Step-by-Step Calculation
To solve the expression \( \frac{3}{5} \times \frac{2}{7} + \frac{6}{5} \times \frac{2}{7} - \frac{8}{7} \times \frac{2}{7} \), we can break it down using properties of fractions and arithmetic operations.
1. Distributive Property
- We notice that \( \frac{2}{7} \) is a common factor in the first two terms.
- Thus, we can factor it out:
\[
\frac{2}{7} \left( \frac{3}{5} + \frac{6}{5} \right) - \frac{8}{7} \times \frac{2}{7}
\]
2. Simplifying Inside the Parentheses
- Now, simplify \( \frac{3}{5} + \frac{6}{5} \):
\[
= \frac{3 + 6}{5} = \frac{9}{5}
\]
3. Substitute Back
- Substitute back into the expression:
\[
\frac{2}{7} \times \frac{9}{5} - \frac{8}{7} \times \frac{2}{7}
\]
4. Multiplying the Fractions
- Now, compute \( \frac{2}{7} \times \frac{9}{5} \):
\[
= \frac{2 \times 9}{7 \times 5} = \frac{18}{35}
\]
- Next, compute \( \frac{8}{7} \times \frac{2}{7} \):
\[
= \frac{8 \times 2}{7 \times 7} = \frac{16}{49}
\]
5. Final Calculation
- Now, we have:
\[
\frac{18}{35} - \frac{16}{49}
\]
- To subtract these fractions, find a common denominator:
- The least common multiple of 35 and 49 is 245.
- Convert each fraction:
- \( \frac{18}{35} = \frac{18 \times 7}{35 \times 7} = \frac{126}{245} \)
- \( \frac{16}{49} = \frac{16 \times 5}{49 \times 5} = \frac{80}{245} \)
- Now perform the subtraction:
\[
\frac{126}{245} - \frac{80}{245} = \frac{46}{245}
\]
Conclusion
- The final result of the expression \( \frac{3}{5} \times \frac{2}{7} + \frac{6}{5} \times \frac{2}{7} - \frac{8}{7} \times \frac{2}{7} \) is:
\[
\frac{46}{245}
\]