Question

Correct any answers that have the incorrect number of significant figures. (a) \(78.9+890.43-23=9.5 \times 10^{2}\) (b) \(9354-3489.56+34.3=5898.74\) (c) \(0.00407+0.0943=0.0984\) (d) \(0.00896-0.007=0.00196\)

Short answer

Corrected answers: (a) \(9.46 \times 10^2\), (b) 5899, (c) 0.098, (d) 0.002.

Step-by-step solution

- Review Significant Figures

Review the rules for significant figures (sig figs) in order to correct the answers. In addition, remember that addition and subtraction should be reported with the least number of decimal places found in any number in the set, while multiplication and division should be reported with no more sig figs than the number with the least total sig figs in the problem.

- Correcting (a) for Significant Figures

First, perform the operation: 78.9 + 890.43 - 23 = 946.33. Now, we round to the least number of decimals in the original numbers, which is 1 decimal place due to the first number, 78.9. Therefore, we round to 946.3. Since the original answer was given in scientific notation, convert 946.3 into scientific notation, which is \(9.463 \times 10^2\). However, based on sig figs in the original numbers, we should have only 3 sig figs, therefore the corrected answer in scientific notation should have only 3 sig figs, which is \(9.46 \times 10^2\).

- Correcting (b) for Significant Figures

Perform the calculation: 9354 - 3489.56 + 34.3 = 5898.74. Here, since we are dealing with addition and subtraction, we need to look at the decimal places. The original number with the least number of decimal places is 9354 (no decimal places shown). Therefore, the result should be rounded to the nearest whole number: 5899.

- Correcting (c) for Significant Figures

Add the two values: 0.00407 + 0.0943 = 0.09837. However, when looking at the original numbers, we can see that the first number has the least number of significant decimal places (2 decimal places). Therefore, the answer should be rounded to two decimal places to give 0.098.

- Correcting (d) for Significant Figures

Subtract the two values: 0.00896 - 0.007 = 0.00196. The number with the fewest decimal places after the decimal is the second number (0.007), which has 3 decimal places. Thus, our answer must be rounded to three decimal places past the decimal, giving us 0.002.

Key concepts

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a concise form. It is commonly used in science and engineering to make calculations and comparisons easier. A number in scientific notation is written as the product of a number (between 1 and 10) and a power of 10. For example, the number 93,500 can be written as \(9.35 \times 10^{4}\).
To convert a number into scientific notation, you need to move the decimal point to the position after the first non-zero digit, and then count the number of places you moved it. If you moved the decimal to the left, the power of 10 will be positive; if to the right, negative. This compact format not only simplifies dealing with extreme values but also clearly indicates the number of significant figures a value has.
In our exercise problem (a), the number 946.3 has been correctly converted into scientific notation as \(9.46 \times 10^{2}\), indicating three significant figures.

Addition and Subtraction Rules for Significant Figures

When adding or subtracting numbers, it's essential to pay attention to the significant figures in the decimal places. The rule for determining the number of significant figures after an addition or subtraction is to round the result to the least precise decimal place of any number in the operation.
For instance, when given the numbers 78.9 (one decimal place) and 890.43 (two decimal places), and subtracting 23 (no decimal places), the result should be rounded to one decimal place, the least precise among the three numbers. Therefore, the correct answer to problem (a) is \(946.3\) or \(9.46 \times 10^{2}\) in scientific notation reflecting the precision of the original numbers.

Rounding in Significant Figures

Rounding is a technique used to reduce the number of significant figures in a result to match the precision of the input data. When rounding, if the digit to the right of your last significant figure is equal to or more than 5, you increase the last significant figure by one. If it's less than 5, you leave the last significant figure as is. It's crucial to round off the value after you complete all the calculations to avoid rounding errors in intermediate steps.

• Any zeros to the left of the first non-zero digit are not considered significant.
• Zeros between non-zero digits are always significant.
• Zeros to the right of the decimal and at the end of the number are significant if one of the digits to the left is non-zero.

For example, when adjusting the solution for problem (b), the answer needed to be rounded to the nearest whole number as the least significant figure was in the whole number place, resulting in an answer of 5899.

Multiplication and Division Rules for Significant Figures

Multiplication and division rules for significant figures differ from addition and subtraction. When you multiply or divide numbers, the result should have as many significant figures as the number with the fewest significant figures used in the calculation.
For example, if you multiply 2.5 (two significant figures) by 3.45 (three significant figures), your answer should have two significant figures because 2.5 has the fewest significant figures. It’s important to complete all multiplications or divisions before rounding off the final result to the correct number of significant figures to avoid compounding rounding errors. Adhering to these rules ensures that the precision indicated by significant figures is maintained throughout your calculations.