The age of the Earth is approximately 1.3 * 10^(17) seconds, and one year is approximately 3.2 * 10^(7) seconds....

1. Translate the problem requirements: We need to convert the Earth's age from seconds to years by dividing total seconds (\(1.3 \times 10^{17}\)) by the number of seconds in one year (\(3.2 \times 10^{7}\))
2. Set up the division for unit conversion: Organize the calculation as \((1.3 \times 10^{17}) \div (3.2 \times 10^{7})\) to convert from seconds to years
3. Apply scientific notation division rules: Separate the calculation into coefficient division (\(1.3 \div 3.2\)) and exponent subtraction (\(17 - 7\))
4. Approximate and compare to answer choices: Calculate the approximate result and match it to the closest given option

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to do in everyday terms. Imagine you have a huge pile of seconds that represents the Earth's age, and you want to know how many years that pile represents. To do this, you need to figure out how many "year-sized portions" you can make from all those seconds.

Think of it like having a giant bag of candy (the Earth's age in seconds) and wanting to know how many smaller bags (years) you can fill from it. If each smaller bag holds a certain number of pieces (seconds per year), you divide the total pieces by the pieces per bag.

In our problem:

- Total seconds representing Earth's age = \(1.3 \times 10^{17}\) seconds
- Seconds in one year = \(3.2 \times 10^{7}\) seconds
- We need: Earth's age in years = Total seconds ÷ Seconds per year

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical relationship

2. Set up the division for unit conversion

Now let's organize our calculation properly. When we divide quantities with units, the units help us check our work:

(Earth's age in seconds) ÷ (seconds per year) = Earth's age in years

Notice how the "seconds" units will cancel out, leaving us with "years" - exactly what we want!

Our calculation becomes:
\((1.3 \times 10^{17} \text{ seconds}) \div (3.2 \times 10^{7} \text{ seconds/year}) = ? \text{ years}\)

This gives us the mathematical setup:
\((1.3 \times 10^{17}) \div (3.2 \times 10^{7})\)

3. Apply scientific notation division rules

When dividing numbers in scientific notation, we can break this into two simpler parts:

Part 1: Divide the regular numbers (coefficients)
\(1.3 \div 3.2 \approx 0.41\) (approximately)

Part 2: Handle the powers of 10
When dividing powers of 10, we subtract the exponents:
\(10^{17} \div 10^{7} = 10^{(17-7)} = 10^{10}\)

Combining both parts:
\(0.41 \times 10^{10}\)

To express this in proper scientific notation (where the coefficient is between 1 and 10), we rewrite:
\(0.41 \times 10^{10} = 4.1 \times 10^{9}\)

Process Skill: MANIPULATE - Using scientific notation rules to simplify complex calculations

4. Approximate and compare to answer choices

Our calculated result is \(4.1 \times 10^{9}\) years.

Looking at the answer choices:

1. \(2.5 \times 10^{9}\)
2. \(4.1 \times 10^{9}\) ← This matches our calculation exactly!
3. \(1.9 \times 10^{10}\)
4. \(2.5 \times 10^{11}\)
5. \(4.1 \times 10^{11}\)

Let's verify this makes sense by checking the order of magnitude. The Earth's age in seconds (\(10^{17}\)) is about \(10^{10}\) times larger than the number of seconds in a year (\(10^{7}\)), so we expect an answer around \(10^{10}\) or \(10^{9}\), which matches our result.

Final Answer

The age of the Earth is approximately \(4.1 \times 10^{9}\) years.

Answer: B

Common Faltering Points

Errors while devising the approach

1. Inverting the division setup
Students often get confused about which number should be the numerator and which should be the denominator. They might incorrectly set up the calculation as \((3.2 \times 10^{7}) \div (1.3 \times 10^{17})\), thinking they need to divide "time per unit" by "total time." This conceptual error leads to an extremely small result that doesn't match any answer choice.

2. Misunderstanding unit conversion direction
Some students struggle with the logic of unit conversion and may think they need to multiply instead of divide. They might calculate \((1.3 \times 10^{17}) \times (3.2 \times 10^{7})\), reasoning that they're "converting" from seconds to years without understanding that division is needed to find "how many years fit into the total seconds."

Errors while executing the approach

1. Incorrect scientific notation arithmetic
When dividing \(1.3 \div 3.2\), students often make decimal errors, getting results like \(0.041\) instead of \(0.41\), or even \(4.1\) directly. This leads to final answers that are off by factors of 10, such as \(4.1 \times 10^{8}\) or \(4.1 \times 10^{10}\).

2. Wrong exponent subtraction
Students frequently make errors when subtracting exponents in scientific notation. They might incorrectly calculate \(10^{17} \div 10^{7}\) as \(10^{(7-17)} = 10^{-10}\) instead of \(10^{(17-7)} = 10^{10}\), or even add the exponents to get \(10^{24}\).

3. Improper scientific notation conversion
After getting \(0.41 \times 10^{10}\), students often struggle to convert this to proper scientific notation. They might leave it as \(0.41 \times 10^{10}\) or incorrectly convert it to \(41 \times 10^{8}\) instead of the correct \(4.1 \times 10^{9}\).

Errors while selecting the answer

1. Choosing based on coefficient alone
Students who correctly calculate the coefficient as 4.1 might quickly select any answer choice containing 4.1 without carefully checking the exponent. This could lead them to choose answer E (\(4.1 \times 10^{11}\)) instead of the correct answer B (\(4.1 \times 10^{9}\)), missing the crucial difference in the power of 10.