# Properties of Operations

Surely you know how to perform the basic operations with two numbers (add, subtract, multiply and divide), but what happens when in the same problem we must perform several mathematical operations at the same time.

Example 1: Three friends go to lunch at McDonald’s and ordered 3 hamburgers, 3 sodas, 2 french-fries, 2 ice cream cones and a lemon cake. They have a \$ 5 discount coupon.  They split the cost evenly among them. How much will each friend have to pay?

To solve this problem we have to carry out many mathematical operations. We need to know how to apply the order of operations to facilitate our work in complex problems like this.

Commutative property

This property tells us that in some mathematical operations, it does not matter if the terms to operate are placed in one order or another. Not all operations have the same property since the terms cannot always be inverted. The operations behave as follows: Commutative property of addition:  a + b = b + a

“Changing the order of addends does not change the sum

Commutative property of multiplication:  a × b = b × a

“The order of the factors does not alter the product”

We can multiply the factors in any order, and the product will be the same

Example 2: In the image, there are six colors that are repeated in three marbles (blue, yellow, orange, red, purple and green). We can know how many marbles there are in total of all those colors by doing a multiplication, and its result will be the same regardless of the order:

Number of marbles per color:  3                        Different Colors:  6

3 × 6 = 6 × 3 = 18

You see, it does not matter in what order we order the multiplication, the result is the same.

Associative Property

“We can group large operations into groups of partial operations”

When an operation is very long, with many terms or sub-operations, we can go through them by grouping or “Associating” them in batches. As in the Commutative property, not all mathematical operations can be associated: Associative Property: We can group the factors in different ways, and the products will be the same.

a × b × c = (a × bc = a × (b × c)

Associative PropertyWe can group addends in any order, and the sum will be the same

a + b + c = (a + b )+ c = a + (b + c)

Example 3 Add the green, blue and red marbles from the figure of Example 3 into two groups, associating the colors:

2 + 3 + 4 = 2 + (3 + 4) = 2 + 7 = 9

2 + 3 + 4 = (2 + 3) + 4 = 5 + 4 = 9

Distributive property

“A term that operates over a group of operations can be distributed among sub-operations”

If any term is multiplying or dividing by another addition or subtraction operation, it can be placed by operating on each of the terms and then completing the sub-operation.

Distributive property:        a(b+c)= ab + ac                       a(b-c)= ab – ac The Distributive Property lets us multiply a sum by multiplying each addend separately and then adding the products.

In this way, we can separate a longer operation into several small operations that may be easier to handle.

Example 4:

4 × 12 = 4 × (10+2) = 4 × 10+ 4 × 2 =40+8 = 48 Identity property

When zero is added to a number the result is the number itself

X +0 = X

When a number is multiplied by 1, the result is that number

X × 1 = X

Example 1:  Find the product of     4×8×25×125

We can shuffle computations to make them easier to carry out (commutative property of multiplication)

=(4×25) ×(8×125)

=100×1000

=100,000

Example 2: Find the sum of    19+33+31+17

We can use the associative property of the addition to group the addends in a different order

= (19+31) + (33+17)

=50+50

=100

Example 3:  Find the product of     45×99

We can use the distributive property of the multiplication (multiply by an equivalent expression)

=45×(100-1)

=(45×100) -(45×1)

=4500-45

=4455

Example 4: Find the numerical value of the expression     84×123 + 16×123

We can use the distributive property of the multiplication (multiply by an equivalent expression)

= (84+16) ×123

= 100 ×123

= 12300

Example 5:  Find the product of     55×102

We can use the distributive property of the multiplication (multiply by an equivalent expression)

=55×(100+2)

=(55×100) + (55×2)

=5500+110

=6510

Inverse Property

Inverse property of Multiplication

The product of a number and its reciprocal is always 1

X × (1/X) = 1