Problem 23 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$85 \div 5 \cdot 5-85$$

Step-by-Step Solution

Answer

Answer: The value of the given expression is 0.

Step 1: Division

First, perform the division operation: $$ 85 \div 5 = 17 $$

Step 2: Multiplication

Next, perform the multiplication operation using the result from Step 1: $$ 17 \cdot 5 = 85 $$

Step 3: Subtraction

Finally, perform the subtraction operation using the results from Step 2: $$ 85 - 85 = 0 $$ The value of the given expression is 0.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is crucial in math problem solving. An algebraic expression is a combination of numbers, variables (letters that represent unknown values), and arithmetic operations such as addition, subtraction, multiplication, and division. For instance, the expression \( 85 \div 5 \cdot 5 - 85 \) from the exercise involves a sequence of arithmetic operations carried out on the number 85.

These expressions can represent real-world situations and can vary in complexity. To simplify or evaluate them, one must follow specific rules known as the order of operations. This rule suggests that operations must be performed in a certain sequence: parentheses first, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). This ensures that everyone solves the expression in the same way, obtaining the same result.

Arithmetic Operations

Arithmetic operations are the foundation of all mathematical problem solving. They include addition (+), subtraction (-), multiplication (\(\times\)), and division (\(\div\)). Each operation has its own level of precedence, which is essential when they are combined in an algebraic expression.

In our example \( 85 \div 5 \cdot 5 - 85 \) we see three of these operations in action. Division and multiplication are considered of equal priority, and according to the order of operations, must be performed from left to right. Subtraction comes afterwards. It’s important to apply these operations accurately to get the correct result. For instance, ignoring the rule and subtracting before dividing would lead to an incorrect answer.

Math Problem Solving

Math problem solving is a process that involves understanding the problem, planning a strategy, carrying out that strategy, and then reviewing the solution. The order of operations is an essential part of this process, especially when dealing with complex algebraic expressions.

When faced with a mathematical problem, it is helpful to break it down into smaller, more manageable steps. This was demonstrated in the step-by-step solution of our given exercise. The problem \( 85 \div 5 \cdot 5 - 85 \) was solved by tackling one arithmetic operation at a time, in the correct order. This methodical approach prevents errors that could arise from executing operations out of order. Additionally, checking each step can help catch mistakes early and enhance understanding of the overall process.