Distributive property of Multiplication over Addition

Formula

$a \times (b+c)$ $\,=\,$ $a \times b + a \times c$

An arithmetic property that distributes the multiplication across the addition is called the distributive property of multiplication over addition.

Introduction

$a$, $b$ and $c$ are three literals and represent three terms.

The product of the term $a$ and the sum of the terms $b$ and $c$ is written in mathematical form as follows.

$a \times (b+c)$

The product of them can be evaluated by distributing the multiplication over the addition.

$\implies$ $a \times (b+c)$ $\,=\,$ $a \times b + a \times c$

This distributive property can also be used to distribute the multiplication of a term over the sum of two or more terms.

$\implies$ $a \times (b+c+d+\ldots)$ $\,=\,$ $a \times b + a \times c + a \times d + \ldots$

Proof

Learn how to prove the distributive property of multiplication across addition in algebraic form by geometric method.

Verification

$2$, $3$ and $4$ are three numbers. Find the product of number $2$ and sum of the numbers of $3$ and $4$.

$2 \times (3+4)$

Find the value of this arithmetic expression.

$\implies$ $2 \times (3+4)$ $\,=\,$ $2 \times 7$

$\implies$ $2 \times (3+4) \,=\, 14$

Now, find the sum of the products of $2$ and $3$, and $2$ and $4$.

$2 \times 3 + 2 \times 4$ $\,=\,$ $6+8$

$\implies$ $2 \times 3 + 2 \times 4$ $\,=\,$ $14$

Now, compare the results of both expressions. They are equal.

$\,\,\, \therefore \,\,\,\,\,\,$ $2 \times (3+4)$ $\,=\,$ $2 \times 3 + 2 \times 4$ $\,=\,$ $14$

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