Question

For each of the following, give an answer with the correct number of decimal places: a. \(5.08 \mathrm{~g}+25.1 \mathrm{~g}\) b. \(85.66 \mathrm{~cm}+104.10 \mathrm{~cm}+0.025 \mathrm{~cm}\) c. \(24.568 \mathrm{~mL}-14.25 \mathrm{~mL}\) d. \(0.2654 \mathrm{~L}-0.2585 \mathrm{~L}\)

Short answer

a. 30.2 g, b. 189.79 cm, c. 10.32 mL, d. 0.0069 L

Step-by-step solution

Understand the Rule for Significant Figures in Addition and Subtraction

When adding or subtracting numbers, the answer should have the same number of decimal places as the measured number with the fewest decimal places.

Calculate part a

Given: \(5.08 \text{ g} + 25.1 \text{ g}\) First add the numbers: \[5.08 + 25.1 = 30.18 \text{ g}\] The number with the fewest decimal places is 25.1 (1 decimal place), hence the answer should be rounded to 1 decimal place:\(30.18 \text{ g} \rightarrow 30.2 \text{ g}\)

Calculate part b

Given: \(85.66 \text{ cm} + 104.10 \text{ cm} + 0.025 \text{ cm}\) First add the numbers: \[85.66 + 104.10 + 0.025 = 189.785 \text{ cm}\] The number with the fewest decimal places is 85.66 (2 decimal places), hence the answer should be rounded to 2 decimal places:\(189.785 \text{ cm} \rightarrow 189.79 \text{ cm}\)

Calculate part c

Given: \(24.568 \text{ mL} - 14.25 \text{ mL}\) First subtract the numbers: \[24.568 - 14.25 = 10.318 \text{ mL}\] The number with the fewest decimal places is 14.25 (2 decimal places), hence the answer should be rounded to 2 decimal places:\(10.318 \text{ mL} \rightarrow 10.32 \text{ mL}\)

Calculate part d

Given: \(0.2654 \text{ L} - 0.2585 \text{ L}\) First subtract the numbers: \[0.2654 - 0.2585 = 0.0069 \text{ L}\] The number with the fewest decimal places is 0.2585 (4 decimal places), hence the answer should be rounded to 4 decimal places:\(0.0069 \text{ L}\) remains unchanged with 4 decimal places.

Key concepts

Decimal Places

Decimal places refer to the number of digits that appear to the right of the decimal point in a number. When you perform mathematical operations like addition or subtraction, you need to consider decimal places to ensure your result is accurate.
For example, in the number 5.08, there are two decimal places. When dealing with multiple numbers, it is essential to know how many decimal places each number has to apply the correct rules for rounding and precision. This concept will become clearer as we dive into the rules associated with addition and subtraction.

Addition and Subtraction Rules

When adding or subtracting numbers, the key rule is to match the number of decimal places in the result with the number that has the least decimal places among the values being added or subtracted.
For instance, consider the calculation in part a:

• Given: 5.08 g + 25.1 g
• The sum: 5.08 + 25.1 = 30.18 g
• The number with the fewest decimal places is 25.1 (1 decimal place).
• Therefore, the answer must be rounded to 1 decimal place, resulting in 30.2 g.

This rule ensures the precision of the result matches the least precise measurement.

Rounding Off

Rounding off is the process of reducing the number of decimal places in a number based on the mathematical rules for rounding. If the digit right after the target decimal place is 5 or higher, you increase the last digit you keep by one. If it is lower than 5, you leave the last digit as it is.
For instance, in part b:

• Given: 85.66 cm + 104.10 cm + 0.025 cm = 189.785 cm
• The number with the fewest decimal places is 85.66 (2 decimal places).
• To round 189.785 to 2 decimal places, observe the third decimal: 189.785
• Since the 3rd decimal place is 5, we round the second decimal place up: 189.785 → 189.79 cm

Rounding helps ensure that the representation of your final result is both accurate and appropriate for its precision level.

Measurement Precision

Measurement precision refers to the consistency and exactness of measurements. When handling scientific data, always align your results to the least precise measurement used in your calculations.
For example, in part d:

• Given: 0.2654 L - 0.2585 L = 0.0069 L
• The number with the fewest decimal places here is 0.2585 (4 decimal places).
• As a result, the final answer also needs to be in 4 decimal places: 0.0069 L, which remains unchanged.

Adhering to precision ensures that the final results are scientifically valid and reflect the accuracy of the measurements taken.