Problem 114 - Prealgebra

Question

Match each statement on the left with the property that justifies it on the right. a. Distributive property b. Associative property c. Commutative property d. Commutative and associative properties $$3+(4+1)=(3+4)+1$$

Step-by-Step Solution

Answer

Associative property.

Step 1: Identify the equation

The given equation is \(3+(4+1) = (3+4) + 1\). We need to determine which property of operations this equation demonstrates.

Step 2: Understand associative property

The associative property states that the way numbers are grouped in an operation (addition or multiplication) does not change their sum or product. In other words, \((a + b) + c = a + (b + c)\) for addition and similar for multiplication.

Step 3: Apply associative property

In the equation \(3+(4+1) = (3+4) + 1\), the parentheses around \((4+1)\) are moved to \((3+4)\), but the order of numbers remains the same. This is an application of the associative property of addition.

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra that combines addition and multiplication in a specific way. It is represented as:

• For any numbers, a, b, and c, it states that: \( a(b + c) = ab + ac \)
• This shows how a single term can be 'distributed' across terms inside a parenthesis.

This property allows you to expand expressions and simplify calculations efficiently. For example, if you were to multiply 2 by the sum of 3 and 4, you can do it directly or use the distributive property:

• Direct: \( 2 \times (3 + 4) = 2 \times 7 = 14 \)
• Using Distributive Property: \(2 \times 3 + 2 \times 4 = 6 + 8 = 14 \)

Both give you the same result, showing the versatility of this property. It's especially useful in complex algebraic expressions and equations, giving a simpler pathway to solving them.

Commutative Property

The commutative property is a basic principle of arithmetic, applicable to addition and multiplication. This property states that changing the order of the numbers in an operation does not change the result. Specifically:

• For addition: \(a + b = b + a\)
• For multiplication: \(a \times b = b \times a\)

This is a handy tool in simplifying expressions by rearranging terms to make calculations more straightforward or manageable. Consider the following example:

• Addition: If you have 5 + 2, it is the same as 2 + 5, both equal to 7.
• Multiplication: \(3 \times 4\) is the same as \(4 \times 3\), both equaling 12.

The commutative property does not apply to subtraction or division, where the order of numbers makes a difference. Understanding and applying the commutative property can make algebraic work more flexible and quite often, easier.

Order of Operations

Mathematics involves a specific set of rules for performing calculations, particularly when expressions involve multiple operations like addition, subtraction, multiplication, and division. The method for determining which operations to perform first is known as the 'Order of Operations'.
To remember the order, one can use the acronym PEMDAS:

• P - Parentheses: Resolve expressions inside parentheses first.
• E - Exponents: Solve exponents or powers.
• MD - Multiplication and Division: Next, perform multiplication and division from left to right.
• AS - Addition and Subtraction: Finally, tackle addition and subtraction from left to right.

This ensures accuracy, avoiding confusion and errors in complex calculations. For example, in the expression \(3 + 5 \times 2\), multiplication occurs before addition according to PEMDAS, resulting in \(3 + 10 = 13\). Following the order of operations is crucial in mathematics for consistently correct results.