Question

Divide the following measurements and round off the answer (a) \(26.0 \mathrm{~cm}^{2} / 10.1 \mathrm{~cm}\) (b) \(9.95 \mathrm{~cm}^{3} / 0.15 \mathrm{~cm}^{2}\) (c) \(131.78 \mathrm{~cm}^{3} / 19.25 \mathrm{~cm}\) (d) \(131.78 \mathrm{~cm}^{3} / 19.2 \mathrm{~cm}\)

Short answer

(a) 2.57 cm, (b) 66 cm, (c) 6.845 cm², (d) 6.86 cm².

Step-by-step solution

Calculate the Division for Part (a)

For part (a), divide 26.0 cm² by 10.1 cm to get \[\frac{26.0 \text{ cm}^2}{10.1 \text{ cm}} = 2.574257425742574 \text{ cm}\]

Round Off the Result for Part (a)

Since 26.0 has three significant figures and 10.1 also has three significant figures, the quotient should be rounded to three significant figures. So, 2.574257425742574 cm rounds to 2.57 cm.

Calculate the Division for Part (b)

For part (b), divide 9.95 cm³ by 0.15 cm² to get \[\frac{9.95 \text{ cm}^3}{0.15 \text{ cm}^2} = 66.3333333333333 \text{ cm}\]

Round Off the Result for Part (b)

Since 9.95 has three significant figures and 0.15 has two significant figures, the quotient should be rounded to two significant figures. So, 66.3333333333333 cm rounds to 66 cm.

Calculate the Division for Part (c)

For part (c), divide 131.78 cm³ by 19.25 cm to get \[\frac{131.78 \text{ cm}^3}{19.25 \text{ cm}} = 6.845032679738562 \text{ cm}^2\]

Round Off the Result for Part (c)

Since 131.78 has five significant figures and 19.25 has four significant figures, the quotient should be rounded to four significant figures. So, 6.845032679738562 cm² rounds to 6.845 cm².

Calculate the Division for Part (d)

For part (d), divide 131.78 cm³ by 19.2 cm to get \[\frac{131.78 \text{ cm}^3}{19.2 \text{ cm}} = 6.863541666666666 \text{ cm}^2\]

Round Off the Result for Part (d)

Since 131.78 has five significant figures and 19.2 has three significant figures, the quotient should be rounded to three significant figures. So, 6.863541666666666 cm² rounds to 6.86 cm².

Key concepts

Rounding Off Measurements

When working with numbers in scientific contexts, rounding off measurements is essential to ensure precision and accuracy in your results. Rounding off helps to present the data without overemphasizing less significant digits. This is particularly important in experiments where measurements are approximations due to instrument limitations.

To round a number, first, determine the place to which you want to round. This typically matches the lowest number of significant figures in your measurement set. Significant figures are the digits in a measurement that contribute to its accuracy. When rounding:

• If the digit immediately after your rounding place is less than 5, keep the rounding place digit unchanged.
• If the next digit is 5 or greater, increase the rounding digit by one.

For example, if you have calculated a value like 2.574257425742574 cm and need to round it off to three significant figures, it becomes 2.57 cm.

It's all about maintaining consistency and reliability in your results, preventing overstatements of precision.

Division in Chemistry

In chemistry, division is frequently used in calculations such as when finding concentrations, determining density, or when dealing with ratios of measurements. When dividing measurements, retaining and understanding the units is crucial. This is because units provide insight into the nature of what you are calculating.

For instance, dividing area (like cm²) by length (cm) gives you another length (cm), but dividing volume (cm³) by area (cm²) also results in a length (cm). Essentially, you're using division to reduce dimensions by cancelling out units appropriately. This reinforces the importance of carefully performed calculations in maintaining dimensionally consistent answers.

It's paramount to apply the correct rules for significant figures in division, which state that the result should be reported with the number of significant figures equal to the measurement with the least number of significant figures. For example, in the expression \(\frac{9.95 \text{ cm}^3}{0.15 \text{ cm}^2}\), the result of 66.3 cm is rounded to 66 cm to match the two significant figures in the measurement with the fewest significant figures.

Dimensional Analysis

Dimensional analysis is a powerful method used to check the correctness of equations, convert units, and ensure dimensional consistency in calculations. By using dimensions (units like length, time, or mass), you can systematically work through a problem to verify that equations make sense and that the final answer is in the desired units.

Whenever you perform any mathematical operation involving physical quantities, using dimensional analysis helps prevent errors by ensuring that units properly align. This method involves using conversion factors, which are fractions that represent the equivalence between two different units (like 1 inch = 2.54 cm). By cancelling out units step by step in a problem, you ensure each step makes sense.

For instance, if dividing \(\frac{131.78 \text{ cm}^3}{19.25 \text{ cm}}\) results in \(6.845 \text{ cm}^2\), dimensional analysis confirms that the dimension (in this case, cm²) in the outcome fits logically within the context of the calculation. This strengthens the reliability of your results and also aids in identifying potential mistakes in your arithmetic or logical approach.