Problem 89 - Algebra 1: Concepts and Skills

Question

Estimate the answer. Then evaluate the expression. $$ 0.3-0.03 $$

Step-by-Step Solution

Answer

The result of the expression \(0.3 - 0.03\) is \(0.27\).

Step 1: Align Decimal Points

Place the numbers in a vertical layout, ensuring the decimal points are aligned:\[ \begin{align*} &0.3 \ - &0.03 \ \end{align*} \]

Step 2: Perform Subtraction

Carry out the subtraction as if working with whole numbers. Remember to place the decimal point in the result at the same vertical position as those in the terms you are subtracting. The answer to \(0.3 - 0.03\) is \(0.27\).

Step 3: Evaluate Results

Check if the result makes sense. Since 0.03 is a small fraction of 0.3, the result should be a little bit less than 0.3. So, the result of 0.27 seems plausible. Always good to think logically about the results to validate it.

Key Concepts

Estimating Answers

Estimating is a useful skill that helps us quickly get a rough idea of what an answer might be. When dealing with decimal subtraction, estimating can make it easier to understand if your final answer is reasonable. For instance, in the exercise of subtracting 0.03 from 0.3, we can round the numbers to the nearest simple value.

• 0.3 rounds to 0.30, which is already a pretty rounded value.
• 0.03 rounds to 0.00.

Subtracting these estimates (0.30 - 0.00) gives us 0.30. Knowing this, our final answer should be close to 0.30 but slightly lower because 0.03 is removed from 0.3. Estimation gives us a benchmark, so when we calculate an exact answer (0.27), we can check if it feels right.

Decimal Alignment

Decimal alignment means lining up the numbers by their decimal points before performing operations like addition or subtraction. It's crucial because it ensures that each digit is in its correct place value position. In our example:\[\begin{align*} &0.3 \ - &0.03 \end{align*}\]We make sure that the key positions like tenths, hundredths, and so on, stack above each other. This alignment is vital because it keeps the subtraction operation organized, allowing accurate calculation. Imagine stacking bricks: everything crumbles if not properly aligned! Moving forward with aligned decimal points makes subtracting smaller parts from larger whole parts straightforward and keeps our place values straight.

Place Value

Understanding place value is fundamental to accurately performing operations with decimals. Each digit in a decimal number has a specific place value—the position that gives it meaning. For example, in 0.3:

• The 3 is in the tenths place, meaning three tenths or 0.3.

And in 0.03:

• The 3 is in the hundredths place, meaning three hundredths or 0.03.

Keeping track of place values lets us determine which section of the number we’re subtracting. When we subtract 0.03 from 0.3, we're essentially removing 3 hundredths from 3 tenths, resulting in 0.27. By respecting place values, we maintain accuracy throughout our calculations.

Logical Reasoning

Logical reasoning adds an additional layer of understanding and verification to math calculations. It involves using reasoned thinking to evaluate whether an answer seems reasonable or needs re-evaluation. For decimal subtraction, like in 0.3 - 0.03: - You know 0.03 is a small part of 0.3. - Therefore, subtracting it should result in a number slightly less than 0.3. - The calculated answer, 0.27, should be deemed correct because it's reasonably close to the estimate of 0.30. Logical reasoning also helps in cross-checking with estimates or intuition. If something seems off, it prompts further review. This habit of sanity-checking your results ensures that even if a computational mistake occurs, you can catch it before settling on the final answer.