Simplifying Rational Numbers and Identifying Multiplicative Properties

In this problem, we are asked to simplify three expressions using the distributive property of multiplication over addition for rational numbers.

The distributive property states that a times the sum of b and c is equal to a times b plus a times c.

Let's solve part a. We have negative three eighths times the sum of four sevenths and negative eleven sevenths.

Using the distributive property, we multiply negative three eighths with each term inside the parentheses.

Now we simplify each term. For the first term, negative three times four is negative twelve, and eight times seven is fifty-six. Similarly, for the second term, negative three times negative eleven is positive thirty-three, and eight times seven is fifty-six.

Since they have a common denominator, we add the numerators. Negative twelve plus thirty-three leaves us with twenty-one over fifty-six.

We can simplify this by dividing both numerator and denominator by their greatest common divisor, which is seven. This gives us three eighths.

Moving on to part b, we have negative two fifths times the difference of three eighths and twenty-five. Since subtraction is adding a negative, we can still use the distributive rule.

Distributing negative two fifths, we get negative two fifths times three eighths, minus negative two fifths times twenty-five.