The Distributive Property for Arithmetic
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The Distributive Property for Arithmetic


It’s Professor Dave; let’s introduce the
distributive property. As we learn math, we will see that there are
certain properties of numbers that are very important. We already learned about the commutative property
and the associative property, and we saw how addition and multiplication abide by these
properties, while subtraction and division do not. Now let’s learn another property, the distributive
property. To see how this works, let’s look at some
more apples. Say we have five piles of seven apples. Five piles times seven apples per pile gives
us thirty-five apples. What if we split each pile up into a group
of three and a group of four? There are a few ways we could represent this. Let’s take the seven and split it up into
three plus four, in parentheses. This is clearly the same value we once had,
because three plus four is seven, which makes sense because rearranging the apples shouldn’t
change the number of them that are present. But we could also say that we have five piles
of three and five piles of four. That means five times three plus five times
four. Visually, we are just describing what we see,
but mathematically, we have just derived the distributive property. When we have a number that is multiplied by
a sum or difference, we can distribute that number across each value within the sum or
difference. In this case, that means five times three
plus five times four. That gives us fifteen plus twenty, which is
indeed still thirty-five. So whether we add the numbers in the parentheses
first and then multiply by the other number, or distribute the number across the sum, we
will get the same result, and that is the essence of the distributive property, which
will be very important once we get to algebra. We can even use this property in reverse. Let’s say we have five times seventeen. If we don’t want to write down a calculation,
we can easily do this in our heads if we split seventeen up into ten plus five plus two. Now, we can distribute the five across the
sum, and rewrite this as five times ten plus five times five plus five times two. These sums can be done easily in your head,
and we get fifty plus twenty-five plus ten, and that’s eighty-five. This sort of mental trickery is very useful
for multiplying large numbers, so let’s move on to that topic next.

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