Problem 88 - Chemistry Matter and Change

Question

Which number will produce the same number when rounded to three significant figures: 3.456, 3.450, or 3.448?

Step-by-Step Solution

Answer

The two numbers that produce the same number when rounded to three significant figures are \(3.450\) and \(3.448\), as both of them round to \(3.450\).

Step 1: Understanding Significant Figures and Rounding Rules

When rounding numbers to a certain number of significant figures, keep the following rules in mind: 1. If the digit after the rounding position is less than 5, round down (leave the digit in the rounding position unchanged). 2. If the digit after the rounding position is 5 or greater, round up (increase the digit in the rounding position by 1). Since we want to round the numbers to three significant figures, our rounding position will be the third digit after the decimal point.

Step 2: Round 3.456 to Three Significant Figures

Since 3.456 already has three significant figures, we don't need to round it. So, the first number is: \(3.456\).

Step 3: Round 3.450 to Three Significant Figures

The number 3.450 also has three significant figures, so no rounding is necessary. The second number is: \(3.450\).

Step 4: Round 3.448 to Three Significant Figures

To round 3.448 to three significant figures, we need to check the fourth digit which is 8. Since the digit is greater than 5, we need to round up the third digit, which is 4. So, when rounding 3.448 to three significant figures, we get: \(3.448 \approx 3.450\).

Step 5: Identify the Numbers Rounded to the Same Number

Now, we compare the rounded numbers from Steps 2, 3, and 4. The numbers are: 1. 3.456 2. 3.450 3. 3.450 (from rounding 3.448) The two numbers that produce the same number when rounded to three significant figures are \(3.450\) and \(3.448\), because after rounding, both of them are equal to \(3.450\).

Key Concepts

Rounding Numbers

Rounding numbers is a valuable and practical mathematical technique used to simplify numbers, making them easier to work with. This process involves adjusting numbers to reflect a particular level of precision that you desire, which can often make calculations more straightforward and results more comprehensible. For instance, having a number like 3.456 can be rounded to 3.46 if two decimal places are needed, or to 3.5 if only one is required.

There are key reasons for rounding numbers:

• To make numbers shorter and easier to understand.
• When the exact number is not necessary, rounding provides a meaningful estimate.
• To maintain consistency when numbers need to have the same precision for comparison or reporting purposes.

Rounding is not arbitrary, but follows a specific set of rules to ensure accuracy and consistency, which leads us to explore the rules of number rounding and how they play a critical role in this process.

Number Rounding Rules

When rounding numbers, you rely on established rules that determine whether a number will round up or down. These rules prevent inconsistency and make sure that everyone arrives at the same result when rounding the same data.

The fundamental rules of rounding are:

• If the number immediately after the digit you want to round is less than 5, you round the number down. For example, rounding 7.42 to one decimal place gives 7.4 because the digit in the second decimal place (2) is less than 5.
• If the number immediately after the digit you want to round is 5 or greater, you round the number up. For example, rounding 7.46 to one decimal place gives 7.5 because the digit in the second decimal place (6) is greater than 5.

Understanding these rules allows you to apply them not only in specific exercises or homework problems, but also when interpreting data from real-world scenarios in fields like engineering, finance, and science.

It is important to note that rounding is often applied in terms of significant figures, not just decimal places. This distinction often matters when you're dealing with numbers that include a variety of decimal places, which we explore in the next section.

Decimal Places

Decimal places are the positions of numbers after the decimal point in a decimal number. They are crucial in maintaining precision and accuracy for mathematical calculations and scientific experiments. For example, in the number 3.456, '4', '5', and '6' are the numbers in the first, second, and third decimal places respectively.

In many cases, you'll find yourself needing to specify how many decimal places you require for your data. This depends largely on the context in which you're working. For instance, financial records often require two decimal places to account for cents in monetary values.

When rounding numbers, deciding how many decimal places you need is a key step:

• More decimal places increase the detail and precision of a number.
• Fewer decimal places can make data easier to read and interpret.

Getting the right number of decimal places ensures that your data is both precise enough to be valid, and concise enough to be practical. It's a balance that sometimes involves rounding to achieve, as seen in the exercise above where understanding rounding in terms of decimal places and significant figures helped determine the final answer.