PROPERTIES OF REAL NUMBERS
                 

Name the correct property following equations with the property being demonstrated :

Problem 1 :

9 + (4 + 11) = (9 + 4) + 11

Solution :

Addition of real numbers is associative.

If 9, 4 and 11 are real numbers.

then, 

9 + (4 + 11) = 9 + (15) = 24

(9 + 4)+ 11 = (13) + 11 = 24

Hence, 9 + (4 + 11) = (9 + 4) + 11

Problem 2 :

8 + 0 = 8

Solution :

The sum of any real number and zero is the real number itself.

If '8' is real number, then

8 + 0 = 0 + 8 = 8

Zero is the additive identity for real numbers.

Problem 3 :

9 + (4 + 11) = (4 + 11) + 9

Solution :

Addition of real numbers is associative.

If 9, 4 and 11 are real numbers.

then, 

9 + (4 + 11) = 9 + (15) = 24

(4 + 11) + 9 = (15) + 9 = 24

Hence, 9 + (4 + 11) = (4 + 11) + 9

Problem 4 :

17 ⋅ 30 = 30 ⋅17

Solution :

17 ⋅ 30 = 30 ⋅ 17

Multiplication of real numbers is commutative.

If 17 and 30 are any two real numbers, then

17 ⋅ 30 = 510

30 ⋅ 17 = 510

Hence, 17 ⋅ 30 = 30 ⋅ 17

Therefore, Commutative Property is true for multiplication of real numbers.

Problem 5 :

19(1) = 19

Solution :

The product of any real number and 1 is rational number itself. 'One' is the multiplicative identity for real numbers..

If 19 is real number, then

19(1) = 1(19)= 19 

Problem 6 :

y + xz = xz + y

Solution :

Addition of two real numbers is commutative.

If y and xz are two real numbers,

Then, 

y + xz = xz + y

Problem 7 :

2 ⋅ (3 ⋅ 4) = (2 ⋅ 3) ⋅ 4

Solution :

Multiplication of real numbers is associative.

If 2, 3 and 4 are three real numbers, then 

2 ⋅ (3 ⋅ 4) = 2 ⋅ 12 = 24

(2 ⋅ 3) ⋅ 4 = 6 ⋅ 4 = 24

Hence, 2 ⋅ (3 ⋅ 4) = (2 ⋅ 3) ⋅ 4

Therefore, associative Property is true for multiplication.

Problem 8 :

st + v = ts + v

Solution :

Addition of two real numbers is commutative.

If st and v are any two real numbers,

Then, st + v = ts + v.

Problem 9 :

-4 + 0 = -4

Solution :

The sum of any real number and zero is the real number itself .

If '-4' is real number, then

-4 + 0 = 0 - 4 = -4

Zero is the additive identity for real numbers.

Problem 10 :

80(9) = 9(80)

Solution :

80(9) = 9(80)

Multiplication of real numbers is commutative.

If 80 and 9 are two real numbers, then

80(9) = 720

9(80) = 720

Hence, 80(9) = 9(80)

Problem 11  :

Miss Collins is going shopping. She needs to buy a new bottle of shampoo for $4, a bag of dog food for $26, and a book for the beach for $14. Show two different methods to figure out her total using the Associative Method.

Solution :

Bottle of shampoo cost = $4

Food cost = $26

Book cost = $14

Way 1 :

= $(4 + 26) + $14

= $30 + $14

= $44

Way 2 :

= $4 + $(26 + 14)

= $4 + $40

= $44

Problem 12 :

Use mental math to simplify each expression. Use mental math to simplify. Think about how to best group the numbers to simplify.

a) (98 + 3) + 2          b) (20 ⋅ 16) ⋅ 5

Solution :

a) (98 + 3) + 2

Addition of real numbers is associative.

If 98, 3 and 2 are real numbers.

then, 

(98 + 3) + 2 = (101) + 2 = 103

98 + (3 + 2) = 98 + (5) = 103

Hence, (98 + 3) + 2 = 98 + (3 + 2)

b) (20 ⋅ 16) ⋅ 5

Multiplication of real numbers is associative.

If 20, 16 and 5 are three real numbers, then 

(20 ⋅ 16) ⋅ 5 = 320 ⋅ 5 = 1600

20 ⋅ (16 ⋅ 5) = 320 ⋅ 5 = 1600

Hence, (20 ⋅ 16) ⋅ 5 = 20 ⋅ (16 ⋅ 5)

Therefore, associative Property is true for multiplication.

Problem 13 :

Suppose you buy snacks costing $0.45, $0.65, and $1.55. Use mental math to find the cost of these snacks. Think about how to best group the numbers to simplify.

Solution :

Snacks cost = $0.45 + $0.65 + $1.55 

= $2.65

Therefore, cost of these snacks is $2.65.