Question

How do you use the distributive property with fractions?

Answer 1

Hint: All the numbers and algebraic expressions have certain properties, so we have generalized most of them for making the calculations easier. The three properties namely, commutative, associative and distributive property are widely used. The distributive property helps to make the multiplications a lot easier, so it is also called the distributive law of multiplication. According to this property, the product of a number (a) with a sum or difference of other two numbers is equal to the sum of the product of the number (a) with first number (b) and the product of the number (a) with the second number (c), that is, $$a(b+c)=ab+ac$$ .

Complete step by step answer:
Let a fraction $({\displaystyle \frac{a}{b}})$ be in multiplication with the sum of other two fractions $({\displaystyle \frac{c}{d}}+{\displaystyle \frac{e}{f}})$ , so the answer to this expression using distributive property is
$\Rightarrow {\displaystyle \frac{a}{b}}\times {\displaystyle \frac{c}{d}}+{\displaystyle \frac{a}{b}}\times {\displaystyle \frac{e}{f}}$ .
Hence, the distributive property for fractions is
$\Rightarrow {\displaystyle \frac{a}{b}}({\displaystyle \frac{c}{d}}+{\displaystyle \frac{e}{f}})={\displaystyle \frac{a}{b}}\times {\displaystyle \frac{c}{d}}+{\displaystyle \frac{a}{b}}\times {\displaystyle \frac{e}{f}}$
and vice-versa.

Note: While multiplying the fractions using the distributive property, we multiply the numerator of one fraction with the numerator of the other fraction, and the denominator of one fraction with the denominator of the other fraction. And we will also simplify the fraction if possible.
For example let a given expression be – ${\displaystyle \frac{3}{2}}({\displaystyle \frac{4}{6}}+{\displaystyle \frac{2}{5}})$ , applying the distributive property, we get –
$$\begin{gathered} \Rightarrow {\displaystyle \frac{3}{2}}({\displaystyle \frac{4}{6}}+{\displaystyle \frac{2}{5}})={\displaystyle \frac{3}{4}}\times {\displaystyle \frac{4}{6}}+{\displaystyle \frac{3}{4}}\times {\displaystyle \frac{2}{5}} \\ \Rightarrow {\displaystyle \frac{3}{2}}({\displaystyle \frac{4}{6}}+{\displaystyle \frac{2}{5}})={\displaystyle \frac{12}{24}}+{\displaystyle \frac{6}{20}} \end{gathered}$$
Now, on the prime factorization of the numerator and the denominator, we get the simplified form as –
$\Rightarrow {\displaystyle \frac{3}{2}}({\displaystyle \frac{4}{6}}+{\displaystyle \frac{2}{5}})={\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{10}}$
We can further solve this fraction by taking the LCM of the denominator and then applying the given arithmetic fraction. So,
$\Rightarrow {\displaystyle \frac{3}{2}}({\displaystyle \frac{4}{6}}+{\displaystyle \frac{2}{5}})={\displaystyle \frac{5+3}{10}}={\displaystyle \frac{8}{10}}={\displaystyle \frac{4}{5}}$