Problem 32 - Prealgebra

Question

Perform the following operations according to the rule for order of operations. $$0.04(0.07+0.09)$$

Step-by-Step Solution

Answer

The result is 0.0064.

Step 1: Evaluate the Expression Inside the Parenthesis

Look at the expression inside the parenthesis, which is \(0.07 + 0.09\). Add these two numbers together to simplify this part of the expression. Therefore, \(0.07 + 0.09 = 0.16\).

Step 2: Multiply the Result by the Number Outside the Parenthesis

Now that you have simplified the expression inside the parenthesis to \(0.16\), multiply this result by the factor outside the parenthesis, which is \(0.04\). Therefore, \(0.04 \times 0.16 = 0.0064\).

Key Concepts

Understanding Parentheses

In mathematics, parentheses are used as a tool to signify which operations should be performed first in a given expression. They act as a kind of highlighter, drawing attention to expressions that need priority.

Let's say you come across an expression like this: \(0.04(0.07 + 0.09)\). The numbers inside the parentheses, \(0.07 + 0.09\), should be addressed first according to the order of operations. This is the part where the calculations begin.

Once you've worked out what is inside the parentheses, you can then move on to the operations outside them. In this case, once the sum inside the parentheses is calculated to be \(0.16\), you can proceed to multiply it with \(0.04\). This structured approach helps in organizing the steps and enhances clarity when dealing with complex expressions.

Simplifying with Multiplication

Multiplication is a fundamental arithmetic operation that represents the idea of repeated addition. It's important to multiply only after simplifying the contents within the parentheses, as seen in our example: \(0.04(0.07 + 0.09)\).

After determining the sum within the parentheses, which is \(0.16\), the next step is to multiply this by \(0.04\). This multiplication will combine these values into a single product. In mathematical terms, you calculate \(0.04 \times 0.16\) to get \(0.0064\).

• Focus on values outside and inside the parentheses: perform operations within first.
• Multiplication distributes over addition, meaning you multiply each term inside the parentheses by the number outside, if needed.
• Always revisit multiplication as it acts as a bridge to simplify and reduce expressions to a single number easier to interpret.

Using Decimals Effectively

Decimals are numbers that lie between integers, giving us a more precise value. They are essential in performing more accurate calculations, especially in financial and scientific fields. Let's discuss how decimals operate in our example:

When adding two decimals, such as \(0.07\) and \(0.09\), each has its fractional value based on its place value.

• Ensure that when adding decimals, you align the decimal points to maintain proper accuracy.

After adding them, you find the decimal sum \(0.16\).

Then, multiplying this result by another decimal, \(0.04\), involves multiplying as if they were whole numbers, then counting and applying the total number of decimal places from both original numbers to the final product. This gives us \(0.0064\).

• Practice using decimals to develop a more refined numerical intuition.
• Remember that each decimal digit represents a further subdivision—knowing this boosts your calculations' precision.

Decimals ensure that results remain within necessary precision limits, providing greater insights into comparisons and estimates.