Problem 39 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$3.4-\frac{1}{2}(0.76)$$

Step-by-Step Solution

Answer

The simplified expression is 3.02.

Step 1: Understand the Expression

First, we need to understand the expression given: \[3.4 - \frac{1}{2}(0.76)\] This expression involves a subtraction and a multiplication operation. Our goal is to simplify it as much as possible.

Step 2: Multiply 0.76 by 1/2

The first operation to perform is the multiplication of 0.76 by \( \frac{1}{2} \). Calculating this gives:\[ \frac{1}{2} \times 0.76 = \frac{0.76}{2} = 0.38 \]So, the multiplication yields 0.38.

Step 3: Perform the Subtraction

Next, we substitute 0.38 back into the expression to perform the subtraction:\[3.4 - 0.38\]This subtraction can be done by aligning the decimal points and subtracting:\[3.4 - 0.38 = 3.02\]

Step 4: Write the Final Answer

After performing all operations, we find that the simplified expression is 3.02.

Key Concepts

Decimal Operations

Decimals are numbers that have a whole number part and a fractional part, separated by a decimal point. These numbers allow us to precisely express values that are not whole numbers.

When working with decimals, it is important to remember:

• Align decimal points when adding or subtracting.
• When multiplying, count the total number of decimal places in both numbers for the final answer.
• Division with decimals often involves converting to fractions or ensuring the divisor is a whole number.

Decimal operations are essential in solving real-world problems as they enable us to perform calculations involving measurements, currency, and other applications with accuracy. Practicing with decimals helps you become comfortable with moving beyond whole numbers and understanding precise values.

Subtraction

Subtraction is the mathematical operation where you remove a certain quantity from another. When dealing with decimal numbers in subtraction, there are certain practices to ensure accuracy:

• Always line up the decimal points vertically in the numbers you are subtracting.
• If necessary, add zeros to the end of a decimal to ensure the numbers have the same number of decimal places.
• Subtract each column starting from the rightmost digit, borrowing if necessary.

In the exercise \(3.4 - 0.38\), aligning the numbers gives you a clear path to subtract efficiently. It involves looking at the smaller pieces (tenths, hundredths) and subtracting them just as you would with whole numbers. This ensures that the subtraction of decimal numbers is just as straightforward as working with whole numbers.

Multiplication

Multiplication involves combining quantities to find a product. When multiplying decimals, there are unique steps to follow:

• Multiply the numbers as if they were whole numbers, ignoring the decimal points initially.
• Count the total number of decimal places in both of the numbers you are multiplying.
• In the result, place the decimal point so that the product has the same total number of decimal places counted from step two.

The exercise demonstrates this with \(\frac{1}{2} \times 0.76\), where \(0.76\) is a decimal being multiplied by a fraction, which effectively halves it. Performing these operations accurately is crucial; it ensures the results make sense in the context you're working with, which could range from scientific calculations to financial transactions.

Algebra Basics

Algebra involves using symbols and letters to represent numbers and quantities in equations and expressions. It's a powerful tool for solving problems and simplifying expressions.

In the exercise, simplifying the expression\(3.4 - \frac{1}{2}(0.76)\) is about managing two important algebraic processes:

• Recognizing the need to perform operations in the correct order, adhering to the rule of remembering PEMDAS/BODMAS (Parentheses/Brackets, Orders, Division/Multiplication, Addition/Subtraction).
• Simplifying means applying operations to reduce an expression to its simplest form.

By simplifying expressions, unknowns become clearer, allowing us to find solutions efficiently. Understanding these basics builds a foundation for more advanced mathematical concepts and applications.