Informative line

Properties Of Decimals

Inverse Property

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

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

Illustration Questions

Which one of the following illustrates the inverse property of addition?

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

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

C \(1.8+0=1.8\)

D \(1.8×1=1.8\)

×

According to inverse property of addition, when we add a number to its opposite number, we get zero.

Only in option (B), sum of two opposite numbers is zero.

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

Hence, option (B) is correct.

Which one of the following illustrates the inverse property of addition?

A

\(1.8+1=1+1.8\)

.

B

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

C

\(1.8+0=1.8\)

D

\(1.8×1=1.8\)

Option B is Correct

Additive Identity

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

\(2.5+0=2.5\)

Examples: 

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

Multiplicative identity

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

Examples: 

\(2.5×1=2.5\)

\(4.2×1=4.2\)

\(6.8×1=6.8\)

Illustration Questions

Which one of the following illustrates the additive identity property?

A \(12.2+0=12.2\)

B \(12.2×0=0\)

C \(12.2+1=13.2\)

D \(12.2+12.2=24.4\)

×

According to the additive identity property, a number does not change when zero is added to that number.

Among the options, only the equation,

 \(12.2+0=12.2\)

involves the addition with \(0,\) and the number \((12.2)\), still remains the same.

So, it illustrates the additive identity property.

Hence, option (A) is correct.

Which one of the following illustrates the additive identity property?

A

\(12.2+0=12.2\)

.

B

\(12.2×0=0\)

C

\(12.2+1=13.2\)

D

\(12.2+12.2=24.4\)

Option A is Correct

Commutative Property

  • 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:

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

Illustration Questions

Which one of the following equations shows the commutative property?

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 \(5.5×6.5=6.5×2.5\)

×

According to the commutative property, the numbers can be added or multiplied in any order, the result remains the same.

In option (B), the order of the addends is changed,

\(5.5+6.5=12\)

\(6.5+5.5=12\)

but the result is still the same.

Hence, option (B) is correct.

Which one of the following equations shows the commutative property?

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

\(5.5×6.5=6.5×2.5\)

Option B is Correct

Associative Property

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.

Illustration Questions

Which one of the following illustrates the associative property of addition?

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

×

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.

So, only option (C), i.e.

 \(6.2+(4.2+3.4)=(6.2+4.2)+3.4\)

represents the associative property.

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

\(6.2+7.6=10.4+3.4\)

\(13.8=13.8\)

Hence, option (C) is correct.

Which one of the following illustrates the associative property of addition?

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

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

Option C is Correct

Decimal Equivalent Expression

  • 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 \(4.5(13.3)\).

Illustration Questions

Which group of expressions is equivalent to \(0.50(1.2-2.8)\)?

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

×

Given: \(0.50(1.2-2.8)\)

According to distributive property,

\(0.50(1.2-2.8)\)   \(0.50(1.2-2.8)\)
  or \(0.50×1.2-0.50×2.8\)
\(0.50(-1.6)\)   \(0.60-1.4\)

\(\therefore\;0.50(1.2-2.8)\) is equivalent to \(0.50(-1.6)\) and \(0.60-1.4\).

Hence, option (C) is correct.

Which group of expressions is equivalent to \(0.50(1.2-2.8)\)?

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

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

Option C is Correct

Distributive Property

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.

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

Illustration Questions

Rewrite the expression: \(2.7(8.4+4.8)\) using distributive property.

A \(8.4(2.7+4.8)\)

B \(2.7(3.6)\)

C \(2.7(8.4+4.8)\)

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

×

According to the distributive property, multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.

\(a(b+c)=ab+ac\)

Thus, the expression

\(2.7(8.4+4.8)\) can be written as 

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

Hence, option (D) is correct.

Rewrite the expression: \(2.7(8.4+4.8)\) using distributive property.

A

\(8.4(2.7+4.8)\)

.

B

\(2.7(3.6)\)

C

\(2.7(8.4+4.8)\)

D

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

Option D is Correct

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