Question

Subtract the following measurements and round off the answer: (a) \( \begin{aligned} 22.10 \mathrm{~cm} & \text { (b) } & 10.0 \mathrm{~cm} \\\ -10.5 \mathrm{~cm} & & &-0.15 \mathrm{~cm} \\ \hline \end{aligned} \)

Short answer

Measurement (a) results in 11.60 cm; Measurement (b) results in 9.9 cm.

Step-by-step solution

Setup the Subtraction for Measurement (a)

Start by aligning the numbers vertically for Measurement (a), which is subtracting 10.5 cm from 22.10 cm. Write it down as a subtraction problem.\[\begin{array}{r}22.10 \, \text{cm} \-10.50 \, \text{cm} \\hline\end{array}\]

Perform the Subtraction for Measurement (a)

Subtract each digit starting from the right, moving to the left, for 22.10 cm minus 10.50 cm.\[\begin{array}{r}\,\,21 \, \text{cm} \\hline11.60 \, \text{cm}\end{array}\]

Round the Result for Measurement (a)

Round 11.60 to the correct number of significant figures. Since the number we started with (22.10 cm) has four significant figures, the result should also have four significant figures. Rounded Result: 11.60 cm

Setup the Subtraction for Measurement (b)

Now, align the numbers for Measurement (b), which involves subtracting 0.15 cm from 10.0 cm. Set them up for subtraction.\[\begin{array}{r}10.00 \text{ cm} \- 0.15 \text{ cm} \\hline\end{array}\]

Perform the Subtraction for Measurement (b)

Subtract each digit from right to left for 10.0 cm minus 0.15 cm.\[\begin{array}{r}\,\,\,9.85 \text{ cm} \\hline9.85 \text{ cm}\end{array}\]

Round the Result for Measurement (b)

Round 9.85 to the appropriate number of decimal places based on the numbers we started with. Since 10.0 cm has the least decimal places in this operation, we round the result to one decimal place. Rounded Result: 9.9 cm

Key concepts

Measurement Subtraction

When working with measurements, precision is essential. Subtraction of measurements involves carefully aligning the numbers based on their decimal points before performing the computation. To understand this process better, follow these simple steps:

• Align the Numbers: Always ensure that the numbers you are subtracting are aligned vertically by their decimal points. This is crucial for accurate subtraction, as it guarantees that each place value corresponds correctly.


• Subtract from Right to Left: Similar to regular subtraction, work from the rightmost digit to the left. In doing so, ensure you handle any necessary borrowing from the higher place values effectively. This preserves the integrity of the significant figures during the subtraction process.

When you initially line up numbers like 22.10 cm and 10.5 cm, make sure that the decimal points are perfectly aligned. This alignment makes it easier to visualize and execute the subtraction process.

Rounding Numbers

Rounding is crucial when presenting the final result of a measurement calculation, ensuring that it reflects the accuracy of the original data. The rules for rounding are straightforward and generally involve adjusting your number to match the precision limit set by your initial figures:

• Determine Significant Figures: Before rounding, decide how many significant figures or decimal places your final answer should contain based on the numbers used in the calculation. In operations with measurements, the number with the least precision dictates the number of significant figures or decimal places in the result.


• Consider Rounding Rules: When the digit to be dropped is less than 5, leave the last kept digit as it is. If it's 5 or greater, increase the last kept digit by one.

In the earlier example, after subtracting measurements, 11.60 was rounded to four significant figures, as determined by the number of digits in 22.10.

Decimal Places

Decimal places are the digits that appear after the decimal point in a number. They represent parts of a whole and are crucial in expressing the precision of a measurement calculation. Managing decimal places correctly helps in maintaining the accuracy of your results, which is particularly important in scientific and mathematical applications:

• Initial Precision Matters: When considering how many decimal places to keep in your final result, look at the original measurements. The number with the fewest decimal places dictates your final answer's precision. For instance, if one number has one decimal place and another has two, the result should have one decimal place.


• Consistent Rounding Practices: When rounding to a certain number of decimal places, follow the same rules as when rounding whole numbers. This consistency ensures that your results are valid and aligned with the data from which they originate.

In the example subtraction of 10.0 cm and 0.15 cm, the final result was rounded to one decimal place, mirroring the least precise measurement used in the subtraction. This approach preserves the integrity of the calculations and ensures the reliability of the final results.