To solve the expression \( \frac{1}{4} \times \frac{6}{7} + \frac{6}{8} - \frac{2}{9} \div 712 \), 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:**

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

2. **Simplify the second fraction:**

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

3. **Division of the third fraction by 712:**

   The division can be interpreted as:

   \[
   \frac{2}{9} \div 712 = \frac{2}{9} \times \frac{1}{712} = \frac{2}{9 \times 712} = \frac{2}{6408}
   \]

   To simplify this, divide the numerator and denominator by their greatest common divisor (which is 2):

   \[
   \frac{2}{6408} = \frac{1}{3204}
   \]

4. **Combine all parts:**

   Now, add the results:

   \[
   \frac{3}{14} + \frac{3}{4} - \frac{1}{3204}
   \]

5. **Find a common denominator:**

   The least common multiple of 14, 4, and 3204 is 3204. Convert each fraction:

   \[
   \frac{3}{14} = \frac{3 \times 229}{14 \times 229} = \frac{687}{3204}
   \]

   \[
   \frac{3}{4} = \frac{3 \times 801}{4 \times 801} = \frac{2403}{3204}
   \]

   The fraction \(\frac{1}{3204}\) is already in terms of the common denominator.

6. **Add and subtract fractions:**

   \[
   \frac{687}{3204} + \frac{2403}{3204} - \frac{1}{3204} = \frac{687 + 2403 - 1}{3204} = \frac{3089}{3204}
   \]

7. **Simplify the fraction (if possible):**

   The fraction \(\frac{3089}{3204}\) cannot be simplified further because 3089 and 3204 are coprime (their greatest common divisor is 1).

### Final Answer

\[
\frac{1}{4} \times \frac{6}{7} + \frac{6}{8} - \frac{2}{9} \div 712 = \frac{3089}{3204}
\]

Or in decimal form, approximately:

\[
0.963
\]

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

### Related Questions:

1. How do you find the least common multiple of multiple denominators?
2. What are some other examples of using the order of operations?
3. How do you simplify complex fractions step-by-step?
4. How do you convert fractions to decimals?
5. What are some tips for adding and subtracting fractions with different denominators?

**Tip:** Always simplify fractions at each step to make calculations easier.

Learn how to solve the fractional expression 1/4 * 6/7 + 6/8 - 2/9 / 712 with a detailed step-by-step solution. This problem covers key mathematical concepts such as fractions and the order of operations. Suitable for students in Grades 6-8.

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