Problem 87 - Prealgebra

Question

Determine which of the two given statements is true. 18.62192 < 18.6293549 or 18.62192 > 18.6293549

Step-by-Step Solution

Answer

18.62192 < 18.6293549 is the true statement.

Step 1: Understand the Problem

We need to compare the two decimal numbers: 18.62192 and 18.6293549, to determine which statement is true: 18.62192 < 18.6293549 or 18.62192 > 18.6293549.

Step 2: Compare Whole Number Part

Both numbers have '18' as the whole number part, which means they are the same. Thus, we need to compare the decimal parts to decide which number is larger.

Step 3: Compare Decimal Parts Digit by Digit

Start comparing the decimal parts digit by digit from left to right. The first decimal digit in each number is '6', so they are the same. The second digit is '2', also the same. The third digit is '1' in 18.62192 and '9' in 18.6293549. Since 1 is less than 9, 18.62192 is less than 18.6293549.

Step 4: Decide the True Statement

Since in the decimal comparison at the third decimal place 1 < 9, 18.62192 < 18.6293549. Therefore, the true statement is 18.62192 < 18.6293549.

Key Concepts

Step-by-step problem solving

When solving a math problem, especially one involving comparisons, following a systematic and structured approach helps in reaching the correct answer effectively.
Step-by-step problem solving begins with a clear understanding of the problem. In our given exercise, the task is to determine which of two decimal statements is true. This means breaking down the problem into manageable parts and dealing with each part methodically.

• First, identify the goal: comparing two decimals.
• Next, dissect the numbers into their components: whole number and decimal parts.
• Finally, proceed with comparing these components systematically, ensuring clarity in each step.

Taking a problem-solving approach that is step-by-step not only avoids confusion but also makes the process more manageable, highlighting each reason behind a decision made at any stage.

Decimal numbers

Decimal numbers extend our number system beyond whole numbers, allowing us to express fractional amounts precisely. They are composed of two parts: a whole number part and a fraction part separated by a decimal point.
In the exercise, both numbers start with 18, making them identical in whole number value. The focus then shifts to the digits following the decimal point. It's crucial to understand that

• Each position after the decimal point represents a fraction of a power of ten: tenths, hundredths, thousandths, and so on.
• When comparing decimals, you inspect each digit in order from left to right after the decimal point, stopping as soon as a difference is found.

In our example, this process revealed a difference at the third decimal place—crucial for determining that 18.62192 is indeed less than 18.6293549.

Mathematical reasoning

Mathematical reasoning is the logical thought process used to determine the truth of a statement or solution. This problem demonstrates such reasoning through careful comparison and analysis.
As per the step-by-step solution, we practiced mathematical reasoning by

• First, noting the identical whole number part, which motivates us to check the decimal parts for differences.
• Then, analyzing each decimal digit sequentially ensures consistency in comparison, highlighting logical decision-making.

Reasoning helps us formulate the decision at the third decimal place where the digit '1' of 18.62192 is compared to '9' of 18.6293549, logically leading us to conclude the correct inequality: 18.62192 < 18.6293549. This structured reasoning ensures clarity and correctness.