Problem 40 - Chemistry Matter and Change

Question

Add and subtract as indicated. Round off when necessary. a. \(43.2 \mathrm{cm}+51.0 \mathrm{cm}+48.7 \mathrm{cm}\) \(\quad\) b. \(258.3 \mathrm{kg}+257.11 \mathrm{kg}+253 \mathrm{kg}\)

Step-by-Step Solution

Answer

a. 142.9 cm; b. 768 kg.

Step 1: Calculate Total Length (a)

To find the total length, we add all the given measurements: \(43.2\, \mathrm{cm}\), \(51.0\, \mathrm{cm}\), and \(48.7\, \mathrm{cm}\). First, align the decimal points and add:\[43.2 \+ 51.0 \+ 48.7 \\hline \142.9 \\hline\]The total length is \(142.9\, \mathrm{cm}\).

Step 2: Calculate Total Weight Before Rounding (b)

Now, add the weights: \(258.3\, \mathrm{kg}\), \(257.11\, \mathrm{kg}\), and \(253\, \mathrm{kg}\).Align the decimal points and add:\[258.30 \+257.11 \+253.00 \\hline \768.41 \\hline\]So, the total weight before rounding is \(768.41\, \mathrm{kg}\).

Step 3: Rounding Total Weight (b)

Since the problem may ask for rounding, check if you need to round based on significant figures or units. Assuming two decimal places:\(768.41\, \mathrm{kg}\) is already rounded to two decimal places.If not specified, it's common to round to the least number of decimal places provided in the problem, which is none. Thus:Rounded total weight is \(768\, \mathrm{kg}\).

Key Concepts

Rounding

Rounding is an essential aspect of numerical calculations. It simplifies numbers by reducing them based on the place value you decide to retain. For example, if you have the number 142.958 and you wish to round it to one decimal place, you would look at the second decimal digit.

• If the digit is 5 or higher, round up.
• If the digit is 4 or lower, round down.

For example, rounding 142.958 to one decimal place gives 143.0 as the 5 in the second decimal place bumps the 2 up to a 3.
In chemistry, rounding is especially important because it reflects the precision of your measurements. This way, you maintain the integrity and reliability of your data, always rounding off according to the least precise measurement involved in your calculations.

Addition and Subtraction in Chemistry

When performing addition and subtraction in chemistry, significant figures play a crucial role. Significant figures are the digits in a measurement that contribute to its precision. In these operations, the precision of your result should not exceed the least precise measurement.
Imagine you're adding 43.2 cm, 51.0 cm, and 48.7 cm:

• Recognize the decimal places: 43.2 has one decimal place, 51.0 has one, and 48.7 has one.
• Compute the addition: 43.2 + 51.0 + 48.7 = 142.9.
• The result, therefore, retains one decimal place, consistent with each of the numbers.

Unlike multiplication or division, which observe total significant figures, addition and subtraction preserve the decimal place as the control for precision, ensuring clarity and correctness in your chemical calculations.

Decimal Alignment

Aligning decimals correctly is critical when performing addition and subtraction calculations. This approach ensures that all numerical operations are done with a proper understanding of place values, which dictates the accuracy of the calculations.
Start by listing the numbers in a column, making sure that the decimal points are directly in line with each other. For instance, if you need to add 258.3 kg, 257.11 kg, and 253 kg:

• Align them vertically by their decimal points to get:
• 258.3 aligns with 258.30 (adding a zero for consistency)
• 257.11 remains as it is
• 253 aligns with 253.00

This ensures that each digit is added or subtracted with its equivalent place value, thereby maintaining the integrity of the operation within your calculated results. Proper decimal alignment is particularly vital in chemistry as it impacts the final outcomes of quantitative experiments and problem-solving.