5.02 The distributive property

The distributive property tells us that multiplying the sum or difference of two or more numbers inside a grouping symbol (such as parentheses) by a number will give the same result as multiplying each number (inside the grouping symbol) individually by the number and then adding the products together.

For example, if we want to multiply 15(10+3) we can relate this into groups.

So 15(10+3) is 15 groups of 10 and 15 groups of 3.

To find an equivalent expression, instead of adding the numbers inside the parentheses before multiplying to 15, we can do the following:

We can see that multipying 15 by 10 and adding this to the product of 15 and 3 is a way to simplify expressions more easily than multiplying 15 by 13 directly.

Examples

For example, let's say we wanted to evaluate 72-48. First, we can find the greatest common factor (GCF) between the two numbers.

The factors of 48 are:

1,\,2,\,3,\,4,\,6,\,8,\,12,\,16,\,24,\,48

The factors of 72 are:

1,\,2,\,3,\,4,\,6,\,8,\,9,\,12,\,18,\,24,\,36,72

The numbers that appear in both factor lists are:

1,\,2,\,3,\,4,\,6,\,8,\,12,\,24

The largest number in this list is the GCF, 24.

Now, we can rewrite the expression as an equivalent multiplication by using the distributive property.

\begin{aligned} 48&=24\times 2\\ 72&=24\times 3\\ 72-48&=24\times \left(3-2\right) \end{aligned}

Finally, we multiply the two integers to find our answer.

\begin{aligned} 24\times \left(3-2\right)&=24\times 1\\ 24\times 1&=24 \end{aligned}

So, 72-48=24.

And that is another way to find the sum or difference between to numbers.

Examples