Distributive Property Of Addition & Multiplication (With Examples)

When you are learning algebra and you are looking at complex mathematical equations, you may be scratching your head. It helps greatly to break the equations down into smaller parts to solve the equation. The distributive property law is a tool to help you do that. It is used in advanced multiplication, addition and algebra.

​Tip:​ The distributive property of addition and multiplication states that:

\(a × (x + y) = ax + ay\)

Or to give a concrete example:

\(3 × (4 + 5) = 3 × 4 + 3 × 5\)

The distributive property allows you to in essence, to move some numbers around in complex mathematical equations of all types. If a number is multiplied by two numbers in parentheses, you can work this out by multiplying the first number by the ones in parentheses separately, and then completing the addition. For example:

\(a × (x + y) = ax + ay\)

Or, using numbers:

\(3 × (4 + 5) = 3 × 4 + 3 × 5\)

Breaking down a complex equation into smaller pieces makes it easier to solve the equation and makes it easier to digest the information in smaller amounts.

The distributive property is usually first approached by students when they start advanced multiplication problems, meaning when adding or multiplying, you have to carry a one. This can be problematic if you have to solve it in your head without working the problem out on paper. In addition and multiplication, you take the larger number and round it down to the nearest number that is divisible by 10, then multiply both numbers by the smaller number. For example:

\(36 × 4 = ?\)

This can be expressed as:

\(4 × (30 + 6) = ?\)

Which allows you to use the distributive property of multiplication and answer the question as follows:

\((4 × 30) + (4 × 6) = ?\)
\(120 + 24 = 144\)

The same rule of moving some of the numbers around to solve an equation is used in simple algebra. This is done by eliminating the parenthesis portion of the equation. For instance, the equation ​a​ × (​b​ + ​c​) = ? shows that both letters in parenthesis need to be multiplied by the letter on the outside of the parenthesis, so you distribute the multiplication of a between both ​b​ and ​c​. The equation can also be written as: (​ab​) + (​ac​) = ? For example:

\(3 × (2 + 4) = ?\)
\((3 × 2) + (3 × 4) =?\)
\(6 + 12 = 18\)

You can also combine some numbers to make it easier to solve an equation. For example:

\(16 × 6 + 16 × 4 = ?\)
\(16 × (6 + 4) = ?\)
\(16 × 10 = 160\)

For another example, watch the video below:

\(a × (b + c) = ?\)

Where ​a​ = 3, ​b​ = 2 and ​c​ = 4

\(6 × (2 + 4) =?\)
\(5 × (6 + 2)= ?\)
\(4 × ( 7 + 2 + 3) =?\)
\(6 × (5 + 4) = ?\)