Informative line

### Properties Of Decimals

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:

$$2.5×1=2.5$$

$$4.2×1=4.2$$

$$6.8×1=6.8$$

#### 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:

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

#### 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

• 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.

#### 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

# Distributive Property

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.

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

#### 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

# Applications of Properties (Explicitly or Implicitly)

An implicit expression can be solved as an explicit expression with the help of properties.

• A decimal number can be written as a combination of the whole and the decimal part using the addition operation.

$$1.5=1+0.5$$

• Using the distributive property, we can multiply the numbers separately.

$$1.5(2.5+3.5)=1.5(2.5)+1.5(3.5)$$

• Using the associative property, we can change the grouping of the numbers.

$$1.5+(2.5+3.5)=(1.5+2.5)+3.5$$

• Using the commutative property, we can change the order of the numbers.

$$1.5+2.5=2.5+1.5$$

For example: Solving $$800×25.5$$

• The decimal number $$25.5$$ can be written as :

$$(25+.5)$$

• Rewriting the expression gives,

$$800(25+.5)$$

• Using the distributive property, we multiply $$800$$ to each of the addends and then adding the products together.

$$800(25)+800(.5)$$

$$20000+400$$

$$=20,400$$

#### Use the fact $$22×17=374$$ to find $$22×1.7=\;?$$

A $$37.4$$

B $$0.370$$

C $$3.74$$

D $$3740$$

×

Given fact:  $$22×17=374$$

We need to convert $$1.7$$ into $$17$$,

So, we split $$1.7$$ into the product of  $$17$$ and $$0.1$$.

Thus, $$1.7$$ can be written as $$17×0.1$$

Rewriting the problem,

$$22×(17×0.1)$$

Using the associative property,

$$\left [ a×(b×c)=(a×b)×c \right]$$

$$22×(17×0.1)$$ can be written as

$$(22×17)×0.1$$

Solving the problem,

$$(22×17)×0.1$$

$$=(374)×0.1$$

$$=37.4$$

Thus, $$22×1.7=37.4$$

Hence, option (A) is correct.

### Use the fact $$22×17=374$$ to find $$22×1.7=\;?$$

A

$$37.4$$

.

B

$$0.370$$

C

$$3.74$$

D

$$3740$$

Option A is Correct