**Inverse Property of Addition:**

- Inverse property of addition says that if we add a number \(\text{(a)}\) to its opposite number, (\(-\text{a } \) i.e. additive inverse) we get zero.

**Example:** \(\text{a +(}-\text{a)}=0\)

\(5.25+(-5.25)=0\)

**Inverse Property of Multiplication:**

- Inverse property of multiplication says that if we multiply a number \(\text{(a)}\) by its opposite number, (\(\dfrac{1}{\text{a}}\) i.e. multiplicative inverse) we get one.

**Example:** \(\text{a} \left( \dfrac{1}{\text{a}} \right)=1\)

\(5.25× \left( \dfrac{1}{5.25} \right) =1\)

A \(1.8+1=1+1.8\)

B \(1.8+(-1.8)=0\)

C \(1.8+0=1.8\)

D \(1.8×1=1.8\)

The additive identity property says that a number does not change when zero is added to that number.

\(2.5+0=2.5\)

**Examples:**

\((1)\;4.2+0=4.2\\ (2)\;8.9+0=8.9\\ (3)\;11.52+0=11.52\)

The multiplicative identity property says that a number does not change when \(1\) is multiplied to that number.

**Examples:**

\(2.5×1=2.5\)

\(4.2×1=4.2\)

\(6.8×1=6.8\)

A \(12.2+0=12.2\)

B \(12.2×0=0\)

C \(12.2+1=13.2\)

D \(12.2+12.2=24.4\)

- The commutative property says that the numbers can be added or multiplied in any order, and we still get the same result.

For addition: \(1.2+2.5=2.5+1.2\)

For multiplication: \(1.2×2.5=2.5×1.2\)

**For example:**Suppose we are adding \(1.6\) and \(1.8\), then

\(1.6+1.8=3.4\)

or

\(1.8+1.6=3.4\)

Thus, the answer is still the same.

**Examples:**

\((1)\;2.14×1.28=1.28×2.14\\ (2)\;3.15+7.45=7.45+3.15\\ (3)\;10.2×5.5=5.5×10.2\)

A \(5.5×6.5=5.5×6.5\)

B \(5.5+6.5=6.5+5.5\)

C \(5.5+6.5=5.5+6.5\)

D \(5.5×6.5=6.5×2.5\)

**Associative property of addition:**

- The associative property says that it does not matter how we group the numbers when we add more than two numbers, the sum remains the same.
- The grouping of the addends (numbers being added) does not change the sum.

\((1.2+2.3)+3.4=1.2+(2.3+3.4)\)

**Associative property of multiplication:**

- The associative property says that it does not matter how we group the numbers when we multiply more than two numbers, the product remains the same.
- The grouping of the factors (numbers being multiplied) does not change the product.

\((1.2×2.3)3.4=1.2(2.3×3.4)\)

**For example:**

\((3.2+5.6)+4.1=3.2+(5.6+4.1)\)

\(8.8+4.1=3.2+9.7\)

\(12.9=12.9\)

In both the groupings, the sum of the addition does not change.

A \(6.2+(4.2+3.4)=6.2+(4.2+3.4)\)

B \(6.2+(4.2+3.4)=(6.2+3.4)+3.4\)

C \(6.2+(4.2+3.4)=(6.2+4.2)+3.4\)

D \((6.2+4.2)+3.4=6.2+(4.2+4.2)\)

- If two expressions give the same result, then both the expressions are equivalent.
- Using properties and operations, we can find the equivalent expression.
- For better understanding, we can use properties.
- According to commutative property,

**Example:** \(4.5+15.6\) can be written as \(15.6+4.5\)

- Therefore, \(4.5+15.6\) is equivalent to \(15.6+4.5\)
- According to associative property,

**Example:** \((4.5+15.6)+2.3\) can be written as \(4.5+(15.6+2.3)\)

- Therefore, \((4.5+15.6)+2.3\) is equivalent to \(4.5+(15.6+2.3)\)
- According to distributive property,

**Example:**

\(4.5(15.6-2.3)\) | \(4.5(15.6-2.3)\) | |

can be written as |
or |
can be written as |

\(4.5×15.6-4.5×2.3\) | \(4.5(13.3)\) | |

\(70.2-10.35\) |

Therefore, \(4.5(15.6-2.3)\) is equivalent to \(70.2-10.35\) and \(4.5(13.3)\).

A \(0.50(4);\;0.6-2.8\)

B \(0.50(1.2)-2.8;\;0.50(1.6)\)

C \(0.50(-1.6);\;0.60-1.4\)

D \(0.50(4);\;0.60-1.4\)

**Distributive property over addition:**

The distributive property over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.

\(4.2(3.1+1.2)=(4.2×3.1)+(4.2×1.2)\)

\(4.2×4.3=13.02+5.04\)

\(18.06=18.06\)

**Distributive property over subtraction:**

Either we find out the difference first and then multiply, or we first multiply with each number and then subtract, the result remains the same.

\(4.2(3.1-1.2)=(4.2×3.1)-(4.2×1.2)\)

\(4.2×1.9=13.02-5.04\)

\(7.98=7.98\)

**Note**: In case, if we change the order of subtraction, then the result also changes.

\(4.2(3.1-1.2)\neq(4.2×1.2)-(4.2×3.1)\)

A \(8.4(2.7+4.8)\)

B \(2.7(3.6)\)

C \(2.7(8.4+4.8)\)

D \(2.7(8.4)+2.7(4.8)\)