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 × b )× c = a × (b × c)
Associative Property: We 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
Inverse Property of Addition
The sum of a number and its opposite is always 0
X + (-X) = 0
Properties of Operations:
Commutative Property For Addition |
The order in which numbers are added does not affect the sum. |
Example: 6+4 = 4+6 |
Commutative Property For Multiplication |
The order in which numbers are multiplied does not affect the product. |
Example; 8·3 = 3·8 |
Associative Property For Addition |
The way in which numbers are grouped does not affect the sum. |
Example: 7 + (3+2) = (7+3) + 2 |
Associative Property For Multiplication |
The way in which numbers are grouped does not affect the product. |
Example: (5·2) ·4 = 5· (2·4) |
Distributive Property |
To multiply a sum of numbers, either (1) multiply the numbers separately, then add the products. |
Example: 4 · (6 + 3) = (4·6) + (4·3) =24+ 12=36 |
Or (2) add the numbers in parentheses and multiply the sum; |
Example: 4· (6+3) = 4· (9) =36 |
Identity Property for Addition |
The sum of any number and 0 (zero) is that number. |
Example: 7 + 0 = 7, |
486 + 0 = 486 |
Identity Property for Multiplication |
The product of any number and 1 (one) is that number. |
Example: |
5840 · 1 = 5840 |
Opposites Property |
If the sum of two numbers is 0 (zero), then each number is the opposite of the other. |
Example: 4 is the opposite of -4 because 4 +(-4) = 0 |
Zero Property |
The sum of 0 (zero) and any number is that number. |
Example: 0+5=5 and 5+0=5 |
The product of 0 (zero) and any number is 0 (zero). |
Example: 0·6=O and 6·O=0 |
Equation Properties |
When adding or subtracting the same number or multiplying or dividing by the same number on both sides of an equation, the result is still an equation. |
Examples: n – 6 = 7 |
n – 6+6= 7+6 |
n = 13 |
4n=24 |
(4n)/4 = 24/4 |
n=6 |