Multiplying And Dividing Sig Figs Practice

Mastering Significant Figures: A Comprehensive Guide to Multiplication and Division Practice

Significant figures (sig figs) are the backbone of accurate scientific calculations. Understanding and applying the rules for sig figs, particularly in multiplication and division, is crucial for ensuring your results reflect the precision of your measurements. This comprehensive guide provides a detailed explanation of the rules, numerous practice problems with step-by-step solutions, and advanced tips to master this essential skill.

Understanding Significant Figures

Before diving into calculations, let's refresh our understanding of what constitutes a significant figure. Significant figures are the digits in a number that carry meaning contributing to its precision. They include all non-zero digits, zeros between non-zero digits, trailing zeros after a decimal point, and leading zeros in numbers less than one. Zeros used solely for placeholding are not significant.

Examples:

• 3.14159: All digits are significant (6 sig figs).
• 0.0025: Only 2 and 5 are significant (2 sig figs). The leading zeros are placeholders.
• 100.0: All digits are significant (4 sig figs). The trailing zero indicates precision.
• 1200: Ambiguous – could be 2, 3, or 4 sig figs depending on the precision of the measurement. Scientific notation avoids this ambiguity (e.g., 1.2 x 10³ has 2 sig figs).

Multiplication and Division with Significant Figures

The rule for multiplication and division is straightforward: the final answer should have the same number of significant figures as the measurement with the fewest significant figures.

Let's illustrate this with several examples.

Example 1:

Calculate 2.5 cm × 3.45 cm.

• 2.5 cm has two significant figures.
• 3.45 cm has three significant figures.

The calculation yields 8.625 cm². However, we must round the answer to two significant figures because 2.5 cm has the fewest sig figs. Therefore, the final answer is 8.6 cm².

Example 2:

Calculate 12.00 g / 2.0 mL.

• 12.00 g has four significant figures.
• 2.0 mL has two significant figures.

The calculation gives 6.0 g/mL. Since 2.0 mL has the fewest significant figures (two), the answer is rounded to 6.0 g/mL.

Example 3:

A more complex calculation: (45.67 m × 2.3 m) / 1.234 s

• 45.67 m has four significant figures.
• 2.3 m has two significant figures.
• 1.234 s has four significant figures.

First, perform the multiplication: 45.67 m × 2.3 m = 105.041 m²

Then, perform the division: 105.041 m² / 1.234 s = 85.11 m²/s

Finally, round the answer to two significant figures (determined by the 2.3 m measurement): 85 m²/s

Practice Problems: Multiplication and Division

Let's tackle several more practice problems to solidify your understanding. Remember to always identify the number of significant figures in each value before performing the calculation and round your final answer accordingly.

Problem 1: Calculate 15.23 mL × 0.87 g/mL

Solution:

1. Identify significant figures: 15.23 mL (4 sig figs), 0.87 g/mL (2 sig figs)
2. Perform the calculation: 15.23 mL × 0.87 g/mL = 13.25 g
3. Round to two significant figures (due to 0.87 g/mL): 13 g

Problem 2: Calculate 4567 kg / 23.4 m³

Solution:

1. Identify significant figures: 4567 kg (4 sig figs), 23.4 m³ (3 sig figs)
2. Perform the calculation: 4567 kg / 23.4 m³ = 195.128 kg/m³
3. Round to three significant figures (due to 23.4 m³): 195 kg/m³

Problem 3: (67.89 cm × 12.3 cm × 0.55 cm)

Solution:

1. Identify significant figures: 67.89 cm (4 sig figs), 12.3 cm (3 sig figs), 0.55 cm (2 sig figs)
2. Perform the calculation: 67.89 cm × 12.3 cm × 0.55 cm = 461.2 cm³
3. Round to two significant figures (due to 0.55 cm): 460 cm³

Problem 4: (234.56 g / 12.3 mL) / 1.0005 g/mL

Solution:

1. Identify significant figures: 234.56 g (5 sig figs), 12.3 mL (3 sig figs), 1.0005 g/mL (5 sig figs)
2. Perform the calculation: (234.56 g / 12.3 mL) / 1.0005 g/mL =19.09 g/mL
3. Round to three significant figures (due to 12.3 mL): 19.1 g/mL

Problem 5: (1.234 x 10⁻⁵ m × 5.67 x 10² m) / (9.87 x 10⁻³ s)

Solution:

1. Identify significant figures: 1.234 x 10⁻⁵ m (4 sig figs), 5.67 x 10² m (3 sig figs), 9.87 x 10⁻³ s (3 sig figs)
2. Perform the calculation: (1.234 x 10⁻⁵ m × 5.67 x 10² m) / (9.87 x 10⁻³ s) = 7.11 x 10⁻¹ m/s
3. Round to three significant figures (due to 5.67 x 10² m and 9.87 x 10⁻³ s): 7.11 x 10⁻¹ m/s

Advanced Tips and Considerations

• Scientific Notation: Using scientific notation is highly recommended, especially when dealing with very large or very small numbers. It clarifies the number of significant figures and avoids ambiguity.

• Intermediate Calculations: When performing multiple steps, retain extra significant figures during intermediate calculations and only round to the final answer. This minimizes rounding errors.

• Exact Numbers: Treat exact numbers (e.g., 12 inches in a foot, 2.54 cm per inch, counting numbers) as having an infinite number of significant figures. They do not affect the number of sig figs in your final answer.

• Addition and Subtraction: Remember that the rules for addition and subtraction with sig figs differ from those for multiplication and division. In addition and subtraction, the answer should have the same number of decimal places as the measurement with the fewest decimal places.

Conclusion

Mastering significant figures is essential for accurate scientific work. By consistently applying the rules for multiplication and division, and practicing with various examples, you can ensure your calculations are precise and reflect the true accuracy of your measurements. Remember to carefully consider the number of significant figures in each value, use scientific notation where appropriate, and practice regularly to develop a strong understanding of this important concept. Through consistent effort and the application of these guidelines, you'll confidently navigate the world of scientific calculations.