To solve the expression $\frac{2}{5} \times \left(-\frac{3}{7}\right) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5}$, we will evaluate each term individually and then combine the results.

Step 1: Evaluate each multiplication

1. $\frac{2}{5} \times \left(-\frac{3}{7}\right) = \frac{2 \times -3}{5 \times 7} = \frac{-6}{35}$
2. $\frac{1}{6} \times \frac{3}{2} = \frac{1 \times 3}{6 \times 2} = \frac{3}{12} = \frac{1}{4}$
3. $\frac{1}{14} \times \frac{2}{5} = \frac{1 \times 2}{14 \times 5} = \frac{2}{70} = \frac{1}{35}$

Step 2: Combine the results

The expression becomes: $\frac{-6}{35} - \frac{1}{4} + \frac{1}{35}$

Step 3: Simplify the expression

1. Combine the fractions with a common denominator. The common denominator for $\frac{-6}{35}$ and $\frac{1}{35}$ is 35: $\frac{-6 + 1}{35} = \frac{-5}{35} = \frac{-1}{7}$

2. Now we have: $\frac{-1}{7} - \frac{1}{4}$ To combine these fractions, find the common denominator, which is 28: $\frac{-1}{7} = \frac{-4}{28} \quad \text{and} \quad \frac{1}{4} = \frac{7}{28}$ So, the expression becomes: $\frac{-4}{28} - \frac{7}{28} = \frac{-11}{28}$

Final Answer:

$\boxed{\frac{-11}{28}}$

Would you like further details or have any questions?

Here are five related questions:

1. How would you solve an expression with three different fractions with unlike denominators?
2. What is the process of finding a common denominator?
3. Can you explain why multiplying a fraction by a negative number changes the sign?
4. How do you simplify fractions after multiplication?
5. How do you add or subtract fractions with different denominators?

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