Problem 12 - Beginning and Intermediate Algebra

Question

Evaluate the following expressions. $$180 \div(15 \div 3) \text { and }(180 \div 15) \div 3$$ Does it appear that division is associative?

Step-by-Step Solution

Answer

The results are 36 and 4, showing division is not associative.

Step 1 Title - Simplify Inside Parentheses for 180 ÷ (15 ÷ 3)

First, simplify the expression inside the parentheses: 15 ÷ 3 = 5

Step 2 Title - Perform Division for 180 ÷ (15 ÷ 3)

Next, divide 180 by the result from Step 1: 180 ÷ 5 = 36

Step 3 Title - Simplify Inside Parentheses for (180 ÷ 15) ÷ 3

Simplify the expression inside the parentheses: 180 ÷ 15 = 12

Step 4 Title - Perform Division for (180 ÷ 15) ÷ 3

Finally, divide the result from Step 3 by 3: 12 ÷ 3 = 4

Step 5 Title - Compare Both Results

Compare the results from Step 2 and Step 4: 36 is not equal to 4. Therefore, division is not associative.

Key Concepts

Order of Operations

In mathematics, the order in which we perform operations can significantly change the outcome of an expression. This is commonly known by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Always begin with calculations inside parentheses, then move on to exponents, followed by multiplication and division, and finish with addition and subtraction.
For example, in the expression \(180 \div (15 \div 3)\):

• Start by calculating inside the parentheses: \(15 \div 3 = 5\).
• Then divide the result from the parentheses by 180: \(180 \div 5 = 36\).

In contrast, the expression \((180 \div 15) \div 3\) needs to simplify inside its parentheses first:

• Simplify \(180 \div 15 = 12\).
• Then, divide that result by 3: \(12 \div 3 = 4\).

Both expressions follow the correct order of operations but produce different results, showing the importance of correctly applying PEMDAS.

Evaluating Expressions

Evaluating expressions involves simplifying them to find their value. In the given exercise, you have two different expressions that need to be evaluated.
The first expression, \(180 \div (15 \div 3)\), is evaluated as:

• Simplify inside the parentheses: \(15 \div 3 = 5\).
• Then perform the division with the result: \(180 \div 5 = 36\).

The second expression, \((180 \div 15) \div 3\), is evaluated as:

• Simplify inside the parentheses: \(180 \div 15 = 12\).
• Then perform the division with the result: \(12 \div 3 = 4\).

Evaluating these expressions correctly demonstrates the significance of simplifying expressions step by step and ensures you get the correct final answer.

Division Properties

The associative property, which holds true for addition and multiplication, does NOT apply to division. This means rearranging the grouping of numbers can change the result.
To check if division is associative, consider the expressions given:

• For \(180 \div (15 \div 3)\), you first simplify the inner parentheses: \(15 \div 3 = 5\), then divide: \(180 \div 5 = 36\).
• For \((180 \div 15) \div 3\), you first simplify inside the parentheses: \(180 \div 15 = 12\), and then divide: \(12 \div 3 = 4\).

The different results (36 and 4) clearly show that rearranging divisions changes the outcome, thereby proving that division is not associative. Understanding such properties helps avoid mistakes when solving division problems in mathematics.