The distributive property: multiplication is the operation we use to make repetitive addition simpler.

Suppose you are adding the same number several times. 

For example, you make $6.50 per hour and you are figuring your pay for 8 hours.  Then, you could add $6.50 eight times:

$6.50  +  $6.50  +  $6.50  +  $6.50  +  $6.50  +  $6.50  +  $6.50  +  $6.50  = $52

 

Well, you learned sometime about third grade that we can learn to multiply those amounts and it becomes shorter.

8 * $6.50 = $52

 

Multiplication is an operation that was developed to ease counting.  You might say that learning to count was natural but multiplication is a learned operation.  Similarly, the distributive property was developed to make multiplication of two amounts easier.

In the example above, suppose that you were paid $5 per hour but also were given a bonus of $1.50 per hour for performing your job well.  Assuming that you worked 8 hours and earned the bonus each hour, you could look at the calculation in two ways:

One, 8 * $5 + 8 * $1.50 would give us $40 + $12 for a total of $52.

Or, you could look at it as 8 * ($5 + $1.50) = 8 * $6.50 = $52.

Obviously, in this example, you are able to see that both ideas are simple and you probably could do both in your head.

 

Sometimes, though, we are asked to calculate when we don’t know all amounts.  The best we can do is to set up a pattern that can be completed later.

Suppose that you are paid $5 per hour but the bonus amount is unknown.  Then the problem becomes

.  This calculation results in a partial result pending the amount of the bonus becoming known later.  To simplify your partial answer you could write it as 8 * $5 + 8 * b which gives you $40 + 8b.  Then , when you find the amount of the bonus you multiply it by 8 and add to $40.

 

This idea is known as the distributive property.

 

 

 

The rules for signed numbers apply: