Simplifying the given expression: 2/5 × (-3/7) - 1/6 1/14 × 2/5

First, let's simplify each multiplication separately:

1. 2/5 × (-3/7):
To multiply fractions, we multiply the numerators together and the denominators together. So, 2/5 × (-3/7) = (2 × -3) / (5 × 7) = -6/35.

2. 1/6 1/14 × 2/5:
To multiply mixed numbers, we convert them to improper fractions. 1/6 1/14 can be written as (6/6 + 1/6) × (14/14 + 1/14) = 7/6 × 15/14 = (7 × 15) / (6 × 14) = 105/84.

Now, let's substitute these values back into the original expression:

-6/35 - 105/84.

Next, we need to find a common denominator to combine these fractions.

Finding a common denominator:
The least common multiple (LCM) of 35 and 84 is 420. Therefore, we need to convert both fractions to have a denominator of 420.

Multiplying the first fraction by (12/12) and the second fraction by (5/5), we get:

(-6/35) × (12/12) = -72/420
(105/84) × (5/5) = 525/420

Now, we can combine the fractions:

-72/420 - 525/420.

Subtracting the fractions:

To subtract fractions with the same denominator, we subtract the numerators and keep the common denominator.

(-72/420) - (525/420) = (-72 - 525)/420 = -597/420.

Simplifying the fraction:

To simplify the fraction, we divide both the numerator and denominator by their greatest common divisor (GCD), which is 3.

(-597/3)/(420/3) = -199/140.

Therefore, the simplified expression is -199/140.

In conclusion, the value of the expression 2/5 × (-3/7) - 1/6 1/14 × 2/5 is -199/140.