Problem 41 - Prealgebra

Question

Perform the following operations according to the rule for order of operations. $$4.23-5(0.04+0.09)$$

Step-by-Step Solution

Answer

The result of the operation is 3.58.

Step 1: Solve Inside the Parentheses

First, solve the expression inside the parentheses: \(0.04 + 0.09 = 0.13\).

Step 2: Multiply by the Number Outside the Parentheses

Next, multiply the result from Step 1 by the number outside the parentheses:\(5 \times 0.13 = 0.65\).

Step 3: Subtract to Find the Final Answer

Finally, subtract the result from Step 2 from the first number in the expression:\(4.23 - 0.65 = 3.58\).

Key Concepts

Understanding Parentheses in Math

In mathematics, parentheses play a crucial role in determining the order in which operations should be performed. Parentheses are used to group numbers or expressions together, indicating that the operations inside them need to be completed first. This is essential to avoid confusion and to ensure that calculations are done in the correct sequence.

Whenever you encounter a mathematical expression, look for parentheses first. Solving operations inside these brackets can significantly change the outcome of the calculation if not addressed first. In the expression given, \(0.04 + 0.09\) is in parentheses, so we need to solve this part before doing anything else.

When you calculate what's inside, you get \(0.13\). Only then do you move to the next steps. This ensures that you're following the correct order of operations and getting the right results.

The Importance of Multiplication

Multiplication is one of the fundamental arithmetic operations and is signified by symbols like \(\times\), \(*\), or by juxtaposition (simply placing numbers and/or variables side by side). When dealing with expressions that involve parentheses, any number outside the parentheses must be multiplied by the result inside.

In our exercise, the number 5 positioned right outside the parentheses (which contained \(0.04 + 0.09\)) indicates that once the inside calculation gives us \(0.13\), we must multiply \(5 \times 0.13\). The result is \(0.65\), and this step is crucial because multiplying first instead of adding or subtracting could completely change the final answer.

Multiplication often occurs in sequences before addition and subtraction in expressions, unless parentheses dictate otherwise. Keeping this order is key to performing correct calculations.

Breaking Down Addition

Addition is an arithmetic operation that involves combining numbers to get a sum. It's straightforward, but its place in the order of operations is essential. Addition usually takes a back seat to parentheses, multiplication, and division.

In our example, addition occurs inside the parentheses with \(0.04 + 0.09\), leading to a result of \(0.13\) before it is used in further calculations like multiplication.

This approach highlights why understanding how addition fits into the order of operations is important. Handling addition on its own is simple, but when in the order of operations context, you need to know when it can be calculated to ensure that other operations like multiplication aren't disrupted.

The Role of Subtraction

Subtraction is used to find the difference between numbers, and in the order of operations, it often comes last after handling any parentheses, exponents, multiplication, and addition steps.

In the given expression, subtraction is the final step. After simplifying what's in parentheses and multiplying by 5, you are left with \(4.23 - 0.65\). Solving this gives the final result of \(3.58\). It's important to subtract only after all other operations are properly settled, as performing it too early might lead to incorrect results.

Understanding where subtraction fits within the sequence of operations ensures your calculations remain accurate and consistent.