Q70. - Algebra 2

Question

CRITICAL THINKING

Is the Distributive Property also true for division? In other words, does $\frac{b+c}{a}=\frac{b}{a}+\frac{c}{a},a\ne 0$? If so, give an example and explain why it is true. If not true, give a counterexample.

Step-by-Step Solution

Answer

Yes, Distributive property is true for division.

One example is $\frac{4+6}{2}=\frac{4}{2}+\frac{6}{2}$ 

Example is true because on simplifying left hand side and right hand side separately each terms gives same result

LHS	RHS
$\begin{array}{c}\frac{4+6}{2}=\frac{10}{2}\\ =5\end{array}$	$\begin{array}{c}\frac{4}{2}+\frac{6}{2}=2+3\\ =5\end{array}$

Step 1 - Check Distributive Property is true for division of not

Let $a,b,\text{ and }c$ be real numbers with $a\ne 0$ such that a divides sum of $b\text{ and }c$

Since, denominator of a fraction can be separated with each term of numerator so distributive property is true for division.

Step 2 - Substitution for exapmle

To give an example substitute 2 for a, 4 for b and 6 for c into expression

$\frac{b+c}{a}=\frac{b}{a}+\frac{c}{a}$ 

Step 3 - Verify 4 + 6 2 = 4 2 + 6 2

First evaluate left hand side by adding terms of numerator then dividing it by denominator as shown below

$\begin{array}{c}\frac{4+6}{2}=\frac{10}{2}\\ =5\end{array}$

Now to evaluate right hand side, first simplify both fractions then add the resultant terms

$\begin{array}{c}\frac{4}{2}+\frac{6}{2}=2+3\\ =5\end{array}$

Thus, $\frac{4+6}{2}=\frac{4}{2}+\frac{6}{2}$