Significant Figures: In Calculations: Videos & Practice Problems

In calculations involving significant figures, multiplication and division yield results based on the least number of significant figures, while addition and subtraction depend on the least number of decimal places. For mixed operations, follow the order of operations (PENDES: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). This ensures accuracy in results, such as determining the final answer for a calculation involving significant figures and decimal places, which is crucial in chemistry for precise measurements and reactions.

Significant Figures in Multiplication and Division

In chemistry, significant figures play a crucial role in ensuring that calculations reflect the precision of the measurements involved. When performing multiplication or division, the final result must contain the same number of significant figures as the measurement with the least significant figures. This principle ensures that the precision of the least accurate measurement dictates the precision of the final answer.

To illustrate this, consider the multiplication of three values: 3.16, 0.003027, and \(5.7 \times 10^{-3}\). First, we determine the number of significant figures in each value:

• For 3.16, the first non-zero digit is 3, and counting from there gives us 3 significant figures.
• For 0.003027, the first non-zero digit is 3, leading to 4 significant figures.
• For \(5.7 \times 10^{-3}\), the coefficient is 5. Counting from the first non-zero digit gives us 2 significant figures.

With these counts, we see that the least number of significant figures among the values is 2. Therefore, our final answer must also be expressed with 2 significant figures.

After performing the multiplication, we initially obtain \(5.4522324 \times 10^{-5}\). To round this to 2 significant figures, we look at the first two digits, which are 5 and 4. The next digit is 2, which means we do not round up. Thus, the final answer is \(5.5 \times 10^{-5}\).

This example highlights the importance of significant figures in multiplication and division. The next step in understanding significant figures involves addition and subtraction, where different rules apply. In those operations, the final answer is determined by the least precise decimal place rather than the number of significant figures.

Significant Figures in Addition and Substraction

When performing calculations involving addition and subtraction, it is essential to determine the final answer based on the least number of decimal places among the numbers involved. This principle directly influences the significant figures in the result. For instance, consider the calculation of \( 402.09 - 2.12 + 2.671 \). Here, the numbers have the following decimal places: 402.09 has 2 decimal places, 2.12 has 2 decimal places, and 2.671 has 3 decimal places. Since the least number of decimal places is 2, the final answer must also be rounded to 2 decimal places.Calculating the expression yields \( 402.09 - 2.12 + 2.671 = 402.09 - 2.12 + 2.671 = 402.641 \). Rounding this to 2 decimal places results in \( 402.64 \). In terms of significant figures, the first non-zero digit is counted from left to right. In this case, \( 402.64 \) has 5 significant figures. Therefore, the final answer, considering both decimal places and significant figures, is \( 402.64 \) with 5 significant figures.When mixing operations such as multiplication and division with addition and subtraction, the rules for determining significant figures and decimal places remain crucial. Understanding how to apply these rules ensures accuracy in scientific calculations.

Significant Figures in Mixed Operations

When performing calculations that involve mixed operations such as multiplication, division, addition, and subtraction, it is essential to adhere to the order of operations. A helpful mnemonic for remembering this order is PENDES, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This structure ensures that calculations are performed correctly and consistently.

For example, consider the expression 1.89 \times 10^6 \times 3.005 and 5.21^3 \div (8.829 - 6.5 + 2.920). The first step is to evaluate any expressions within parentheses. In this case, we first multiply the numbers in brackets:

1.89 \times 10^6 \times 3.005 = 5679450. However, when reporting this result, we must consider significant figures. The number 1.89 \times 10^6 has 3 significant figures, while 3.005 has 4. Therefore, our final answer should also have 3 significant figures, resulting in 5.68 \times 10^6 when expressed in scientific notation.

Next, we calculate 5.21^3, which means multiplying 5.21 \times 5.21 \times 5.21. The initial result is 141.420761, but since all factors have 3 significant figures, we round this to 141.

Now, we address the subtraction and addition in the denominator: 8.829 - 6.5 + 2.920. The subtraction yields 2.329, which has 3 decimal places, while 2.920 has 3 decimal places as well. Therefore, the final result of this operation must be rounded to 1 decimal place, giving us 2.3.

Next, we add 2.3 + 2.920, which results in 5.220. Since 2.3 has 1 decimal place and 2.920 has 3, the final answer must also have 1 decimal place, resulting in 5.2.

Now, we multiply the results from the numerator and denominator. The multiplication of 5.68 \times 10^6 and 141 gives 8.0088 \times 10^8. Since both numbers have 3 significant figures, we round this to 8.01 \times 10^8.

Finally, we divide 8.01 \times 10^8 by 5.2. The result is 1.54 \times 10^8, but since 5.2 has 2 significant figures, we round our final answer to 1.5 \times 10^8.

In summary, when dealing with mixed operations, always follow the order of operations and apply the rules for significant figures: for multiplication and division, use the least number of significant figures; for addition and subtraction, use the least number of decimal places.

Perform the following calculation to the right number of sig figs:
[(1.7 × 10⁶) ÷ (2.63 × 10⁵)] + 6.96

• A. 13.46
• B. 13.0
• C. 13.5
• D. 14.2
• E. 4.471

Answer: (C)

What answer should be reported, with the correct number of significant figures, for the following calculation?

• A. 0.00822
• B. 0.0845
• C. 0.0921
• D. 0.0947

Answer: (D)

Here’s what students ask on this topic:

What are the rules for determining significant figures in multiplication and division?

When performing multiplication or division, the number of significant figures in the final answer is determined by the measurement with the fewest significant figures. For example, if you multiply 3.16 (3 significant figures) by 0.003027 (4 significant figures) and by 5.7 × 10⁻³ (2 significant figures), the final answer must have 2 significant figures, as 2 is the smallest number of significant figures among the inputs. Always round your result to match this precision.

How do significant figures affect addition and subtraction calculations?

In addition and subtraction, the precision of the final answer is determined by the number with the fewest decimal places. For example, if you calculate 402.09 (2 decimal places) − 12.2 (1 decimal place) + 2.671 (3 decimal places), the result must have only 1 decimal place, as that is the least precise measurement. After performing the calculation, round the result to match the required number of decimal places.

What is the process for handling significant figures in mixed operations?

For mixed operations, follow the order of operations (parentheses, exponents, multiplication/division, addition/subtraction) and apply the rules for significant figures at each step. For multiplication and division, use the least number of significant figures. For addition and subtraction, use the least number of decimal places. For example, if you calculate (1.89 × 10⁶ × 3.005) ÷ (8.829 − 6.5) + 2.920, handle each operation step-by-step, applying the appropriate significant figure rules at each stage to ensure accuracy.

How do you round a number to the correct number of significant figures?

To round a number to the correct number of significant figures, identify the digit at the cutoff point and check the digit immediately to its right. If the digit to the right is 5 or greater, round up; otherwise, keep the digit unchanged. For example, if you need to round 5.4522324 × 10⁻⁵ to 2 significant figures, focus on the first two digits (5.4). Since the next digit is 5, round up to 5.5 × 10⁻⁵.

Why is it important to follow the rules of significant figures in chemistry calculations?

Following the rules of significant figures ensures that your calculations reflect the precision of the measurements used. This prevents overestimating the accuracy of your results and maintains consistency in scientific reporting. For example, if one measurement is precise to 2 significant figures, the final result should not imply greater precision than the data allows. This is crucial for reliable and reproducible results in chemistry.