Problem 60 - Prealgebra and Introductory Algebra

Question

Evaluate \(a \div b c\) and \(a \div(b c)\) for \(a=16, b=2,\) and \(c=-4 .\) Explain why the answers are not the same.

Step-by-Step Solution

Answer

The results of evaluating the expressions \(a \div b c\) and \(a \div (b c)\) for \(a=16, b=2, c=-4\) are -32 and -2 respectively. The difference arises because of the order of operations, which mandates that operations inside parentheses are performed before the others, and that multiplication and division are performed from left to right.

Step 1: Evaluate \(a \div b c\)

Here we evaluate \(a \div b c\) applying the order of operations. As multiplication and division are on the same level of operation, they are evaluated from left to right. Hence, first we divide \(a\) by \(b\), then multiply the result by \(c\). This gives us \((16 \div 2) \times -4 = 8 \times -4 = -32\).

Step 2: Evaluate \(a \div (b c)\)

Here, the operations are performed differently because of the parentheses. First, evaluate the expression within the parentheses \((b c)\), which is \(2 \times -4 = -8\). Then, divide \(a\) by the result of the expression in the parentheses. So, \(16 \div -8 = -2\).

Step 3: Discuss why the Results are Different

The results for the expressions are different due to the order of operations. In the first expression, \(a \div b c\), \(a\) is divided by \(b\) first then multiplied by \(c\) because of the left-to-right rule for operations of the same precendence. However, in the second expression, \(a \div (b c)\), multiplication is performed before division because the operation within the parentheses should be done first as per the order of operations.

Key Concepts

Division

Understanding division in mathematics is essential because it helps us split a whole into parts. In any expression that involves division, the number you are dividing is called the dividend. The number you are dividing by is the divisor. The outcome is known as the quotient.

• For example, in the expression \(16 \div 2\), 16 is the dividend, 2 is the divisor, and the result is the quotient.
• Division is tricky because it is both a standalone operation and part of more complex evaluations like our exercise.

Division has the same priority as multiplication, which means when both appear together, you'll evaluate them in the order they come from left to right unless parentheses indicate otherwise.

Multiplication

Multiplication is the operation of scaling one number by another. It can be seen as repeated addition. But in the context of expressions with multiple operations, it shares the same level of priority as division.

• In expressions like \(b \times c\) or \(b \cdot c\), you take the product of \(b\) and \(c\).
• Multiplication is usually straightforward, but its order with division can change outcomes as shown in our problem.

Always remember in complex expressions like \(a \div b c\), if parentheses are absent, division and multiplication are treated equally and evaluated from left to right.

Parentheses

Parentheses are used in mathematics to denote operations that should be performed first in an expression. They are crucial for establishing the correct order of operations.

• Consider the expression \(a \div (b c)\). The parentheses around \(b c\) means that the multiplication inside ought to be performed before anything else.
• This changes the outcome of the calculation considerably in comparison to the expression without parentheses.

Using parentheses can be thought of as providing a 'safe space' for operations to be evaluated independently of others. It underscores the importance of grouping in ensuring accurate mathematical outcomes.

Expression Evaluation

Evaluating expressions involves performing operations in a specific sequence to arrive at a correct result. This usually means careful adherence to the order of operations, often remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• In \(a \div b c\) the operations are performed left to right, first dividing then multiplying.
• In \(a \div (b c)\) the multiplication inside the parentheses happens first, influencing the order in which calculations proceed.

Recognizing and accurately applying operations in the correct order ensures accurate results, as seen where applying division before multiplication vs. grouping them results in different answers. Proper expression evaluation is key to solving mathematical problems correctly and avoiding errors.