frac{6}{3} = \frac{6}{7} \times \frac{3}{6} \]
Now, we can simplify:
\[ \frac{6}{7} \times \frac{3}{6} = \frac{3}{7} \]

### Step 3: Simplify the multiplication
Next, we simplify the multiplication \( \frac{2}{3} \times \frac{4}{3} \):
\[ \frac{2}{3} \times \frac{4}{3} = \frac{8}{9} \]

### Step 4: Substitute back into the expression
Now we can substitute these simplified results back into the original expression:
\[ \frac{1}{3} + \frac{3}{7} + \frac{8}{9} - \frac{8}{9} \]
Notice that \( \frac{8}{9} - \frac{8}{9} = 0 \), so the expression simplifies to:
\[ \frac{1}{3} + \frac{3}{7} \]

### Step 5: Find a common denominator
To add \( \frac{1}{3} \) and \( \frac{3}{7} \), we need a common denominator. The least common multiple of 3 and 7 is 21.

Convert each fraction:
\[ \frac{1}{3} = \frac{7}{21} \quad \text{(multiply numerator and denominator by 7)} \]
\[ \frac{3}{7} = \frac{9}{21} \quad \text{(multiply numerator and denominator by 3)} \]

### Step 6: Add the fractions
Now we can add the fractions:
\[ \frac{7}{21} + \frac{9}{21} = \frac{16}{21} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{16}{21} \]

---