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Commutative property for addition | Arithmetic properties | Pre-Algebra | Khan Academy


Rewrite the expression
below applying the commutative and associative
properties of addition. And then show that
both expressions yield the same result. So one, we could just evaluate
the expression the way it’s written. And then we can
kind of mess around with it using the commutative
and associative properties of addition. So let’s first add 17.5 plus 3. So that’s going to give us 20.5. So this is going to be 20.5. And then to that, we’re
going to add negative 7.5. Now, adding negative 7.5
is the exact same thing as subtracting 7.5. So this is going to be equal
to– the 0.5’s cancel out. And then 20 minus 7 is 13. So that’s our first way
of getting the answer. And we kind of adhered
to the parentheses. Now let’s use the
commutative property. The commutative
property tells us that the order doesn’t matter. We can commute these
numbers around. They can move, commute. It’s like you’re going to work. They can move around. So let’s just move
the numbers around. Let’s make it–
actually, we could do all sorts of crazy things. We could just change
the order here. We could make this negative
7.5 plus 17.5 plus 3. We could keep the
parentheses, just like that. So we would have
essentially just changed the order of this
expression right over here. But let’s use both
the commutative and the associative properties. So now we’ve commuted
everything around. And now we can
re-associate everything. So instead of putting the
parentheses like that, we could put the
parentheses like that. So that’s what the
associative property tells us. So let me write this down. So associative
property of addition tells us that a plus b plus
c– we do a plus b first– is the same thing as a plus b
plus c, where you do b plus c first. The commutative
property tells us that a plus b is
equal to b plus a, that you can move
these guys around. So let’s evaluate this one. We actually got here
using both the commutative and the associative property. So we get negative
7.5 plus 17.5. This is the exact same
thing as 17.5 minus 7.5. It might be easier for
you to realize, OK, I’m adding two numbers
of different signs, so what I could do is take the
difference between the two. And since the larger
number is positive, I have a larger number. Or you could just view
this as 17.5 minus 7.5. So 17.5 minus 7.5–
the 0.5’s cancel out. 17 minus 7 is 10. So this part right
here becomes 10. And then we still
have the plus 3 there. And that, once again, is
going to be equal to 13. And we could keep
commuting this around and keep re-associating it. No matter how we do it,
we are going to get 13.

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