Answer :
Step-by-step explanation:
Let's simplify the expression step by step using properties of fractions:
1. \( \frac{2}{5} \times \left(- \frac{3}{7}\right) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5} \)
2. Multiplying fractions:
\( \frac{2 \times (-3)}{5 \times 7} - \frac{1 \times 3}{6 \times 2} + \frac{1 \times 2}{14 \times 5} \)
3. Simplify the numerators and denominators:
\( \frac{-6}{35} - \frac{3}{12} + \frac{2}{70} \)
4. Convert the fractions to have a common denominator:
\( \frac{-6 \times 2}{35 \times 2} - \frac{3 \times 35}{12 \times 35} + \frac{2 \times 7}{70 \times 7} \)
5. Calculate the numerators:
\( \frac{-12}{70} - \frac{105}{420} + \frac{14}{490} \)
6. Simplify the fractions:
\( \frac{-12}{70} - \frac{105}{420} + \frac{14}{490} \)
7. Find the LCD of the denominators, which is 420:
\( \frac{-12 \times 6}{70 \times 6} - \frac{105 \times 1}{420 \times 1} + \frac{14 \times 6}{490 \times 6} \)
8. Calculate the numerators again:
\( \frac{-72}{420} - \frac{105}{420} + \frac{84}{2940} \)
9. Combine the fractions:
\( \frac{-72 - 105 + 84}{420} \)
10. Calculate the numerator:
\( \frac{-93}{420} \)
11. Simplify the fraction:
\( -\frac{31}{140} \)
So, the value of \( \frac{2}{5} \times (- \frac{3}{7}) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5} \) using properties of fractions is \( -\frac{31}{140} \).