To solve the expression \( \frac{2}{3} - \frac{5}{7} + \frac{3}{4} \div 7 \times \frac{5}{14} \), we follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)):

1. **Division:**
\[ \frac{3}{4} \div 7 = \frac{3}{4} \times \frac{1}{7} = \frac{3}{28} \]

2. **Multiplication:**
\[ \frac{3}{28} \times \frac{5}{14} = \frac{3 \times 5}{28 \times 14} = \frac{15}{392} \]

3. **Addition and Subtraction:**
\[ \frac{2}{3} - \frac{5}{7} + \frac{15}{392} \]

To combine these fractions, we need a common denominator. The least common multiple (LCM) of 3, 7, and 392 is 8232.

Convert each fraction to have the common denominator of 8232:

\[
\frac{2}{3} = \frac{2 \times 2744}{3 \times 2744} = \frac{5488}{8232}
\]
\[
\frac{5}{7} = \frac{5 \times 1176}{7 \times 1176} = \frac{5880}{8232}
\]
\[
\frac{15}{392} = \frac{15 \times 21}{392 \times 21} = \frac{315}{8232}
\]

Now, perform the addition and subtraction:

\[
\frac{5488}{8232} - \frac{5880}{8232} + \frac{315}{8232} = \frac{5488 - 5880 + 315}{8232} = \frac{-77}{8232}
\]

Simplify the fraction:

\[
\frac{-77}{8232} = \frac{-1}{107}
\]

Thus, the solution is:

\[
\boxed{\frac{-1}{107}}
\]

Would you like more details on any part of the solution, or do you have any questions?

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Here are 8 related questions that expand on this information:

1. How do you find the least common multiple (LCM) of a set of numbers?
2. What are the steps to convert a fraction to have a common denominator?
3. How do you simplify a fraction to its lowest terms?
4. Can you explain the order of operations (PEMDAS/BODMAS) in more detail?
5. How do you handle division of fractions?
6. What is the process to add and subtract fractions with different denominators?
7. How do you multiply fractions?
8. What are some common mistakes to avoid when solving fractions?

**Tip:** When working with fractions, always simplify your final answer to its lowest terms for clarity and precision.

Learn how to solve the complex fraction expression 2/3 - 5/7 + 3/4 ÷ 7 of 5/14 with a detailed step-by-step solution. This problem covers key concepts in fractions, order of operations (PEMDAS/BODMAS), and finding common denominators. Suitable for students in Grades 7-9.

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