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3. Distributivity of multiplication over addition for rational numbers

The distributive property states that multiplying a sum by a number gives the same result as multiplying each added by the number and then adding the products together.

For any number \(a\), \(b\) and \(c\), we have \(a × b + c = a × b + a × c\).

Consider the number \(4\), \(3\) and \(2\) we have

\(4 × 3 + 2 = 4 × 3 + 4 × 2\). (LHS \(=\) RHS)

LHS: \(4 × 2 + 3 = 4 × 5\)

\(= 20\).

RHS: \(4 × 3 + 4 × 2\)

\(= 12 + 8\)

\(= 20\).

Thus, \(4 × 3 + 2 = 4 × 3 + 4 × 2\). (LHS \(=\) RHS)

For any three rational numbers

\frac{a}{b}

\frac{c}{d}

and

\frac{e}{f}

, we have:

\frac{a}{b}

\(×\)

(\frac{c}{d} + \frac{e}{f})

\(=\)

(\frac{a}{b} \times \frac{c}{d})

\(+\)

(\frac{a}{b} \times \frac{e}{f})

Consider the rational numbers

\frac{- 3}{4}

\frac{2}{3}

and

\frac{- 5}{6}

we have

\frac{- 3}{4}

\(×\)

{\frac{2}{3} + \frac{- 5}{6}}

\(=\)

(\frac{- 3}{4} \times \frac{2}{3})

\(+\)

(\frac{- 3}{4} \times \frac{- 5}{6})

. (LHS \(=\) RHS)

LHS:

\frac{- 3}{4}

\(×\)

{\frac{2}{3} + \frac{- 5}{6}}

\frac{- 3}{4}

\(×\)

{\frac{2}{3} + \frac{- 5}{6}}

\(=\)

\frac{- 3}{4}

\(×\)

{\frac{2}{3} \times \frac{2}{2} + \frac{- 5}{6}}

(LCM was taken to get the same denominator)

\(=\)

\frac{- 3}{4}

\(×\)

{\frac{(4 + (- 5))}{6}}

\(=\)

\frac{- 3}{4}

\(×\)

\frac{- 1}{6}

\(=\)

\frac{3}{24}

\(=\)

\frac{1}{8}

RHS:

(\frac{- 3}{4} \times \frac{2}{3})

\(+\)

(\frac{- 3}{4} \times \frac{- 5}{6})

(\frac{- 3}{4} \times \frac{2}{3})

\(=\)

\frac{- 3 \times 2}{4 \times 3}

\(=\)

- \frac{6}{12}

\(=\)

- \frac{1}{2}

And

(\frac{- 3}{4} \times \frac{- 5}{6})

\(=\)

\frac{5}{8}

Therefore

(\frac{- 3}{4} \times \frac{2}{3})

\(+\)

(\frac{- 3}{4} \times \frac{- 5}{6})

\(=\)

- \frac{1}{2}

\(+\)

\frac{5}{8}

\(=\)

- \frac{1}{2} \times \frac{4}{4}

\(+\)

\frac{5}{8} \times \frac{1}{1}

(LCM was taken to get the same denominator)

\(=\)

\frac{(- 4 + 5)}{8}

\(=\)

\frac{1}{8}

Thus,

\frac{- 3}{4}

\(×\)

{\frac{2}{3} + \frac{- 5}{6}}

\(=\)

(\frac{- 3}{4} \times \frac{2}{3})

\(+\)

(\frac{- 3}{4} \times \frac{- 5}{6})

. (LHS \(=\) RHS)

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3. Distributivity of multiplication over addition for rational numbers

Theory: