2. State whether each of the following mathematical statements is true (T) or false (F). If the statement is true, state the law that was applied.
(a) [tex]$5+4=5 \times 4$[/tex]
(b) [tex]$7+5=5+7$[/tex]
(c) [tex]$3+(8+6)=(3+8)+6$[/tex]
(d) [tex]$4+7+2=(4+7)+2$[/tex]
(e) [tex]$7 \times 4=4 \times 7$[/tex]
(f) [tex]$2 \times 6=6 \div 2$[/tex]
(g) [tex]$2 \times(3 \times 8)=(2 \times 3) \times 8$[/tex]
(h) [tex]$5 \times(3 \times 4)=5 \times(4+3)$[/tex]
Answer (2)
Most of the statements are related to fundamental properties of addition and multiplication, with several demonstrating the commutative and associative laws. Only statements (a), (f), and (h) are false. The remaining statements (b, c, d, e, g) are true.
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Statement (a) is false because 5 + 4 ๎ = 5 ร 4 .
Statement (b) is true due to the Commutative Law of Addition: 7 + 5 = 5 + 7 .
Statement (c) is true due to the Associative Law of Addition: 3 + ( 8 + 6 ) = ( 3 + 8 ) + 6 .
Statement (d) is true: 4 + 7 + 2 = ( 4 + 7 ) + 2 .
Statement (e) is true due to the Commutative Law of Multiplication: 7 ร 4 = 4 ร 7 .
Statement (f) is false because 2 ร 6 ๎ = 6 รท 2 .
Statement (g) is true due to the Associative Law of Multiplication: 2 ร ( 3 ร 8 ) = ( 2 ร 3 ) ร 8 .
Statement (h) is false because 5 ร ( 3 ร 4 ) ๎ = 5 ร ( 4 + 3 ) .
Explanation
Analyzing the Statements Let's analyze each statement to determine if it's true or false and, if true, identify the mathematical law applied.
Statement (a) (a) 5 + 4 = 5 ร 4
LHS: 5 + 4 = 9
RHS: 5 ร 4 = 20
Since 9 ๎ = 20 , the statement is false.
Statement (b) (b) 7 + 5 = 5 + 7
LHS: 7 + 5 = 12
RHS: 5 + 7 = 12
Since 12 = 12 , the statement is true. This demonstrates the Commutative Law of Addition, which states that the order of addends does not affect the sum.
Statement (c) (c) 3 + ( 8 + 6 ) = ( 3 + 8 ) + 6
LHS: 3 + ( 8 + 6 ) = 3 + 14 = 17
RHS: ( 3 + 8 ) + 6 = 11 + 6 = 17
Since 17 = 17 , the statement is true. This demonstrates the Associative Law of Addition, which states that the grouping of addends does not affect the sum.
Statement (d) (d) 4 + 7 + 2 = ( 4 + 7 ) + 2
LHS: 4 + 7 + 2 = 13
RHS: ( 4 + 7 ) + 2 = 11 + 2 = 13
Since 13 = 13 , the statement is true. This is an example of how addition can be grouped; although it explicitly shows the associative law, it's more about the process of addition.
Statement (e) (e) 7 ร 4 = 4 ร 7
LHS: 7 ร 4 = 28
RHS: 4 ร 7 = 28
Since 28 = 28 , the statement is true. This demonstrates the Commutative Law of Multiplication, which states that the order of factors does not affect the product.
Statement (f) (f) 2 ร 6 = 6 รท 2
LHS: 2 ร 6 = 12
RHS: 6 รท 2 = 3
Since 12 ๎ = 3 , the statement is false.
Statement (g) (g) 2 ร ( 3 ร 8 ) = ( 2 ร 3 ) ร 8
LHS: 2 ร ( 3 ร 8 ) = 2 ร 24 = 48
RHS: ( 2 ร 3 ) ร 8 = 6 ร 8 = 48
Since 48 = 48 , the statement is true. This demonstrates the Associative Law of Multiplication, which states that the grouping of factors does not affect the product.
Statement (h) (h) 5 ร ( 3 ร 4 ) = 5 ร ( 4 + 3 )
LHS: 5 ร ( 3 ร 4 ) = 5 ร 12 = 60
RHS: 5 ร ( 4 + 3 ) = 5 ร 7 = 35
Since 60 ๎ = 35 , the statement is false.
Final Answer Here's a summary of our findings: (a) False (b) True, Commutative Law of Addition (c) True, Associative Law of Addition (d) True (e) True, Commutative Law of Multiplication (f) False (g) True, Associative Law of Multiplication (h) False
Examples
Understanding the commutative and associative laws is crucial in various real-life scenarios. For instance, when calculating the total cost of items in a store, the order in which you add the prices (commutative law) doesn't change the total. Similarly, when scaling a recipe (associative law), whether you double the ingredients first and then multiply by another factor, or multiply each ingredient individually, the final result remains the same. These laws ensure consistency and flexibility in calculations, simplifying everyday tasks.
