Question

Compute and express each answer with the proper number of significant figures, rounding as necessary. a) \(1.005+17.88=?\) b) \(56,700-324=?\) c) \(405,007-123.3=?\) d) \(55.5+66.66-77.777=?\)

Short answer

a) 18.89, b) 56,400, c) 404,883.7, d) 44.4

Step-by-step solution

Addition of a and rounding

First, add the numbers in part (a):\[1.005 + 17.88 = 18.885\]The number with the fewest decimal places is 17.88 with two decimal places, so we round the result to two decimal places: 18.89.

Subtraction of b and rounding

Now subtract the numbers in part (b):\[56{,}700 - 324 = 56{,}376\]The significant figures in the result are determined by the number in scientific notation, assuming 56,700 has three significant figures (based on its presentation). Therefore, round to three significant figures: 56,400.

Subtraction of c and rounding

Next, subtract the numbers in part (c):\[405{,}007 - 123.3 = 404{,}883.7\]Here, the number with the least digits after the decimal is 123.3 with one digit. So, we round to one decimal place: 404,883.7.

Combined addition and subtraction for d and rounding

Finally, calculate the combined operation in part (d):\[55.5 + 66.66 - 77.777 = 44.383\]The least number of decimal places in any term is one (55.5), so round the result to one decimal place: 44.4.

Key concepts

Rounding Rules

Rounding is the process of adjusting a number to make it simpler and easier to use, but still keeping its value close to what it was. The key to proper rounding lies in understanding when and how to round.
Rounding is especially important in scientific calculations where maintaining the appropriate level of precision is crucial.

• If the digit following the last significant figure is less than 5, you leave the last significant figure unchanged.
• If it is 5 or more, you increase the last significant figure by one. For example, rounding 2.876 to two significant figures results in 2.9.

In the context of the exercise, rounding was done after calculations to ensure results were expressed with the correct number of significant figures. This maintains the integrity and accuracy of the calculation outcomes.

Addition and Subtraction of Significant Figures

When you're working with addition and subtraction, it's important to keep an eye on the decimal places to determine significant figures. Unlike multiplication and division, we don't count the significant figures in the numbers for these operations.
Instead, here is what you do:

• Align the numbers by their decimal points.
• Identify the number with the fewest decimal places.
• Perform the addition or subtraction and round off the result to the smallest number of decimal places found in the original numbers.

In the given exercise, for each part, the numbers are aligned by their decimal places, and the result is rounded based on the number with the fewest decimal places. This ensures that all results have been reported with the proper level of precision.

Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is often used in science and engineering.

• A number is expressed as the product of a number between 1 and 10, and a power of ten.
• This notation helps in handling significant figures effectively, often simplifying calculations that involve very large or small numbers.

For instance, a number like 5,670,000 can be written as \(5.67 \times 10^6\) in scientific notation, where it can clearly see three significant figures. In the second part of the exercise, understanding significant figures was key, despite scientific notation not being explicitly used.

Precision in Calculations

Precision in calculations refers to how detailed and exact a measurement is in comparison to its true value. The goal is to have confidence that the calculations are correctly representing the measurement.

• More significant figures indicate greater precision.
• Each step in a calculation should maintain the correct amount of precision based upon the inputs.

During the example calculation, each result was carefully expressed using the appropriate number of significant figures to preserve precision. This approach ensures that the results are not misleadingly more precise than the data allows.