Final answer:
When subtracting rational numbers as in the expression ( {2}/{5} - {7}/{10} - (-{3}/{5}) ), we can use the commutative property to reorder terms. Options (a) correctly identifies the commutative property, whereas options (b) and (d) misuse the associative property with subtraction. Finding a common denominator and properly applying addition and subtraction rules are also crucial.
Step-by-step explanation:
Understanding the properties of addition can significantly simplify the process of working with rational numbers. When subtracting rational numbers such as ( {2}/{5} - {7}/{10} - (-{3}/{5}) ), we can apply these properties effectively.
Using the Commutative Property:
The commutative property of addition states that the order in which two numbers are added does not change the sum. For example, a + b = b + a. In our case, option (a) demonstrates this by reordering the terms: ( {2}/{5} - {7}/{10} + {3}/{5} ). Notice that we are essentially adding a positive {3}/{5} rather than subtracting a negative {3}/{5}, which aligns with the intuition that subtracting a negative is equivalent to adding a positive.
Using the Associative Property:
The associative property of addition allows us to group numbers differently without altering the result. For example, (a + b) + c = a + (b + c). However, this property does not apply when subtraction is involved, as subtraction is not associative. Therefore, options (b) and (d) incorrectly use the associative property because the subtraction sign is not taken into account. Additionally, option (c) is not using any property at all as it is identical to the original expression minus the negative sign transformation.
When subtracting rational numbers, it is important to find a common denominator, change subtraction to addition where needed (by changing the sign of the number being subtracted), and then add the numerators while keeping the denominator the same.
To evaluate the given expression correctly, we would first need to find a common denominator for all terms, turn the subtraction of a negative into addition, and then proceed with the actual operation while taking into account that the commutative property can only apply to addition and not to subtraction.