Calculations Using Significant Figures Answer Key

Calculations Using Significant Figures: A Comprehensive Guide with Answers

Significant figures are crucial in scientific calculations, representing the precision and accuracy of a measurement. Understanding how to use them correctly ensures your calculations reflect the true level of certainty in your data. This comprehensive guide will delve into the rules for determining significant figures, performing calculations with them, and providing answers to common practice problems. We'll cover addition, subtraction, multiplication, and division, ensuring you gain a strong grasp of this essential scientific concept.

Understanding Significant Figures

Before we dive into calculations, let's solidify our understanding of what constitutes a significant figure. Significant figures are all digits known with certainty plus one uncertain digit. This uncertain digit is an estimate, reflecting the limitations of the measuring instrument.

Rules for Determining Significant Figures:

1. Non-zero digits are always significant. For example, in the number 234, all three digits are significant.

2. Zeros between non-zero digits are always significant. In 204, the zero is significant.

3. Leading zeros (zeros to the left of the first non-zero digit) are not significant. They only serve to place the decimal point. For instance, 0.0025 only has two significant figures (2 and 5).

4. Trailing zeros (zeros to the right of the last non-zero digit) are significant only if the number contains a decimal point. For example:

   • 2500 has two significant figures.
   • 2500. has four significant figures.
   • 2500.0 has five significant figures.

5. In scientific notation (e.g., 2.5 x 10³), all digits in the coefficient are significant. In this case, both 2 and 5 are significant.

Calculations with Significant Figures

Now, let's tackle how significant figures affect different arithmetic operations.

Addition and Subtraction

The result of addition or subtraction should have the same number of decimal places as the measurement with the fewest decimal places.

Example 1:

12.345 + 6.7 + 101.2 = ?

• 12.345 has three decimal places.
• 6.7 has one decimal place.
• 101.2 has one decimal place.

The least number of decimal places is one. Therefore, the sum is rounded to one decimal place.

12.345 + 6.7 + 101.2 = 120.245 ≈ 120.2

Example 2:

25.00 – 12.3 = ?

• 25.00 has two decimal places.
• 12.3 has one decimal place.

The least number of decimal places is one. Therefore, the difference is rounded to one decimal place.

25.00 – 12.3 = 12.7 ≈ 12.7

Multiplication and Division

In multiplication and division, the result should have the same number of significant figures as the measurement with the fewest significant figures.

Example 3:

2.34 x 5.6 = ?

• 2.34 has three significant figures.
• 5.6 has two significant figures.

The fewest number of significant figures is two. Therefore, the product is rounded to two significant figures.

2.34 x 5.6 = 13.064 ≈ 13

Example 4:

12.345 / 2.1 = ?

• 12.345 has five significant figures.
• 2.1 has two significant figures.

The fewest number of significant figures is two. Therefore, the quotient is rounded to two significant figures.

12.345 / 2.1 = 5.87857 ≈ 5.9

Practice Problems with Detailed Answers

Let's solidify our understanding with a series of practice problems, complete with detailed solutions to illustrate the application of significant figure rules.

Problem 1:

Calculate the following, expressing your answer with the correct number of significant figures:

25.67 + 1.345 – 10.0 = ?

Solution:

1. Perform the addition: 25.67 + 1.345 = 27.015
2. Perform the subtraction: 27.015 – 10.0 = 17.015
3. Determine the number of decimal places: 10.0 has one decimal place, which is the least.
4. Round the final answer to one decimal place: 17.0

Problem 2:

Calculate the following, expressing your answer with the correct number of significant figures:

12.5 x 4.678 / 0.23 = ?

Solution:

1. Perform the multiplication: 12.5 x 4.678 = 58.475
2. Perform the division: 58.475 / 0.23 = 254.239
3. Determine the number of significant figures: 12.5 has three, 4.678 has four, and 0.23 has two.
4. The least number of significant figures is two.
5. Round the answer to two significant figures: 250

Problem 3:

A rectangular field measures 12.34 meters by 5.6 meters. Calculate the area of the field with the correct number of significant figures.

Solution:

1. Multiply the length and width: 12.34 m x 5.6 m = 69.024 m²
2. Determine the number of significant figures: 12.34 has four, and 5.6 has two.
3. The least number of significant figures is two.
4. Round the area to two significant figures: 69 m²

Problem 4:

A student measured the mass of an object three times: 15.2 g, 15.1 g, and 15.3 g. Calculate the average mass with the correct number of significant figures.

Solution:

1. Sum the masses: 15.2 g + 15.1 g + 15.3 g = 45.6 g
2. Divide by the number of measurements: 45.6 g / 3 = 15.2 g
3. The original measurements all have three significant figures, so the average should also have three.
4. The average mass is 15.2 g

Problem 5:

A car travels 125.5 km in 2.5 hours. Calculate its average speed in km/h with correct significant figures.

Solution:

1. Divide the distance by the time: 125.5 km / 2.5 h = 50.2 km/h
2. Determine significant figures: 125.5 has four, and 2.5 has two.
3. The least number of significant figures is two.
4. Round the answer to two significant figures: 50 km/h

Advanced Considerations: Rounding and Scientific Notation

Rounding: When rounding, if the digit to be dropped is 5 or greater, round up. If it is less than 5, round down.

Scientific Notation: Using scientific notation is crucial when dealing with very large or very small numbers, simplifying the process of determining significant figures and performing calculations. Remember that only the digits in the coefficient are significant.

Conclusion

Mastering significant figures is essential for accurate scientific calculations. By understanding the rules for determining significant figures and applying them consistently to addition, subtraction, multiplication, and division, you can ensure your results accurately reflect the precision of your measurements. Consistent practice with problems like those presented here will build your proficiency and confidence in working with significant figures in any scientific context. Remember to always pay attention to the least precise measurement when determining the final number of significant figures in your calculations. This meticulous approach will contribute significantly to the accuracy and reliability of your scientific endeavors.