Problem 49 - Chemistry The Central Science

Question

Carry out the following operations and express the answers with the appropriate number of significant figures. $$ \begin{array}{ll}{\text { (a) } 14.3505+2.65} & {\text { (b) } 952.7-140.7389} \\\ {\text { (c) }\left(3.29 \times 10^{4}\right)(0.2501)} & {\text { (d) } 0.0588 / 0.677}\end{array} $$

Step-by-Step Solution

Answer

(a) \(16.00\) (b) \(812.0\) (c) \(8.22 \times 10^{4}\) (d) \(0.0869\)

Step 1: (a) Addition

First, add the given numbers: \(14.3505 + 2.65 = 16.0005\) The least number of decimal places in the problem is 2 (2.65). Therefore, the answer must have two decimal places as well. The final answer should be rounded to two decimal places: \(16.0005 \approx 16.00\)

Step 2: (b) Subtraction

Subtract the given numbers: \(952.7 - 140.7389 = 811.9611\) The least number of decimal places in the problem is 1 (952.7). Therefore, the answer must have one decimal place as well. The final answer should be rounded to one decimal place: \(811.9611 \approx 812.0\)

Step 3: (c) Multiplication

Multiply the given numbers: \((3.29 \times 10^{4})(0.2501) = 82229.0\) The least number of significant figures in the problem is 3 (3.29). Therefore, the answer must have three significant figures as well. The final answer should be rounded to three significant figures: \(82229.0 \approx 8.22 \times 10^{4}\)

Step 4: (d) Division

Divide the given numbers: \(0.0588 / 0.677 = 0.086838399\) The least number of significant figures in the problem is 3 (0.0588). Therefore, the answer must have three significant figures as well. The final answer should be rounded to three significant figures: \(0.086838399 \approx 0.0869\)

Key Concepts

Addition and Significant Figures

When performing addition, it's crucial to keep track of decimal places, as they determine how precise your answer can be. Each number in an addition operation may have a different number of decimal places. The result of the addition should match the smallest number of decimal places among the numbers being added.

For example, when we add 14.3505 and 2.65, 2.65 has two decimal places while 14.3505 has four. We must, therefore, round our final answer to two decimal places to align with 2.65. This ensures that the level of precision of the numbers is respected in your answer.

By applying this rule, we get:

• Initial sum: 16.0005
• Rounded result: 16.00

This rule helps prevent mistakes that could arise from assuming more precision than the numbers inherently have.

Subtraction and Decimal Places

Much like addition, subtraction requires attention to decimal places to retain the correct level of precision in the answer. Again, you must find the number in the problem that has the least number of decimal places. Your subtraction result should be rounded to this level.

In the example of subtracting 140.7389 from 952.7, the number 952.7 guides the formatting of the final result since it only has one decimal place. After carrying out the subtraction, we ensure the final answer has just one decimal place.

Let's see how it looks in practice:

• Initial difference: 811.9611
• Rounded result: 812.0

This practice avoids overestimating how precisely you know the answer.

Multiplication and Significant Figures

When multiplying numbers, the rule of significant figures takes the lead over decimal places. The result of a multiplication should contain the same number of significant figures as the number with the least significant figures in the operation.

Take the multiplication of (3.29 × 10⁴) by 0.2501. The number 3.29 has the fewest significant figures, that is, three. Therefore, our result must also be expressed in three significant figures.

In this case, the calculation gives:

• Initial product: 82229.0
• Rounded result: 8.22 × 10⁴

Following significant figure rules ensures the accuracy of your result reflects the precision of your measurements or calculations.

Division and Significant Figures

In division, as with multiplication, significant figures dictate how you should present your final answer. Your result should match the number with the fewest significant figures in the original problem.

For instance, divide 0.0588 by 0.677. Here, 0.0588, having three significant figures, determines the precision of our answer. Therefore, the quotient should also be reported with three significant figures.

Here's how it unfolds:

• Initial quotient: 0.086838399
• Rounded result: 0.0869

This approach respects the inherent uncertainty in your calculations, delivering a result that remains consistent with the precision of the input values.