Question

Evaluate each of the following and write the answer to the appropriate number of significant figures. a. \(212.2+26.7+402.09\) b. \(1.0028+0.221+0.10337\) c. \(52.331+26.01-0.9981\) d. \(2.01 \times 10^{2}+3.014 \times 10^{3}\) e. \(7.255-6.8350\)

Short answer

Here is the short answer version: a. \(641.0\) b. \(1.327\) c. \(77.3429\) d. \(3.22 \times 10^{3}\) e. \(0.4200\)

Step-by-step solution

Evaluate Expression (a)

Step 1: Begin by adding the numbers together: \[ 212.2 + 26.7 + 402.09 \] Step 2: Write the numbers one below the other and add like place values, remembering to align decimal points: \[ \begin{array}{@{}c@{\;}c@{}c@{}c@{}c@{}} & 2& 1 & 2. & 2 \\ + & & 2 & 6. & 7 \\ + & 4 & 0 & 2. & 09 \\ \cline{1-5} &&& & \end{array} \] Step 3: Add each column to find the sum: \[ = 640.99 \] The fewest decimal places present in the given numbers is one (in 212.2 and 26.7). So, we must round our final answer to one decimal place. Final answer: \(640.99 \Rightarrow 641.0\)

Evaluate Expression (b)

Step 1: Begin by adding the numbers together: \[1.0028 + 0.221 + 0.10337 \] Step 2: Write the numbers one below the other and add like place values, remembering to align decimal points: \[ \begin{array}{@{}c@{\;}c@{}c@{}c@{}c@{}c} & 1. & 0 & 0 & 28 \\ + & 0. & 2 & 2 & 1 \\ + & 0. & 1 & 0 & 337 \\ \cline{1-5} &&&&& \end{array} \] Step 3: Add each column to find the sum: \[= 1.32717\] The fewest decimal places present in the given numbers is three (in 0.221). So, we must round our final answer to three decimal places. Final answer: \(1.32717 \Rightarrow 1.327 \)

Evaluate Expression (c)

Step 1: Begin by performing the operations as given: \[52.331 + 26.01 - 0.9981\] Step 2: Perform the addition operation first: \[ 52.331 + 26.01 = 78.341\] Step 3: Subtract 0.9981 from the sum obtained in step 2: \[78.341 - 0.9981 = 77.3429 \] The fewest decimal places present in the given numbers is four (in 0.9981). So, we must round our final answer to four decimal places. Final answer: \(77.3429 \Rightarrow 77.3429 \) (no rounding required)

Evaluate Expression (d)

Step 1: Begin by adding the numbers together: \[ 2.01 \times 10^{2} + 3.014 \times 10^{3} \] Step 2: Convert the numbers to the same power of ten by shifting the decimal point: \[ = 201 \times 10^{1} + 3014 \times 10^{1} \] Step 3: Add the numbers and simplify the expression: \[ = (201 + 3014) \times 10^{1} = 3215 \times 10^{1} \] Step 4: Convert the result back to scientific notation: \[ = 3.215 \times 10^{3}\] The fewest significant figures present in the given numbers is three (in 2.01). So, we must round our final answer to three significant figures. Final answer: \(3.215 \times 10^{3} \Rightarrow 3.22 \times 10^{3}\)

Evaluate Expression (e)

Step 1: Begin by subtracting the numbers: \[ 7.255 - 6.8350\] Step 2: Write the numbers one below the other and subtract like place values, remembering to align decimal points: \[ \begin{array}{@{}c@{\;}c@{}c@{}c@{}c@{}} & 7. & 2 & 5 & 50 \\ - & 6. & 8 & 3 & 50 \\ \\ \cline{1-5} &&&& \end{array} \] Step 3: Subtract each column to find the difference: \[= 0.4200\] The fewest decimal places present in the given numbers is three (in 7.255). However, we need to consider the last zero in 6.8350 as significant. So, our final answer should have four decimal places. Final answer: \(0.4200 \Rightarrow 0.4200\) (no rounding required)

Key concepts

Calculating Significant Figures

Understanding how to calculate significant figures is crucial in the precise representation of numbers in scientific and mathematical contexts. Significant figures are the digits in a number that carry meaning contributing to its precision. This includes all non-zero numbers, zeroes between significant digits, and zeroes which are after the decimal point and at the end of the number.

When performing arithmetic operations such as addition or subtraction, the result should have the same number of decimal places as the number with the least decimal places from the original problem. For instance, the sum of 212.2 (one decimal place) and 26.7 (one decimal place) must be reported to one decimal place, leading to a result of 641.0.

It is crucial to first determine which digits are significant to correctly perform calculations and report the final answer with the appropriate number of significant figures.

Rounding Decimal Places

Rounding decimal places is a method used to reduce the number of digits right of the decimal point, simplifying the number while maintaining its overall value within an acceptable range of accuracy. When rounding, if the digit to the right of the last significant place is greater than or equal to five, we increase the last significant digit by one. If it is less than five, the last significant digit remains unchanged.

For example, consider the number 1.32717, which must be rounded to three decimal places because of the least precise term 0.221 in the sum. The fourth decimal place is 7, which is greater than 5, so we round up the third decimal place of 1.32717 from 7 to 8, yielding a rounded number of 1.327.

Rounding is a fundamental skill in proper significant figures handling, and a misstep in rounding can lead to inaccuracies in the final reported figures.

Scientific Notation Arithmetic

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in standard decimal form. In arithmetic, numbers expressed in scientific notation are manipulated differently compared to regular numbers.

When adding or subtracting numbers in scientific notation, their exponents must match. For example, the calculation \(2.01 \times 10^{2} + 3.014 \times 10^{3}\) requires adjusting the exponents to confidently add them, leading to \(3215 \times 10^{1}\).

Once the decimal point has been moved, the significant figures in the answer are determined by the original numbers. If the smallest number of significant figures in the terms being added or subtracted is three, as in the term 2.01 in the example, the result must also be rounded to three significant figures, resulting in \(3.22 \times 10^{3}\).

Scientific notation simplifies the process of working with extremely large or small numbers and ensures that significant figures are accurately maintained throughout arithmetic operations.