The World of Numbers | Exercise 3.3

Question 7

Simplify the following using the distributive property:

$\dfrac{7}{9}\left(\dfrac{6}{7} - \dfrac{3}{4}\right)$

Solution

Understand the Question

The distributive property of multiplication over subtraction states that $a(b - c) = (a \times b) - (a \times c)$ .
To simplify the expression using this property, multiply $\dfrac{7}{9}$ with each fraction inside the parentheses separately and then subtract the two products.

Step 1 · Apply Distributive Property

Using the distributive property $a(b - c) = (a \times b) - (a \times c)$

\begin{aligned} \dfrac{7}{9}\left(\dfrac{6}{7} - \dfrac{3}{4}\right) &= \left(\dfrac{7}{9} \times \dfrac{6}{7}\right) - \left(\dfrac{7}{9} \times \dfrac{3}{4}\right) \\[0.6em] &= \dfrac{42}{63} - \dfrac{21}{36} \\[0.6em] &= \dfrac{2}{3} - \dfrac{7}{12} \end{aligned}

Step 2 · Subtract the Fractions

The $\text{LCM}$ of denominators $3$ and $12$ is $12$ .

Convert $\dfrac{2}{3}$ to have a denominator of $12$ $\dfrac{2}{3} = \dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}$

Now subtract the fractions

\begin{aligned} \dfrac{8}{12} - \dfrac{7}{12} &= \dfrac{8 - 7}{12} \\[0.6em] &= \dfrac{1}{12} \end{aligned}

Answer

$\dfrac{1}{12}$

Common Mistakes

Partial Distribution: Multiplying $\dfrac{7}{9}$ only by the first term $\dfrac{6}{7}$ and forgetting to multiply it by the second term $\dfrac{3}{4}$ .
Sign Error: Adding the products instead of subtracting them, misapplying $a(b - c) = ab - ac$ as $ab + ac$ .
Direct Simplification: Solving the bracket first $\left(\dfrac{6}{7} - \dfrac{3}{4}\right)$ directly without using the distributive property as specifically instructed by the question.

More questions in Exercise 3.3