To solve the expression \( \frac{1}{4} \times \frac{6}{7} + \frac{6}{8} - \frac{2}{9} \div \frac{7}{12} \), we need to follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

### Step-by-Step Solution

1. **Multiplication of the first two fractions:**

   \[
   \frac{1}{4} \times \frac{6}{7} = \frac{1 \times 6}{4 \times 7} = \frac{6}{28} = \frac{3}{14}
   \]

2. **Simplify the second fraction:**

   \[
   \frac{6}{8} = \frac{3}{4}
   \]

3. **Division of the third fraction:**

   The division can be interpreted as:

   \[
   \frac{2}{9} \div \frac{7}{12} = \frac{2}{9} \times \frac{12}{7} = \frac{2 \times 12}{9 \times 7} = \frac{24}{63}
   \]

   Simplify \( \frac{24}{63} \) by dividing the numerator and the denominator by their greatest common divisor (which is 3):

   \[
   \frac{24}{63} = \frac{8}{21}
   \]

4. **Combine all parts:**

   Now, add and subtract the fractions:

   \[
   \frac{3}{14} + \frac{3}{4} - \frac{8}{21}
   \]

5. **Find a common denominator:**

   The least common multiple of 14, 4, and 21 is 84. Convert each fraction:

   \[
   \frac{3}{14} = \frac{3 \times 6}{14 \times 6} = \frac{18}{84}
   \]

   \[
   \frac{3}{4} = \frac{3 \times 21}{4 \times 21} = \frac{63}{84}
   \]

   \[
   \frac{8}{21} = \frac{8 \times 4}{21 \times 4} = \frac{32}{84}
   \]

6. **Add and subtract fractions:**

   \[
   \frac{18}{84} + \frac{63}{84} - \frac{32}{84} = \frac{18 + 63 - 32}{84} = \frac{49}{84}
   \]

7. **Simplify the fraction:**

   The fraction \( \frac{49}{84} \) can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 7:

   \[
   \frac{49}{84} = \frac{7}{12}
   \]

### Final Answer

\[
\frac{1}{4} \times \frac{6}{7} + \frac{6}{8} - \frac{2}{9} \div \frac{7}{12} = \frac{7}{12}
\]

Or in decimal form, approximately:

\[
0.5833
\]

Would you like more details, or do you have any questions?

### Related Questions:

1. How do you multiply and divide fractions?
2. What is the least common multiple and how do you find it for multiple numbers?
3. How do you simplify fractions?
4. How can you convert between fractions and decimals?
5. What are some strategies for solving complex arithmetic expressions?

**Tip:** Remember that dividing by a fraction is the same as multiplying by its reciprocal.

Learn how to solve the fractional expression 1/4 * 6/7 + 6/8 - 2/9 / 7/12 with a detailed step-by-step solution. This problem covers key concepts in fractions and order of operations. Suitable for students in Grades 6-8.

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