Question

What will be the answer in appropriate significant figures as a result of addition of \(3.0223\) and \(5.041 ?\) (a) \(80.633\) (b) \(8.0633\) (c) \(8.063\) (d) \(806.33\)

Short answer

The answer, rounded to the appropriate number of significant figures, is option (c) 8.063.

Step-by-step solution

- Review the rules for addition with significant figures

When adding numbers, the sum should be reported with the same number of decimal places as the measurement with the least number of decimal places. In this case, since 3.0223 has four decimal places and 5.041 has three, the final answer should be rounded off to three decimal places.

- Perform the Addition

Add the two numbers normally, ignoring the rule about significant figures momentarily. So, you would calculate 3.0223 + 5.041 which equals 8.0633.

- Round to the Correct Number of Significant Figures

Since the number with the least number of decimal places (5.041) has three, we round our answer from Step 2 to three decimal places, which gives us 8.063.

Key concepts

Rules for Addition with Significant Figures

Understanding the correct application of significant figures in addition is crucial for the precision and accuracy of measurements in chemistry. When you add two or more numbers, the result should not have more significant figures after the decimal point than the least precise measurement. In simpler terms, look at the number of decimal places each addend has and use the one with the least amount as a guide for your answer.

For example, if you are adding 123.456 (three decimal places) and 7.89 (two decimal places), the result should only have two decimal places, since that's the least precise measurement. It's like respecting the weakest link in a chain. This ensures that the inherent precision (or lack thereof) of the original measurements is not overstated in the final calculation.

Rounding Off Significant Figures

Once the result is obtained, it might be necessary to round off the number to maintain the proper count of significant figures. Rounding is done by looking at the number immediately to the right of the last significant figure you want to keep. If this digit is five or greater, you would round up the last significant figure. If less than five, you keep it the same. However, remember to keep in mind the context and conventions specific to certain fields of study or measurement standards.

Example:

If you have the sum 12.3456 and it needs to be rounded to three decimal places, you would round it to 12.346. But if it needs to be rounded to two decimal places, it becomes 12.35. Precision is not just about detailing a measurement; it's about reflecting the reliability of the experimental data.

Performing Addition with Significant Figures

Let's apply these concepts to an actual problem. When instructed to add 3.0223 and 5.041, consider the precision of each number. The former has four decimal places while the latter has three. Align the numbers by their least precise measurement, three decimal places, and add them together.

Addition process:

Start with the sum ignoring significant figures: 3.0223 + 5.041 = 8.0633. Next, apply the rules of significant figures by rounding the answer to three decimal places, yielding 8.063 — even though the mathematical sum provides a fourth decimal place. It's like framing a picture; you cut the excess to fit the frame, considering the original image's boundaries dictated by the least precise number.