Question

Which of the following is an example of the distributive property of multiplication over addition for rational numbers.

• A. $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left[-\left(\frac{1}{4}\right)\times \left(\frac{2}{3}\right)\right]+\left[-\left(\frac{1}{4}\right)\times \left(\frac{-4}{7}\right)\right]$
• B. $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left(\frac{2}{3}\right)+\left[-\left(\frac{1}{4}\right)\times \left(\frac{-4}{7}\right)\right]$
• C. $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left[\left(\frac{1}{4}\right)\times \left(\frac{2}{3}\right)\right]-\left[\left(\frac{-4}{7}\right)\right]$
• D. $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}-\left(\frac{1}{4}\right)$

Answer 1

The correct option is A

$-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left[-\left(\frac{1}{4}\right)\times \left(\frac{2}{3}\right)\right]+\left[-\left(\frac{1}{4}\right)\times \left(\frac{-4}{7}\right)\right]$

Explanation for the correct answer:

Option A. $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left[-\left(\frac{1}{4}\right)\times \left(\frac{2}{3}\right)\right]+\left[-\left(\frac{1}{4}\right)\times \left(\frac{-4}{7}\right)\right]$

The distributive property splits the expression $a\times \left(b+c\right)$ into $a\times b+a\times c$.

So, the split expression for $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}$ is shown below

$\left[-\left(\frac{1}{4}\right)\times \left(\frac{2}{3}\right)\right]+\left[-\left(\frac{1}{4}\right)\times \left(\frac{-4}{7}\right)\right]$

This corresponds to option A.

So, Option A is correct.

Explanation for incorrect answers:

Option B. $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left(\frac{2}{3}\right)+\left[-\left(\frac{1}{4}\right)\times \left(\frac{-4}{7}\right)\right]$

The expression $a\times b+a\times c$ involves the product of the term outside the bracket with each term inside the bracket.

However, in the expression $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left(\frac{2}{3}\right)+\left[-\left(\frac{1}{4}\right)\times \left(\frac{-4}{7}\right)\right]$ , the term $-\frac{1}{4}$ is not multiplied by each term inside the bracket.

So, the option (B) is incorrect.

Option C. $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left[\left(\frac{1}{4}\right)\times \left(\frac{2}{3}\right)\right]-\left[\left(\frac{-4}{7}\right)\right]$

The expression $a\times b+a\times c$ involves the product of the term outside the bracket with each term inside the bracket.

However, in the expression$-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left[\left(\frac{1}{4}\right)\times \left(\frac{2}{3}\right)\right]-\left[\left(\frac{-4}{7}\right)\right]$, the term $-\frac{1}{4}$ is not multiplied by each term inside the bracket.

So, the option (C) is incorrect.

Option D. $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}-\left(\frac{1}{4}\right)$
The expression $-\left(\frac{1}{4}\right)\times \left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}=\left\{\left(\frac{2}{3}\right)+\left(\frac{-4}{7}\right)\right\}-\left(\frac{1}{4}\right)$ does not involve any multiplication.

So, option (D) is incorrect.

Hence, the correct option is A.