Problem 41 - Fundamentals of Mathematics

Question

For the following 4 problems, perform the indica ted operations $$ -0.012-(-0.111)-(0.035) $$

Step-by-Step Solution

Answer

The result is 0.064.

Step 1: Simplify the Expression

The given expression is \(-0.012 - (-0.111) - 0.035\). First, let's handle the double negative in the expression. Subtracting a negative is equivalent to adding a positive. Therefore, \(-0.012 - (-0.111)\) becomes \(-0.012 + 0.111\). Now, the expression simplifies to \(-0.012 + 0.111 - 0.035\).

Step 2: Add the First Two Numbers

Add \(-0.012\) and \(0.111\) together. This is equivalent to: \(0.111 - 0.012 = 0.099\). So after the first two numbers, we have \(0.099\).

Step 3: Subtract the Third Number

Now, subtract \(0.035\) from the result of Step 2. So, \(0.099 - 0.035 = 0.064\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are used to represent a lack or deficit of something. In mathematics, negative numbers are often found in operations involving direction, subtraction, or when dealing with debts. The negative sign (-) in front of a number signifies its direction on the number line, pointing towards values less than zero.

When working with negative numbers, especially in expressions or equations, remember these key points:

• Opposite Sign Rule: When you subtract a negative number, you are essentially adding the positive equivalent of that number. For instance, \(-(-5)\) becomes \(+5\).
• Combining Negatives: When both numbers you are working with are negative, the signs combine to make a positive. This concept helps in simplifying expressions like double negatives.
• Number Line Visual: Picture a number line to understand the direction of numbers and operations better. Moving left indicates subtraction (or negative), while moving right implies addition (or positive).

Understanding these basic properties will help make calculations with negatives more intuitive and manageable.

Addition of Decimals

Adding decimals involves lining up the decimal points and adding each column of numbers like you would in whole numbers. Decimals are simply parts of whole numbers spoken about in tenths, hundredths, thousandths, etc.

Here’s how you can add decimals effectively:

• Align Decimal Points: Always line up the decimal points before you begin to add. This ensures that you are adding correspondingly similar values, such as tenths with tenths and hundredths with hundredths.
• Fill in Zeros: If you are adding decimals of varying lengths, fill in zeros so each number has the same number of decimal places. For example, in the sum \(0.5 + 0.05\), write it as \(0.50 + 0.05\).
• Add like Whole Numbers: Once lined up and ready, add as you would with whole numbers, keeping the decimal directly aligned in your answer.

By following these steps, adding decimals becomes a straightforward process, just like adding whole numbers but with a focus on decimal placement to ensure accuracy.

Subtraction of Decimals

Subtraction of decimals is similar to addition, with the focus on aligning decimal points and careful borrowing when necessary. When subtracting, it is crucial to ensure numbers are in the same place value position, meaning the subtraction happens digit by digit from right to left.

Consider the following steps to subtract decimals:

• Align Decimal Points: Just like addition, line up the decimals in a column to maintain place value alignment.
• Borrow When Necessary: If a digit in the top number is smaller than the corresponding digit below, borrowing from the next column is necessary, similar to whole number subtraction. For instance, in \(1.50 - 0.075\), borrow as needed for accurate results.
• Subtract Digit by Digit: After ensuring proper alignment and borrowing, subtract each column individually, starting from the right and moving to the left.

Careful attention to each step will help ensure that decimal subtraction leads to correct and reliable answers.