-(-) = -1 x - = +

+(-) = +1 x - = -

-15 - (-20) + (-10) = -15 + 20 - 10

-15 - (-20) + (-10) = -15 - 10 + 20

-15 - (-20) + (-10) = -25 + 20

-15 - (-20) + (-10) = -5

Commutative Property

States that changing the order of addends does not change the sum.

For example, 3 + 5 = 5 + 3, thus three plus five equals five plus 3.

Associative Property

States that changing the grouping of addends does not change the sum.

For example, (2 + 5) + 1 = 2 + (5 + 1), thus when you group the first two terms and when you group the last two terms and the answer is the same, then its associative property

From the question,(6 + x) + 5 = 6 + (x + 5), the first two terms are group at the left and its the same as grouping the last two terms at the right

Difference Between Commutative & Associative Property

The commutative property states that it's okay to change of the order of the numbers in addition and multiplication operations because the result will be the same, no matter the order but the associative property, on the other hand, states that the result will be the same, no matter how you group the number or variables in addition/multiplication operations.

Distributive Property

To "distribute" means to divide something or give a share or part of something.

According to this property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

Example: a(b+c) = a x b + a x c, a(b-c) = a x b - a x c

This indicates that operand a is shared between the other two operands.

Associative Property

This property states that when three or more are added (or multiplied), the sum (or the ) is the same regardless of the grouping of the (or the multiplicands).

Associative property gets its name from the word "Associate" and it refers to grouping of numbers.

Grouping means the use of parentheses or brackets to group numbers.

Associative property involves 3 or more numbers.

The numbers that are grouped within a parenthesis or bracket become one unit.

Associative property can only be used with addition and multiplication and not with subtraction or division.

Example of Associative Property for Addition

Examples of Associative Property for Multiplication:

The above examples indicate that changing the grouping doesn't make any changes to the answer.

The associative property is helpful while adding or multiplying multiple numbers. By grouping, we can create smaller components to solve. It makes the calculations of addition or multiplication of multiple numbers easier and faster.

Example Addition:

17 + 5 + 3 = (17 + 3) + 5
           = 20 + 5
           = 25

Here, adding 17 and 3 gives 20. Then, adding 5 to 20 gives 25. The grouping helped to find the answer easily and quickly.

Example Multiplication:
3 × 4 × 25 = (25 × 4) × 3
           = 100 × 3
           = 300

Here, multiplying 25 by 4 gives 100. Then, 3 can be easily multiplied by 100 to get 300.

However, we cannot apply the associative property to subtraction or division. When we change the grouping of numbers in subtraction or division, it changes the answer and hence, this property is not applicable.

Example Subtraction:

10 – (5 – 2) = 10 - 3 = 7
(10 – 5) – 2 = 5 – 2 = 3
So, 10 – (5 – 2) ≠ (10 – 5) – 2

Example Division:

(24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3
24 ÷ (4 ÷ 2) = 24 ÷ 2 = 12
So, (24 ÷ 4) ÷ 2 ≠ 24 ÷ (4 ÷ 2)

Commutative Property

The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division.

The above examples clearly show that we can apply the commutative property on addition and multiplication. However, we cannot apply commutative property on subtraction and division. If you move the position of numbers in subtraction or division, it changes the entire problem.

Therefore, if a and b are two non-zero numbers, then:

The commutative property of addition is:
a + b = b + a

The commutative property of multiplication is:
a × b = b × a

In short, in commutative property, the numbers can be added or multiplied to each other in any order without changing the answer.

Distributive Property

To "distribute" means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

Example

( 5 + 7 + 3 ) x 4 = 15 x 4
                  = 60

This can be solved using the distributive property as:
( 5 + 7 + 3 ) x 4 = 5x4 + 7x4 + 3x4
                  = 20+28+12
                  = 60