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 accurate results, such as determining significant figures in scientific notation and rounding appropriately. Understanding these principles is crucial for precise measurements in chemistry, impacting calculations involving acids, bases, and various chemical reactions.

Significant Figures Calculations Concept 1

In chemistry, understanding significant figures is crucial for accurate calculations, especially when performing multiplication and division. The key principle to remember is that the final answer should reflect the least number of significant figures from the values used in the calculation.

For example, consider the multiplication of three values: 3.16, 0.003027, and \(5.7 \times 10^{-3}\). To determine the number of significant figures in each value, we start counting from the first non-zero digit:

• 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 \(5.7 \times 10^{-3}\), we focus on the coefficient 5.7, which has 2 significant figures.

Now, we identify the least number of significant figures among these values, which is 2. Therefore, our final answer must also contain 2 significant figures.

After performing the multiplication, we initially obtain \(5.4522324 \times 10^{-5}\). To express this with 2 significant figures, we look at the first two digits, 5 and 4. The next digit is 5, which prompts us to round up the 4 to 5. Thus, the final answer is \(5.5 \times 10^{-5}\), ensuring it adheres to the significant figures rule based on the original values.

Having covered multiplication and division, it is essential to explore how significant figures apply to addition and subtraction in subsequent calculations.

Significant Figures Calculations Concept 2

When performing calculations involving addition and subtraction, the final result should reflect the least number of decimal places from the numbers involved. This principle ensures that the precision of the answer is consistent with the least precise measurement. For example, consider the calculation \( 402.09 - 2.12 + 2.671 \). Here, the numbers have different decimal places: 402.09 has 2 decimal places, 2.12 has 1 decimal place, and 2.671 has 3 decimal places. Since 2.12 has the least number of decimal places, the final answer must also be rounded to 1 decimal place.Calculating the expression gives \( 402.09 - 2.12 + 2.671 = 402.09 - 2.12 + 2.671 = 402.641 \). Rounding this to 1 decimal place results in \( 402.6 \). In terms of significant figures, the first non-zero digit is counted from left to right. In this case, \( 402.6 \) has 4 significant figures, as all non-zero digits and any zeros between them are counted. This approach to determining the number of decimal places directly influences the significant figures in the final answer. When mixing operations such as multiplication and division with addition and subtraction, the rules for significant figures and decimal places must be carefully applied to maintain accuracy in the results.

Significant Figures Calculations Concept 3

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, yielding 5679450. However, when dealing with significant figures, we must consider the number of significant figures in the original values. The number 1.89 \times 10^6 has 3 significant figures, while 3.005 has 4. Therefore, the final answer must 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 values have 3 significant figures, we round this to 141.

For the subtraction 8.829 - 6.5, the result is 2.329. Here, we must consider the decimal places: 8.829 has 3 decimal places, while 6.5 has 1. Thus, the final answer must have 1 decimal place, resulting in 2.3.

Next, we add 2.3 + 2.920, which gives 5.220. Again, we apply the rule for decimal places: 2.3 has 1 decimal place, and 2.920 has 3, so the final result is 5.2.

Now, we multiply the results from the top: 5.68 \times 10^6 \times 141. Since both numbers have 3 significant figures, the final product is 8.01 \times 10^8.

Finally, we divide 8.01 \times 10^8 \div 5.2. Here, 8.01 has 3 significant figures, while 5.2 has 2. Therefore, the final answer must have 2 significant figures, resulting in 1.5 \times 10^8.

In summary, when performing mixed operations, always follow the order of operations and apply the rules for significant figures and decimal places appropriately. This systematic approach will lead to accurate and precise results.

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)

Perform the following mathematical operations and express the result to the correct number of significant figures.
(6.404 × 2.91) / (18.7 – 17.1)

• A. 11.65
• B. 11.7
• C. 10
• D. 12
• E. 11.625

Answer: (D)

What answer should be reported, with the correct number of significant figures, for the following calculation?
[(42.00 − 40.914) ⋅ (25.739 − 25.729)] / [(11.50⋅1.001) + (0.00710 ⋅ 700.)]

• A. 7.0 x 10^-4
• B. 6.67 x 10^-4
• C. 6.70 x 10^-4
• D. 6.7 x 10^-4
• E. 6.75 x 10^-4

Answer: (D)

Here’s what students ask on this topic:

How do you determine the number of significant figures in a multiplication or division calculation?

When performing multiplication or division, the final answer should have the same number of significant figures as the value with the least number of significant figures in the calculation. For example, if you multiply 3.16 (3 significant figures) by 0.003027 (4 significant figures) and 5.7 × 10^-3 (2 significant figures), the result should be rounded to 2 significant figures, as 2 is the smallest number of significant figures among the values.

How do you handle significant figures in addition and subtraction?

In addition and subtraction, the final answer should be rounded to the least number of decimal places present in any of the numbers being added or subtracted. For instance, if you have 402.09 (2 decimal places), 2.1 (1 decimal place), and 2.671 (3 decimal places), the result should be rounded to 1 decimal place, as that is the smallest number of decimal places among the values.

What is the order of operations for mixed calculations involving significant figures?

For mixed operations involving significant figures, follow the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS). Perform calculations inside parentheses first, then handle exponents, followed by multiplication and division, and finally addition and subtraction. Apply the rules for significant figures at each step: least number of significant figures for multiplication/division and least number of decimal places for addition/subtraction.

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

To round a number to the correct number of significant figures, identify the digit at the desired place and look at the digit immediately to its right. If this digit is 5 or greater, round up the target digit. If it is less than 5, keep the target digit the same. For example, to round 5.4522324 to 2 significant figures, look at the third digit (4). Since the next digit (5) is 5 or greater, round up to 5.5.

How do significant figures affect scientific notation?

When using scientific notation, the number of significant figures is determined by the coefficient (the number before the exponent). For example, in 5.7 × 10^-3, the coefficient 5.7 has 2 significant figures. When performing calculations, ensure the final result is expressed with the correct number of significant figures based on the rules for multiplication, division, addition, or subtraction.